Novel Approximations to the Multi-Dimensional Fractional Diffusion Models Using the Tantawy Technique and Two Other Transformed Methods

Abstract

This study analyzes the family of one of the most essential fractional differential equations due to its wide applications in physics and engineering: the multidimensional fractional linear and nonlinear diffusion equations. The Caputo fractional derivative operator is used to treat the time-fractional derivative. To complete the analysis and generate more stable and highly accurate approximations of the proposed models, three extremely effective techniques, known as the direct Tantawy technique, the new iterative transform technique (NITM), and the homotopy perturbation transform method (HPTM), which combine the Elzaki transform (ET) with the new iterative method (NIM), and the homotopy perturbation method (HPM), are employed. These reliable approaches produce more stable and highly accurate analytical approximations in series form, which converge to the exact solutions after a few iterations. As the number of terms/iterations in the problems series solution rises, it is found that the derived approximations are closely related to each problem’s exact solutions. The two- and three-dimensional graphical representations are considered to understand the mechanism and dynamics of the nonlinear phenomena described by the derived approximations. Moreover, both the absolute and residual errors for all generated approximations are estimated to demonstrate the high accuracy of all derived approximations. The obtained results are encouraging and appropriate for investigating diffusion problems. The primary benefit lies in the fact that our proposed plan does not necessitate any presumptions or limitations on variables that might affect the real problems. One of the most essential features of the proposed methods is the low computational cost and fast computations, especially for the Tantawy technique. The findings of the present study will be valuable as a tool for handling fractional partial differential equation solutions. These approaches are essential in solving the problem and moving beyond the restrictions on variables that could make modeling the problem challenging.

1. Introduction

Fractional calculus (FC) has been the focus of considerable study for a long time. New techniques and mechanisms are continually being created in the discipline of FC, allowing for the discovery of significant, challenging discoveries, and previously unrecognized connections across numerous fields of physics. Newton and Leibniz established the calculus theory in the seventeenth century. Leibnitz created the integral and derivative notations that are currently in use today. Afterward, the definitions of derivative and integral were expanded to include any real-order. In 1819, Lacroix was the first to propose the concept of a non-integer-order derivative. In 1823, Abel examined the first application [1]. As a result, Fourier, Liouville, Riemann, Grunwald, Letnikov, etc., gave the field above a great deal of attention [2,3]. There are no precise definitions for fractional order differentiation and integration, but fractional differentiation and integration extend the classical order. Numerous scholars have presented various definitions of integrations and arbitrary-order derivatives. The most widely applicable definitions among all of them are the ones provided by Caputo and Riemann-Liouvilli. The non-integer derivative reduces the integer derivative to any order. Therefore, these approaches are just extensions of those already used to handle the integer case models [4,5,6,7,8,9]. As a result, the study of the fractional behavior of the function is commonly referred to as the memory effect. The fractional operators’ nonlocality is used to illustrate the classical derivatives gradually.

Fractional operators are defined for various objects, including complex memories, that can be explored by conventional mathematical techniques such as classical differential calculus. FC is currently a very favorable technique due to its growing popularity in modeling complicated nonlinear events, although the idea is still in its early stages of application in numerous academic domains [10,11,12]. The discrete Grünwald–Letnikov equations and their several variations with the generating functions are the key tools used to create numerical discretization for the Riemann–Liouville integrals and derivatives [13]. For Caputo derivatives, the L1 (for $\sigma \in (0,1)$) and L2 (for $\sigma \in (1,2)$) approximation methods are frequently employed [14]. First, the higher order discretization approach for Caputo derivatives with a convergence order of 2 was examined in [15]. A numerical approach of (3 − $\sigma$)^th-order to Caputo derivatives ($\sigma \in (0,1)$) was recently derived by Li et al. and used to the Caputo type advection–diffusion equations [16]. Very recently, schemes for Caputo derivatives of the (4 − $\sigma$)^th-order and (r + 1 − $\sigma$)^th-order have been developed. These schemes have also been used for Caputo-type advection-diffusion equations, where $\sigma \in (0,1)$ and r correspond to the smoothness of the provided function [17]. You can refer to some more intriguing works on (3 − a)^th-order algorithms [18,19].

In many fields of engineering and science, integer-order differential equations cannot provide a satisfactory explanation for a wide range of real phenomena compared to non-integer (fractional) differential equations. The primary motivation for developing numerical methods for fractional differential equations (FDEs) arises from the rising attraction of fractional derivative models among scientists worldwide. In the study of nonlinear science, nonlinear partial differential equations (PDEs) are widely used to address issues in many fields, including image processing, quantum physics, epidemiology, ecology, and economic systems. PDEs are extensively used in many physical applications, including electronic nerve signaling, population modeling, magnetic resonance imaging, magnetohydrodynamic movement through pipes, supersonic and turbulent flow, computational fluid dynamics, wave dispersion and propagation, and others [20,21,22,23]. An accurate assessment of the number of patients with COVID-19 has confirmed the universal prevalence of PDE [24,25]. As demonstrated in [26], PDE can be used to estimate the form of COVID-19. Furthermore, the fractional PDE is more precise than the integer-order PDE for several interesting situations in these fields. It is crucial to establish numerical solutions for fractional PDEs.

The multi-dimensional fractional diffusion equation is the focus of this work [27,28,29]:

$$D_{\tau }^{\sigma }\mathbb{F}=▽·(D(\mathbb{F},r)▽\left(\mathbb{F}\right)),0<\sigma \le 1,$$

(1)

with the initial condition (IC)

$$\mathbb{F}(r,0)={\mathbb{F}}_0\left(r\right),r\in {\mathbb{R}}^3,$$

(2)

where $D_{\tau }^{\sigma }\mathbb{F}={\displaystyle \frac{{\partial }^{\sigma }\mathbb{F}}{\partial {\tau }^{\sigma }}}\equiv {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}$ indicates the fractional Caputo derivative (FCD) having order $\sigma$, $\mathbb{F}(r,\tau )$ indicates the density of the diffusing material at the position $r=\left(\xi ,\zeta \right)$ for two-dimensional (2D) objects and $r=\left(\xi ,\zeta ,\eta \right)$ for three-dimensional (3D) objects, and $D(\mathbb{F},r)$ indicates the diffusion coefficient for $\mathbb{F}$ at the point r. When the diffusion coefficient is independent of density, that is, $D(\mathbb{F},r)=A$ is a positive constant, then Equation (1) simplifies to the fractional heat equation, $D_{\tau }^{\sigma }\mathbb{F}=A{▽}^2\mathbb{F}$, which sums up the heat distribution in a particular area. Equation (1), in specific, becomes the classical multidimensional diffusion equation, ${\partial }_{\tau }\mathbb{F}=A{▽}^2\mathbb{F}$, for $\sigma =1$ and the constant diffusion coefficient. This equation has been applied extensively in numerous linear and nonlinear systems in chemistry, physics, engineering, ecology, and biology. It can also be used to examine processes that demonstrate diffusive behavior, such as the diffusion of alleles within a population in population genetics. To model real-world subdiffusive issues in fluid flow processes and finance, the fractional-order diffusion Equation (1) has been used [30]. The basic solution was initially derived in 1996 [31] for the one-dimensional case, then for the multidimensional case [32], and later in a simplified form [33].

Generally, no method can provide exact solutions for FDEs. To overcome this problem, various techniques can be utilized extensively to derive high-accuracy analytical and numerical approximations. These encompass the homotopy perturbation method (HPM) [34,35,36], the Adomain decomposition method (ADM) [36,37], the variational iteration method [36,38], the homotopy analysis method [39], the novel iterative method (NIM) [40,41], the Chebyshev spectral approach [42,43], the residual power series method (RPSM) [29,44], the Tantawy technique [44,45,46,47], and many others.

In this investigation, we will apply the Tantawy technique [44,45,46,47], in addition to two hybrid transformed methods, namely, the homotopy perturbation transform method (HPTM) and the new iterative transform method (NITM) for analyzing multidimensional fractional diffusion linear and nonlinear models. The Tantawy technique [44] has successfully analyzed many FDEs, and its results have been compared with some derived approximations using other counterpart techniques. The comparison results have demonstrated the superiority of this technique over other methods. For example, the family of fractional Burgers-type equations was analyzed using the Tantawy technique and two different approaches: the Aboodh new iteration method (ANIM) and Aboodh residual power series method (ARPSM). The absolute errors of all derived approximations using these methods were calculated and compared with each other, and the comparison results demonstrated the superiority of the derived approximations using the Tantawy technique over all other derived approximations using ANIM and ARPSM. In addition, the family of time fractional Fokker–Planck equations was solved using the Tantawy technique and the optimal auxiliary function method (OAFM) [45]. Moreover, the Tantawy technique has been used to effective analyze fourth-order fractional Cahn–Hilliard models and its results have been compared with those obtained using the HPTM and variational iteration transform method (VITM) [46]. The absolute errors of the obtained approximations using the two techniques were estimated and compared with each other, revealing that the approximations provided by the Tantawy technique surpassed those gained from the OAFM, due to the promising results shown by the Tantawy technique in analyzing certain strong nonlinear FDEs [44,45,46]. Consequently, it has recently succeeded in examining fractional evolutionary wave equations (EWEs) to simulate various nonlinear phenomena that emerge and propagate in various plasma models, including the investigation of fractional KdV-solitary waves [47], fractional modified KdV-solitary waves [48] in electronegative plasmas, and fractional KdV-cnoidal waves in a nonthermal plasma [49].

For the first time, Daftardar-Gejji and Jafari [50] used the new iterative method (NIM) to solve and analyze nonlinear functional integer and fractional equations. Subsequently, due to the encouraging outcomes exhibited by this strategy, it has rapidly proliferated among researchers for the analysis of diverse fractional differential equations. This approach has proven effective in analyzing various fractional differential equations and generating highly accurate approximations. Some transforms, such as the Laplace transform (LT), the Aboodh transform (AT), and the Elzaki transform (ET), have been used with this technique to facilitate the calculation process and to produce some new hybrid methods in the form of a new iteration transform method (NITM). For instance, the NIM was employed to address many nonlinear integer and fractional differential equations [51] and fractional boundary value problems [52]. Furthermore, NIM was applied to analyze a fractional logistic equation, and the convergence of series solutions was examined [53]. The authors also contrasted the findings of the homotopy perturbation and Adomian decomposition methods with the NIM approximation [53]. Furthermore, Bhalekar and Daftardar-Gejji established sufficient criteria for converging the obtained analytical approximation utilizing the NIM [54].

Moreover, Madani et al. presented the HPTM for the first time to analyze a wide range of PDEs, including nonhomogeneous PDEs with variable coefficients [55]. The authors also compared their results with exact, standard HPM, and numerical solutions [55]. The authors also found that this method’s main advantage is that it overcomes the deficiency, which is mainly caused by unsatisfied conditions. Khan and Wu [56] used HPTM to analyze both homogeneous and nonhomogeneous PDEs and to derive high approximations. They discovered that the HPTM can solve and analyze nonlinear problems without employing Adomian’s polynomials, which is a significant advantage of this technique compared to the decomposition method. This method identifies the solution without restrictive assumptions or discretization, thus avoiding round-off errors. Owolabi et al. [57] addressed the fractional diffusive predator–prey models and derived analytical approximations for these models by integrating the Laplace transform and the HPM. Moreover, Biazar and Aminikhah [58] investigated the convergence with the generated PDE approximations using HPM. The authors addressed the sufficient condition for the HPM’s convergence [58]. Furthermore, Touchent et al. [59] discussed in detail the convergence analysis of the coupled Sumudu transform with HPM. The authors [59] determined that the obtained results demonstrated that the utilized method is straightforward and effective in analytically addressing various problems involving local fractional derivatives.

The primary motivation and objective of this study are to employ the Tantawy technique [44,45,46,47,48] to address and examine multidimensional fractional diffusion linear and nonlinear models and to derive analytical approximations for these models using this novel technique, which is recognized as one of the most straightforward methods to analyze various types of FDEs. Furthermore, this study aims to combine the local fractional Elzaki transform (ET) [60] with the standard HPM [61,62] and NIM to obtain new combination methods, namely, HPTM and NITM, to analyze multidimensional fractional diffusion linear and nonlinear models and to obtain highly accurate analytical approximations for these fractional models. The ET originates from the standard Fourier integral. This transform was introduced to simplify the time domain solution of partial and ordinary differential equations, depending on the ET’s basic features and simple mathematical formulation. The third approach combines He’s polynomials, the HPM, and the ET. He [61,62] presented HPM, a series expansion method to solve nonlinear PDEs. The HPM utilizes a convergence control parameter to guarantee the convergence of the approximation series within a specific range of physical parameters. The current study is essential because it finds some analytical approximate solutions to the multi-dimensional fractional diffusion linear and nonlinear models using three relatively new and innovative methods.

The outline of our paper is as follows: Section 2 provides the basic definitions of FC. Section 3 briefly summarizes the proposed methods for analyzing fractional PDEs. Section 4 provides various numerical test examples for multidimensional fractional diffusion linear and nonlinear models, derives some analytical approximations for these models, and demonstrates the usefulness of the techniques under consideration. A brief discussion of the main results and conclusions is given in Section 5.

2. Basic Definitions

In this portion, we expressed the FC basic results associated with the current work.

Definition 1. 

The non-integer Riemann-Liouville integration operator for the function $\mathbb{F}\left(\xi ,\tau \right)$ is defined as [63]

$$J^{\sigma }\mathbb{F}\left(\xi ,\tau \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma \right)}}{\mathit{\int }}_0^{\tau }{(\tau -t)}^{\sigma -1}\mathbb{F}\left(\xi ,t\right)dt,\tau >0,0<\sigma <1,$$

(3)

with the subsequent property

$$J^{\sigma }{\tau }^{\delta }={\displaystyle \frac{\mathrm{\Gamma }(\delta +1)}{\mathrm{\Gamma }(\delta +\sigma +1)}}{\tau }^{\delta +\sigma },\delta >-1.$$

(4)

Definition 2. 

The Caputo fractional derivative (CFD) operator to the function $\mathbb{F}\left(\xi ,\tau \right)$ is defined as [64]

$${}^{C}D_{\tau }^{\sigma }\mathbb{F}\left(\xi ,\tau \right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\mathrm{\Gamma }(\delta -\sigma )}}{\int }_0^{\tau }{\displaystyle \frac{{\mathbb{F}}^{\left(\delta \right)}\left(\xi ,t\right)}{{(\tau -t)}^{\sigma -\delta +1}}}dt,\delta -1<\sigma <\delta ,,\delta \in N,\hfill \\ {\displaystyle \frac{d^{\delta }}{d{\tau }^{\delta }}}\mathbb{F}\left(\xi ,\tau \right),\delta =\sigma ,\hfill \end{array}\right.$$

(5)

with ${\mathbb{F}}^{\left(\delta \right)}\left(\xi ,\tau \right)\equiv {\partial }_t^{\delta }\mathbb{F}\left(\xi ,\tau \right)$ and the following properties are fulfilled:

• $J_{\tau }^{\sigma }{D}_{\tau }^{\sigma }\mathbb{F}\left(\xi ,\tau \right)=\mathbb{F}\left(\xi ,\tau \right)-{\sum }_{k=0}^{\delta -1}{\mathbb{F}}^k\left(\xi ,0^{+}\right){\displaystyle \frac{{\tau }^k}{k!}},for\tau >0,\mathrm{\forall } \delta -1<\sigma \le \delta ,\delta \in N.$
• $D_{\tau }^{\sigma }{J}_{\tau }^{\sigma }\mathbb{F}\left(\xi ,\tau \right)=\mathbb{F}\left(\xi ,\tau \right)$.

Definition 3. 

Elzaki transform (ET) to the function $\mathbb{F}(\xi ,\tau )$ is defined as [60]

$$E\left[\mathbb{F}(\xi ,\tau )\right]=G\left(s\right)=s{\int }_0^{\infty }\mathbb{F}(\xi ,\tau )e^{{\displaystyle \frac{-\tau }{s}}}d\tau ,\tau \ge 0.$$

(6)

Definition 4. 

The ET of the CFD operator to the function $\mathbb{F}\left(\xi ,\tau \right)$ is given by [65,66]

$$E\left[D_{\tau }^{\sigma }\mathbb{F}\left(\xi ,\tau \right)\right]=s^{-\sigma }E\left[\mathbb{F}\left(\xi ,\tau \right)\right]-\sum _{k=0}^{\delta -1}{s}^{2-\sigma +k}{\mathbb{F}}^{\left(k\right)}(\xi ,0),\forall \delta -1<\sigma <\delta .$$

(7)

Definition 5. 

The Mittag–Lefer function $E_{\sigma }\left(\tau \right)$ for $\sigma >0$ is given by [67]

$$E_{\sigma }\left(\tau \right)={\displaystyle \sum _{i=0}^{\infty }}{\displaystyle \frac{{\tau }^i}{\mathrm{\Gamma }\left(i\sigma +1\right)}}.$$

3. A Brief Overview of the Proposed Methods for Analyzing the Fractional  PDEs

This section will outline the methodologies employed in the current investigation, including the Tantawy technique, the NITM, and the HPTM. For this purpose, we consider the following general inhomogenous FPDE:

$$D_{\tau }^{\sigma }\mathbb{F}+N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)=h,\forall \tau >0 \& 0<\sigma \le 1,$$

(8)

with the the initial condition (IC)

$$\mathbb{F}\left(\xi ,0\right)\equiv {\mathbb{F}}_0=f\left(\xi \right),$$

(9)

where $\mathbb{F}\equiv \mathbb{F}(\xi ,\tau )$, ${\mathbb{F}}_0\equiv \mathbb{F}\left(\xi ,0\right)$, $h\equiv h\left(\xi ,\tau \right)$ indicates the source term, which is responsible in homogenous term. Here, $M\equiv M\left(\mathbb{F}\left(\xi ,\tau \right)\right)$ and $N\equiv N\left(\mathbb{F}\left(\xi ,\tau \right)\right)$ indicate the linear and nonlinear terms, respectively. Note that we wrote the function $\mathbb{F}\equiv \mathbb{F}(\xi ,\tau )$ for simplicity one but we can also write this function as $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\zeta ,\eta ,\dots ,\tau \right)$.

Now, we proceed to analyze the problem (8) using the suggested approaches.

3.1. The Tantawy Technique for Analyzing FDEs

Many studies have been conducted on FDEs using diverse methods and strategies to elucidate various engineering and physical processes. However, physical researchers frequently encounter difficulties in employing such methods to investigate the diverse EWEs associated with distinct physical phenomena due to some difficulties in application during analysis, particularly when addressing complex and heterogeneous equations. To achieve this goal, we have continually sought to provide a straightforward, easily implemented method without barriers that may challenge novice researchers. As a result, this work produced one of the most succinct and comprehensible approaches to assessing and resolving diverse EWEs, enabling researchers to progress in their investigations on any subject without the hurdles faced previously. The new technique, known as the Tantawy technique [44,45,46,47,48], was recently published in 2025 and has been used to analyze several physical and engineering problems. It has proven its efficiency and, in some cases, superiority over counterpart methods. The most important advantage of this new technique is its ease of application for analyzing any FDE, without any challenges or difficulties. Any junior researcher can easily apply it to analyze various types of fractional differential equations. It is not computationally intensive and does not require extended calculation to achieve superior approximations. This method is straightforward and can be effectively employed by novice researchers to evaluate and resolve any fractional differential (non)linear equation, yielding highly accurate higher-order approximations. Moreover, it is devoid of the intricacies often associated with alternative approaches.

This technique can be elucidated more effectively through the following main steps [44,45,46,47,48]:

Step (1)

According to the Tantawy technique, the analytical approximation to problem (8) is assumed to have the following convergent series form:

$$\mathbb{F}={\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma },$$

(10)

where ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi \right)$ ∀$i=1,2,3,\dots$, are unknown functions in the independent variable $\xi$. It is important to note that this technique can also be employed to investigate multidimensional fractional differential equations; in this case, the unknown functions rely on all independent variables (e.g., ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi ,\zeta ,\eta ,\dots \right)$).

Henceforth, the following notations may be used in all coming calculations:

$${\mathbb{F}}_i^{\prime }\equiv {\partial }_{\xi }{\mathbb{F}}_i,{\mathbb{F}}_i^{″}\equiv {\partial }_{\xi }^2{\mathbb{F}}_i,{\mathbb{F}}_i^{\left(3\right)}\equiv {\partial }_{\xi }^3{\mathbb{F}}_i,\cdots ,{\mathbb{F}}_i^{\left(n\right)}\equiv {\partial }_{\xi }^n{\mathbb{F}}_i.$$

Step (2)

Inserting the Ansatz (10) into problem (8) yields

$$D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+N\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+M\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)=h,$$

(11)

and for $m^{th}$-order, Equation (11) can be reduced to the following form:

$$D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+N\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+M\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)=h.$$

(12)

Step (3)

Using the Mathematica command for the CFD in Equation (12), it can be written in the following form:

$$D_{\tau }^{\sigma }\mathbb{F}\equiv \mathrm{CaputoD}\left[\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right),\left\{\tau ,\sigma \right\}\right].$$

(13)

Step (4)

Inserting the Mathematica command (13) into Equation (12) and collecting the various coefficients of ${\tau }^{i\sigma }$, we have

$$\sum _{i=0}^m{Q}_i{\tau }^{i\sigma }=Q_0+Q_1{\tau }^{\sigma }+Q_2{\tau }^{2\sigma }+Q_3{\tau }^{3\sigma }+\cdots =0,$$

(14)

with

$$\begin{array}{cc}\hfill \mathrm{Coeff}.\left({\tau }^{0\sigma }\right)& :Q_0\equiv Q_0\left({\mathbb{F}}_0,{\mathbb{F}}_0^{\prime },{\mathbb{F}}_0^{″},{\mathbb{F}}_0^{\left(3\right)},\dots ,{\mathbb{F}}_1\right),\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{1\sigma }\right)& :Q_1\equiv Q_1\left({\mathbb{F}}_0,{\mathbb{F}}_0^{\prime },{\mathbb{F}}_0^{″},{\mathbb{F}}_0^{\left(3\right)},\dots ,{\mathbb{F}}_1,{\mathbb{F}}_1^{\prime },{\mathbb{F}}_1^{″},\dots ,{\mathbb{F}}_2\right),\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{2\sigma }\right)& :Q_2\equiv Q_2\left({\mathbb{F}}_0,{\mathbb{F}}_0^{\prime },{\mathbb{F}}_0^{″},{\mathbb{F}}_0^{\left(3\right)},\dots ,{\mathbb{F}}_1,{\mathbb{F}}_1^{\prime },{\mathbb{F}}_1^{″},\dots ,{\mathbb{F}}_2,{\mathbb{F}}_2^{\prime },{\mathbb{F}}_2^{″},\dots ,{\mathbb{F}}_3\right),\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{3\sigma }\right)& :Q_3\equiv Q_3\left({\mathbb{F}}_0,{\mathbb{F}}_0^{\prime },{\mathbb{F}}_0^{″},{\mathbb{F}}_0^{\left(3\right)},\dots ,{\mathbb{F}}_1,{\mathbb{F}}_1^{\prime },{\mathbb{F}}_1^{″},\dots ,{\mathbb{F}}_2,{\mathbb{F}}_2^{\prime },{\mathbb{F}}_2^{″},\dots ,{\mathbb{F}}_3,{\mathbb{F}}_3^{\prime },{\mathbb{F}}_3^{″},\dots ,{\mathbb{F}}_4\right)\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(15)

Step (5)

Equating the coefficients $Q_i$ to zero, we obtain a system of DEs involving ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯. By solving this system, we can finally determine the implicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯ as functions of the IC ${\mathbb{F}}_0$.

Step (6)

Using the given value of the IC ${\mathbb{F}}_0$ in the above obtained system, we finally get the explicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯.

Step (7)

By collecting the obtained values of ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯ in the Ansatz (10), we ultimately obtain an analytical approximate solution to Equation (8) as follows:

$$\mathbb{F}\left(\xi ,\tau \right)={\mathbb{F}}_0+{\mathbb{F}}_1{\tau }^{\sigma }+{\mathbb{F}}_2{\tau }^{2\sigma }+{\mathbb{F}}_3{\tau }^{3\sigma }+\cdots .$$

(16)

3.2. The NITM for Analyzing FDEs

To analyze problem (8) using the NITM, the following brief points are introduced:

Step (1)

By implementing the ET to Equation (8), we get

$$E\left[D_{\tau }^{\sigma }\mathbb{F}\right]=E\left[h\right]-\mathbb{E}\left[N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right].$$

(17)

Step (2)

Using the definition of CFD in Equation (17), we get

$$s^{-\sigma }E\left[\mathbb{F}\right]-\sum _{k=0}^{m-1}{s}^{2-\sigma +k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}=E\left[h\right]-E\left[N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right],$$

(18)

which leads to

$$E\left[\mathbb{F}\right]=\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}+s^{\sigma }E\left[h\right]-s^{\sigma }E\left[N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right].$$

(19)

Step (3)

By implementing the inverse ET to Equation (19), we have

$$\mathbb{F}=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}+s^{\sigma }E\left[h\right]\right]-E^{-1}\left[s^{\sigma }E\left[N\left(\mathbb{F}\right)\right]\right]-E^{-1}\left[s^{\sigma }E\left[M\left(\mathbb{F}\right)\right]\right].$$

(20)

Step (4)

The NITM allows the approximate solution to be expressed as a convergent series solution in the following manner:

$$\mathbb{F}=\sum _{m=0}^{\infty }{\mathbb{F}}_m.$$

(21)

Step (5)

According to the assumption (21), the nonlinear term ($N\left(\mathbb{F}\right)$) in Equation (20) can be expressed in the following form:

$$N\left(\mathbb{F}\right)=N\left(\sum _{m=0}^{\infty }{\mathbb{F}}_m\right)=\sum _{m=0}^{\infty }N\left[{\mathbb{F}}_m\right],$$

(22)

and this term can be decomposed as follows:

$$N\left(\sum _{m=0}^{\infty }{\mathbb{F}}_m\right)=N\left({\mathbb{F}}_0\right)+\sum _{m=1}^{\infty }\left[N\left(\sum _{i=0}^m{\mathbb{F}}_i\right)-N\left(\sum _{i=0}^{m-1}{\mathbb{F}}_i\right)\right].$$

(23)

Step (6)

Using Equations (21)–(23) into Equation (20) yields

$$\begin{array}{cc}\hfill \sum _{m=0}^{\infty }{\mathbb{F}}_m& =E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}+s^{\sigma }E\left[h\right]\right]\hfill \\ \hfill & -E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0(\xi ,\tau )\right)\right]\right]-E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0(\xi ,\tau )\right)\right]\right]\hfill \\ \hfill & -E^{-1}\left[s^{\sigma }E\left[\sum _{m=1}^{\infty }\left[N\left(\sum _{i=0}^m{\mathbb{F}}_i\right)-N\left(\sum _{i=0}^{m-1}{\mathbb{F}}_i\right)\right]\right]\right]\hfill \\ \hfill & -E^{-1}\left[s^{\sigma }E\left[\sum _{m=1}^{\infty }\left[M\left(\sum _{i=0}^m{\mathbb{F}}_i\right)-M\left(\sum _{i=0}^{m-1}{\mathbb{F}}_i\right)\right]\right]\right].\hfill \end{array}$$

(24)

Step (7)

From Equation (24), the following recurrence relation is obtained:

$$\left\{\begin{array}{c}{\mathbb{F}}_0=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}\right],\\ {\mathbb{F}}_1=-E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0(\xi ,\tau )\right)\right]\right]-E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0(\xi ,\tau )\right)\right]\right]+E^{-1}\left[s^{\sigma }E\left[h\right]\right],\\ {\mathbb{F}}_i=-E^{-1}\left[s^{\sigma }E\left[\sum _{m=1}^{\infty }\left[N\left(\sum _{i=0}^m{\mathbb{F}}_i\right)-N\left(\sum _{i=0}^{m-1}{\mathbb{F}}_i\right)\right]\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[\sum _{m=1}^{\infty }\left[M\left(\sum _{i=0}^m{\mathbb{F}}_i\right)-M\left(\sum _{i=0}^{m-1}{\mathbb{F}}_i\right)\right]\right]\right].\end{array}\right.$$

(25)

Step (8)

By aggregating the values of ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ⋯, we ultimately get the analytical approximation in the subsequent form

$$\mathbb{F}=\sum _{m=0}^{\infty }{\mathbb{F}}_m={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+\cdots .$$

(26)

The convergence of the analytical approximate solution derived by the NIM was further analyzed in Refs. [53,54].

3.3. The HPTM for Analyzing FDEs

To analyze problem (8) using the HPTM, the following brief points are introduced:

Step (1)

By implementing the ET to Equation (8), we get

$$E\left[D_{\tau }^{\sigma }\mathbb{F}\right]=E\left[h\right]-\mathbb{E}\left[N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right].$$

(27)

Step (2)

Using the definition of CFD in Equation (27), we get

$$s^{-\sigma }E\left[\mathbb{F}\right]-\sum _{k=0}^{m-1}{s}^{2-\sigma +k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}=E\left[h\right]-E\left[N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right],$$

(28)

which leads to

$$E\left[\mathbb{F}\right]=\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}+s^{\sigma }E\left[h\right]-s^{\sigma }E\left[N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right].$$

(29)

Step (3)

By applying the inverse ET to Equation (29), we obtain

$$\mathbb{F}=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}\right]-E^{-1}\left[s^{\sigma }E\left[-h+N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)\right]\right].$$

(30)

Step (4)

The HPM allows the approximate solution to be expressed as a convergent series solution in the following manner:

$$\mathbb{F}=\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k,$$

(31)

where $ϵ\in [0,1]$ represents the perturbation parameter.

Step (5)

Nonlinear terms are decomposed as follows:

$$N\left(\mathbb{F}\right)=\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{F}}_k\right),$$

(32)

where $H_k$ illustrates He’s polynomials ${\mathbb{F}}_0,{\mathbb{F}}_1,{\mathbb{F}}_2,\dots ,{\mathbb{F}}_k,$ and is expressed as

$$H_k({\mathbb{F}}_0,{\mathbb{F}}_1,\cdots ,{\mathbb{F}}_n)={\displaystyle \frac{1}{k!}}{\displaystyle \frac{{\partial }^k}{\partial {ϵ}^k}}{\left[N\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)\right]}_{ϵ=0}.$$

Step (6)

Using Equations (31) and (32) in Equation (29) yields

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}\right]-ϵE^{-1}\left[s^{\sigma }E\left[-h+M\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{F}}_k\right)\right]\right].$$

(33)

Step (7)

By comparing the coefficients of various order of $ϵ$,

$$\begin{array}{cc}\hfill & {ϵ}^0:{\mathbb{F}}_0={\mathbb{F}}_0\left(\xi ,0\right),\hfill \\ \hfill & {ϵ}^1:{\mathbb{F}}_1=-E^{-1}\left[s^{\sigma }E(-h+M\left({\mathbb{F}}_0\right)+H_0\left(\mathbb{F}\right))\right],\hfill \\ \hfill & {ϵ}^2:{\mathbb{F}}_2=-E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_1\right)+H_1\left(\mathbb{F}\right))\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill & {ϵ}^k:{\mathbb{F}}_k=-E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_{k-1}\right)+H_{k-1}\left(\mathbb{F}\right))\right],k>0,k\in N,\hfill \end{array}$$

(34)

Note that the inhomogenous part is taken only in the first-order approximation.

Step (8)

Now, for $ϵ\to 1$, the approximate solution in the convergence series form reads

$$\mathbb{F}={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+\cdots =\underset{M\to \infty }{lim}\sum _{k=0}^M{\mathbb{F}}_k.$$

(35)

Various published works meticulously scrutinized the convergence study of the employed approaches [53,54,59]. Consequently, we urge the reader to examine these works for additional elucidation and specifics. However, we introduce error functions to analyze the accuracy and performance of the proposed methods. The absolute error $E_n\equiv E_n\left(\xi ,\tau \right)$ in the $n^{th}$-order approximation is as follows, assuming that $\mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)$ is the exact solution for the integer case of the FPDE, i.e., the fractional-order parameter $\sigma =1$ and ${\mathbb{F}}_n\left(\xi ,\zeta ,\eta ,\tau \right)$ is the $n^{th}$-order approximation of $\mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)$ generated by the proposed methods. The final form of the absolute error $E_n$ reads

$$E_n=|\mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)-{\mathbb{F}}_n\left(\xi ,\zeta ,\eta ,\tau \right)|.$$

(36)

Therefore, the maximum value of the absolute error $\left(M{E}_n\right)$ along the study domain can be expressed as follows [68,69,70]:

$$M{E}_n=\underset{\mathsf{\Omega }}{max}|\mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)-{\mathbb{F}}_n\left(\xi ,\zeta ,\eta ,\tau \right)|,$$

(37)

where $\mathsf{\Omega }=\left(\xi ,\zeta ,\eta ,\tau \right)\in [{\xi }_i,{\xi }_f]\times [{\zeta }_i,{\zeta }_f]\times [{\eta }_i,{\eta }_f]\times [{\tau }_i,{\tau }_f]$.

In addition, another type of error can be calculated, which is the maximum value of the residual error “$\mathbb{R}$” over the whole study domain. This is one of the most accurate methods for measuring the accuracy and efficiency of the derived approximations

$$\mathbb{R}=\underset{\mathsf{\Omega }}{max}\left|D_{\tau }^{\sigma }\mathbb{F}+N\left(\mathbb{F}\right)+M\left(\mathbb{F}\right)-h\right|,$$

(38)

where $\mathsf{\Omega }=\left(\xi ,\zeta ,\eta ,\tau \right)\in [{\xi }_i,{\xi }_f]\times [{\zeta }_i,{\zeta }_f]\times [{\eta }_i,{\eta }_f]\times [{\tau }_i,{\tau }_f]$.

A significant characteristic of this error is that, in the absence of the exact solution to the problem at hand, it can serve as a metric to evaluate the correctness of the obtained approximations.

4. Applications and Test Examples

In the present study, we discuss some examples of fractional diffusion equations in two and three dimensions and seek to derive approximations for these models utilizing the offered approaches.

4.1. Example (I)

Here, we consider the following time-fractional two-dimensional heat flow inhomogenous problem [71]:

$$D_{\tau }^{\sigma }\mathbb{F}\equiv {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}={\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+\sin \left(\zeta \right),$$

(39)

with the IC

$$\mathbb{F}(\xi ,\zeta ,0)\equiv {\mathbb{F}}_0=\sin \left(\xi \right)\sin \left(\zeta \right)+\sin \left(\zeta \right),$$

(40)

where $\mathbb{F}\equiv \mathbb{F}(\xi ,\zeta ,\tau )$ and $0<\sigma \le 1$.

For $\sigma =1$, the exact solution to problem (39) reads

$$\mathbb{F}(\xi ,\zeta ,\tau )=e^{-2\tau }\sin \left(\xi \right)\sin \left(\zeta \right)+\sin \left(\zeta \right).$$

(41)

This problem can be analyzed using the suggested methods, as shown below.

4.1.1. Analyzing Example (I) via the Tantawy Technique

Step (1)

According to the Tantawy technique, the analytical approximation to problem (39) is assumed to have the following convergent series form:

$$\mathbb{F}={\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma },$$

(42)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\zeta ,\tau \right)$, ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi ,\zeta \right)$ ∀$i=1,2,3,\cdots$ are unknown functions in the independent variable $\left(\xi ,\zeta \right)$ and ${\mathbb{F}}_0\equiv {\mathbb{F}}_0\left(\xi ,\zeta ,{\tau }_0\right)$.

Step (2)

By inserting the Ansatz (42) into problem (39), we get

$$D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+{\partial }_{\zeta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+\sin \left(\zeta \right),$$

(43)

and for $m^{th}$-order (say $m=3$), Equation (43) reduces to the following form:

$$D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+{\partial }_{\zeta }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+\sin \left(\zeta \right),$$

(44)

Step (3)

Using the following Mathematica command for the CFD in Equation (44),

$$D_{\tau }^{\sigma }\mathbb{F}\equiv \mathrm{CaputoD}\left[\left({\mathbb{F}}_0\left[\xi ,\zeta \right]+\sum _{i=1}^m{\mathbb{F}}_i\left[\xi ,\zeta \right]{\tau }^{i\sigma }\right),\left\{\tau ,\sigma \right\}\right].$$

(45)

Step (4)

After inserting the Mathematica command (45) into Equation (44) and collecting the various coefficients of ${\tau }^{i\sigma }$, we get

$$\sum _{i=0}^m{Q}_i{\tau }^{i\sigma }=Q_0+Q_1{\tau }^{\sigma }+Q_2{\tau }^{2\sigma }+Q_3{\tau }^{3\sigma }+\cdots =0,$$

(46)

with

$$\begin{array}{cc}\hfill \mathrm{Coeff}.\left({\tau }^{0\sigma }\right)& :Q_0=g_1{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\zeta }^2{\mathbb{F}}_0-\sin \left(\zeta \right),\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{1\sigma }\right)& :Q_1=g_2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\zeta }^2{\mathbb{F}}_1,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{2\sigma }\right)& :Q_2=g_3{\mathbb{F}}_3-{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\zeta }^2{\mathbb{F}}_2,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{3\sigma }\right)& :Q_3=g_4{\mathbb{F}}_4-{\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\zeta }^2{\mathbb{F}}_3,\hfill \\ \hfill & ⋮,\hfill \end{array}$$

(47)

and

$$g_i={\displaystyle \frac{\mathrm{\Gamma }\left(i\sigma +1\right)}{\mathrm{\Gamma }\left(\left(i-1\right)\sigma +1\right)}},\forall i=1,2,3,\cdots .$$

(48)

Step (5)

Equating the coefficients $Q_i$ to zero, we obtain a system of DEs involving ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯:

$$\begin{array}{cc}\hfill g_1{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\zeta }^2{\mathbb{F}}_0-\sin \left(\zeta \right)& =0,\hfill \\ \hfill g_2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\zeta }^2{\mathbb{F}}_1& =0,\hfill \\ \hfill g_3{\mathbb{F}}_3-{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\zeta }^2{\mathbb{F}}_2& =0,\hfill \\ \hfill g_4{\mathbb{F}}_4-{\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\zeta }^2{\mathbb{F}}_3& =0,\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(49)

Step (6)

By solving system (49) simultaneously, we can finally determine the implicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯ as functions of the IC ${\mathbb{F}}_0$:

$$\begin{array}{cc}\hfill {\mathbb{F}}_1& ={\displaystyle \frac{{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0+\sin \left(\zeta \right)}{\mathrm{\Gamma }\left(\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_2& ={\displaystyle \frac{{\partial }_{\xi }^4{\mathbb{F}}_0+{\partial }_{\zeta }^4{\mathbb{F}}_0+2{\partial }_{\xi }^2{\partial }_{\zeta }^2{\mathbb{F}}_0-\sin \left(\zeta \right)}{\mathrm{\Gamma }\left(2\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_3& ={\displaystyle \frac{{\partial }_{\xi }^6{\mathbb{F}}_0+{\partial }_{\zeta }^6{\mathbb{F}}_0+3{\partial }_{\xi }^4{\partial }_{\zeta }^2{\mathbb{F}}_0+3{\partial }_{\xi }^2{\partial }_{\zeta }^4{\mathbb{F}}_0+\sin \left(\zeta \right)}{\mathrm{\Gamma }\left(3\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_4& ={\displaystyle \frac{{\partial }_{\xi }^8{\mathbb{F}}_0+{\partial }_{\zeta }^8{\mathbb{F}}_0+4{\partial }_{\xi }^6{\partial }_{\zeta }^2{\mathbb{F}}_0+4{\partial }_{\xi }^2{\partial }_{\zeta }^6{\mathbb{F}}_0+6{\partial }_{\xi }^4{\partial }_{\zeta }^4{\mathbb{F}}_0-\sin \left(\zeta \right)}{\mathrm{\Gamma }\left(4\sigma +1\right)}}.\hfill \end{array}$$

(50)

Step (7)

Using the value of the IC ${\mathbb{F}}_0$ given in Equation (40) in the system (50), we finally get the explicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, $\cdots ,$ as follows:

$${\mathbb{F}}_i={\left(-2\right)}^i\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{1}{\mathrm{\Gamma }(i\sigma +1)}},\forall i=1,2,3,\cdots .$$

(51)

Step (8)

By inserting the obtained values of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, ⋯, into the Ansatz (42), we ultimately obtain an analytical approximate solution to Equation (39) as follows:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1{\tau }^{\sigma }+{\mathbb{F}}_2{\tau }^{2\sigma }+{\mathbb{F}}_3{\tau }^{3\sigma }+\cdots \hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)+\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)\sum _{i=1}^{\infty }{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\hfill \\ \hfill & =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)\sum _{i=0}^{\infty }{\displaystyle \frac{{\left(-2{\tau }^{\sigma }\right)}^i}{\mathrm{\Gamma }(i\sigma +1)}}\hfill \\ \hfill & =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)E_{\sigma }\left(-2{\tau }^{\sigma }\right),\hfill \end{array}$$

(52)

where $E_{\sigma }\left(-2{\tau }^{\sigma }\right)$ indicates the Mittag-Leffler function. Note that the obtained result (52) is complete agreement with the result in Ref. [29] in the absence of inhomogenous “$\sin \left(\zeta \right)$”.

4.1.2. Analyzing Example (I) via the NITM

To analyze Equation (39) utilizing the NITM, we can encapsulate this procedure in the following succinct steps:

Step (1)

By implementing the ET to Equation (39), we obtain

$$E\left[\mathbb{F}\right]-s^2{\mathbb{F}}_0=s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+\sin \left(\zeta \right)\right].$$

(53)

Step (2)

On implementing the inverse ET to Equation (53), we have

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+\sin \left(\zeta \right)\right]\right].$$

(54)

Step (3)

Note that Equation (39) is linear and according to the recurrence relation (25), we get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $$\begin{array}{c}{\mathbb{F}}_0=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\zeta ,\tau )\right|}_{\tau =0}\right]=E^{-1}\left[s^2\mathbb{F}(\xi ,\zeta ,0)\right]=\sin \left(\xi \right)\sin \left(\zeta \right)+\sin \left(\zeta \right).\end{array}$$

  (55)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0\right)+h\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0+\sin \left(\zeta \right)\right]\right]\hfill \\ \hfill & =-2\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (56)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_1\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_1+{\partial }_{\zeta }^2{\mathbb{F}}_1\right]\right]\hfill \\ \hfill & =2^2\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (57)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_2\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_2+{\partial }_{\zeta }^2{\mathbb{F}}_2\right]\right]\hfill \\ \hfill & =-2^3\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.\hfill \end{array}$$

  (58)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_3\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_3+{\partial }_{\zeta }^2{\mathbb{F}}_3\right]\right]\hfill \\ \hfill & =2^4\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}}.\hfill \end{array}$$

  (59)

  and so on.

Step (4)

At the end, the solution in the convergence series form takes the following form:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots \hfill \\ \hfill & =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)\sum _{i=0}^{\infty }{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\hfill \\ \hfill & =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)E_{\sigma }\left(-2{\tau }^{\sigma }\right).\hfill \end{array}$$

(60)

4.1.3. Analyzing Example (I) via the HPTM

To examine Equation (39) using the HPTM, we can summarize this process in the following concise steps:

Step (1)

By implementing the ET to Equation (39), we obtain

$$E\left[\mathbb{F}\right]=s^2{\mathbb{F}}_0+s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+\sin \left(\zeta \right)\right].$$

(61)

Step (2)

On implementing the inverse ET to Equation (53), we have

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[\sin \left(\zeta \right)\right]\right]+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}\right]\right].$$

(62)

Step (3)

Note that Equation (39) is linear, thus, relation (33) according to Equation (62) can be written as follows:

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵE^{-1}\left[s^{\sigma }E\left[h+M\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)\right]\right],$$

(63)

which leads to

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵ\left(E^{-1}\left[s^{\sigma }E\left[\sin \left(\zeta \right)+{\partial }_{\xi }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\zeta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)\right]\right]\right).$$

(64)

Step (4)

By comparing the coefficients of $ϵ$ in Equation (64), we finally get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $${\mathbb{F}}_0=\sin \left(\xi \right)\sin \left(\zeta \right)+\sin \left(\zeta \right).$$

  (65)
• First-order approximation ${\mathbb{F}}_1$:

  $${\mathbb{F}}_1=E^{-1}\left[s^{\sigma }E\left[\sin \left(\zeta \right)+{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0\right]\right]=-2\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}},$$

  (66)
• Second-order approximation ${\mathbb{F}}_2$:

  $${\mathbb{F}}_2=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_1+{\partial }_{\zeta }^2{\mathbb{F}}_1\right]\right]=2^2\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.$$

  (67)
• Third-order approximation ${\mathbb{F}}_3$:

  $${\mathbb{F}}_3=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_2+{\partial }_{\zeta }^2{\mathbb{F}}_2\right]\right]=-2^3\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.$$

  (68)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $${\mathbb{F}}_4=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_3+{\partial }_{\zeta }^2{\mathbb{F}}_3\right]\right]=2^4\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}}.$$

  (69)

Step (5)

Thus, the series convergent solution takes the following final form:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots \hfill \\ \hfill & =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)\sum _{i=0}^{\infty }{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\hfill \\ \hfill & =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right)E_{\sigma }\left(-2{\tau }^{\sigma }\right).\hfill \end{array}$$

(70)

Equations (52), (60), and (70) clearly show that all the generated approximations using the Tantawy technique, the NITM, and the HPTM are completely identical. Therefore, we will analyze one of these approximations to understand the dynamics of the phenomena described by problem (39). Figure 1 demonstrates the effect of the fractionality $\sigma$ on the profile of the approximate solution (52). Moreover, a comparison between the approximation (52) and the exact solution (41) for $\sigma =1$ is considered as illustrated in Figure 2, to evaluate the efficacy and stability of the used approaches. Figure 2 illustrates that the concordance between the approximate and exact solutions increases as the value of $\sigma$ increases, achieving total agreement at $\sigma =1$. This validates the precision of the obtained approximations and the efficacy of the employed approaches. Additionally, the absolute error $E_n=\left|\mathrm{Exact}\mathrm{Sol}.\left(41\right)-\mathrm{Approx}.\left(52\right)\right|$ for the derived approximations (52) is estimated compared to the exact solution (41) at $\sigma =1$, as shown in

Table 1. The absolute error $E_n$ to the approximation (52) is estimated at $\left(\sigma ,\zeta ,\tau \right)=\left(1,1,0.1\right)$.

$\mathit{\xi }$	$\mathbf{Exact}$ (41)	${\mathbf{Approx.}}_{\mathit{\sigma }=1}$ (52)	${\mathit{E}}_{\mathit{n}}$
−10	1.21627	1.21627	1.18118 ×${10}^{-6}$
−8	0.159864	0.159862	2.1481 ×${10}^{-6}$
−6	1.03397	1.03397	0.60667$r\times {10}^{-6}$
−4	1.36286	1.36286	1.64318 ×${10}^{-6}$
−2	0.215021	0.215019	1.97428 ×${10}^{-6}$
0	0	0	0
2	1.46792	1.46792	1.97428 ×${10}^{-6}$
4	0.320081	0.320079	1.64318 ×${10}^{-6}$
6	0.648971	0.64897	0.60667 ×${10}^{-6}$
8	1.52308	1.52308	2.1481 ×${10}^{-6}$
10	0.466674	0.466673	1.18118 ×${10}^{-6}$

. Furthermore, the absolute error for higher-order approximations (fourth- and tenth-order) is calculated as shown in

Table 2. The absolute error $E_n$ for the different order approximations to the generated approximation (52) is estimated at $\left(\sigma ,\zeta ,\tau \right)=\left(1,1,0.1\right)$.

$\mathit{\xi }$	$4^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${10}^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${20}^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$
−10	1.18118 ×${10}^{-6}$	2.77556 ×${10}^{-16}$	0.
−8	2.1481 ×${10}^{-6}$	3.33067 ×${10}^{-16}$	0.
−6	0.60667 ×${10}^{-6}$	1.11022 ×${10}^{-16}$	0.
−4	1.64318 ×${10}^{-6}$	3.33067 ×${10}^{-16}$	0.
−2	1.97428 ×${10}^{-6}$	4.44089 ×${10}^{-16}$	1.11022 ×${10}^{-16}$
0	0	0	0
2	1.97428 ×${10}^{-6}$	4.44089 ×${10}^{-16}$	1.11022 ×${10}^{-16}$
4	1.64318 ×${10}^{-6}$	3.33067 ×${10}^{-16}$	0.
6	0.60667 ×${10}^{-6}$	1.11022 ×${10}^{-16}$	0.
8	2.1481 ×${10}^{-6}$	3.33067 ×${10}^{-16}$	0.
10	1.18118 ×${10}^{-6}$	2.77556 ×${10}^{-16}$	0.

. Furthermore, the maximum residual error for all derived approximations across the whole study domain $\mathsf{\Omega }$ is computed using the following relationship:

$$\begin{array}{ll}\mathbb{F}& =\sin \left(\zeta \right)+\sin \left(\xi \right)\sin \left(\zeta \right){\displaystyle \sum _{i=0}^{20}}{\displaystyle \frac{{\left(-2{\tau }^{\sigma }\right)}^i}{\mathrm{\Gamma }(i\sigma +1)}},\\ \mathbb{R}& ={\displaystyle \underset{\mathsf{\Omega }}{max}}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}-{\partial }_{\zeta }^2\mathbb{F}-\sin \left(\zeta \right)\right|=3.76588\times {10}^{-13},\end{array}$$

where $\mathsf{\Omega }\equiv \left(\xi ,\zeta ,\tau \right)\in \left[-5,5\right]\times \left[0,1\right]\times \left[0,1\right]$ and $\sigma =1$. Note here that $\mathbb{R}=3.76588\times {10}^{-13}$ at $\left(\xi ,\zeta ,\tau \right)=\left(-1.37545,0.524101,0.99416\right)$.

The proposed approaches clearly allow us to compute higher-order approximations and, accordingly, generate high accurate and more stable approximations, confirming the efficiency of the current methods. The obtained results demonstrate how our computed values closely match the exact solution when $\sigma =1$.

4.2. Example (II)

Assume the following time-fractional three-dimensional heat flow homogenous problem [71]:

$$D_{\tau }^{\sigma }\mathbb{F}\equiv {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}={\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}-2\mathbb{F},$$

(71)

with the IC

$$\mathbb{F}\left(\xi ,\zeta ,\eta ,0\right)\equiv {\mathbb{F}}_0=\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right),$$

(72)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)$ and $0<\sigma \le 1$.

For $\sigma =1$, the exact solution to problem (71) reads

$$\mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)=e^{-5\tau }\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right).$$

(73)

This problem can be analyzed using the suggested approaches, as shown below.

4.2.1. Analyzing Example (II) via the Tantawy Technique

Step (1)

The analytical approximation to problem (71) is assumed to have the following convergent series form:

$$\mathbb{F}={\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma },$$

(74)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)$, ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi ,\zeta ,\eta \right)$ ∀$i=1,2,3,\cdots$ are unknown functions in the independent variable $\left(\xi ,\zeta ,\eta \right)$ and ${\mathbb{F}}_0\equiv {\mathbb{F}}_0\left(\xi ,\zeta ,{\tau }_0\right)$.

Step (2)

By inserting the Ansatz (74) into problem (71), we have

$$\begin{array}{cc}\hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+{\partial }_{\zeta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & +{\partial }_{\eta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)-2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right),\hfill \end{array}$$

(75)

and for $m^{th}$-order (say $m=3$), Equation (75) reduces to the following form:

$$\begin{array}{cc}\hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+{\partial }_{\zeta }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & +{\partial }_{\eta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)-2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right).\hfill \end{array}$$

(76)

Step (3)

Using the following Mathematica command for the CFD in Equation (76),

$$D_{\tau }^{\sigma }\mathbb{F}\equiv \mathrm{CaputoD}\left[\left({\mathbb{F}}_0\left[\xi ,\zeta ,\eta \right]+\sum _{i=1}^m{\mathbb{F}}_i\left[\xi ,\zeta ,\eta \right]{\tau }^{i\sigma }\right),\left\{\tau ,\sigma \right\}\right].$$

(77)

Step (4)

After inserting the Mathematica command (77) into Equation (76) and collecting the various coefficients of ${\tau }^{i\sigma }$, we obtain

$$\sum _{i=0}^m{Q}_i{\tau }^{i\sigma }=Q_0+Q_1{\tau }^{\sigma }+Q_2{\tau }^{2\sigma }+Q_3{\tau }^{3\sigma }+\cdots =0,$$

(78)

with

$$\begin{array}{cc}\hfill \mathrm{Coeff}.\left({\tau }^{0\sigma }\right)& :Q_0=g_1{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\zeta }^2{\mathbb{F}}_0-{\partial }_{\eta }^2{\mathbb{F}}_0+2{\mathbb{F}}_0,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{1\sigma }\right)& :Q_1=g_2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\zeta }^2{\mathbb{F}}_1-{\partial }_{\eta }^2{\mathbb{F}}_1+2{\mathbb{F}}_1\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{2\sigma }\right)& :Q_2=g_3{\mathbb{F}}_3-{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\zeta }^2{\mathbb{F}}_2-{\partial }_{\eta }^2{\mathbb{F}}_2+2{\mathbb{F}}_2\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{3\sigma }\right)& :Q_3=g_4{\mathbb{F}}_4-{\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\zeta }^2{\mathbb{F}}_3-{\partial }_{\eta }^2{\mathbb{F}}_3+2{\mathbb{F}}_3\hfill \\ \hfill & ⋮,\hfill \end{array}$$

(79)

where $g_i$∀$i=1,2,3,\dots$ are defined in Equation (48).

Step (5)

Equating the coefficients $Q_i$ to zero, we obtain a system of DEs involving ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯:

$$\begin{array}{cc}\hfill g_1{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\zeta }^2{\mathbb{F}}_0-{\partial }_{\eta }^2{\mathbb{F}}_0+2{\mathbb{F}}_0& =0,\hfill \\ \hfill g_2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\zeta }^2{\mathbb{F}}_1-{\partial }_{\eta }^2{\mathbb{F}}_1+2{\mathbb{F}}_1& =0,\hfill \\ \hfill g_3{\mathbb{F}}_3-{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\zeta }^2{\mathbb{F}}_2-{\partial }_{\eta }^2{\mathbb{F}}_2+2{\mathbb{F}}_2& =0,\hfill \\ \hfill g_4{\mathbb{F}}_4-{\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\zeta }^2{\mathbb{F}}_3-{\partial }_{\eta }^2{\mathbb{F}}_3+2{\mathbb{F}}_3& =0,\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(80)

Step (6)

By solving system (80) simultaneously, we can finally determine the implicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯ as functions of the IC ${\mathbb{F}}_0$:

$$\begin{array}{cc}\hfill {\mathbb{F}}_1& =\left({\mathbb{F}}_0^{(2,0,0)}+{\mathbb{F}}_0^{(0,2,0)}+{\mathbb{F}}_0^{(0,0,2)}-2{\mathbb{F}}_0\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_2& =\left(\begin{array}{c}-4{\mathbb{F}}_0^{(0,0,2)}+{\mathbb{F}}_0^{(0,0,4)}-4{\mathbb{F}}_0^{(0,2,0)}+2{\mathbb{F}}_0^{(0,2,2)}+{\mathbb{F}}_0^{(0,4,0)}\\ -4{\mathbb{F}}_0^{(2,0,0)}+2{\mathbb{F}}_0^{(2,0,2)}+2{\mathbb{F}}_0^{(2,2,0)}+{\mathbb{F}}_0^{(4,0,0)}+4{\mathbb{F}}_0\end{array}\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(2\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_3& =\left(\begin{array}{c}12{\mathbb{F}}_0^{(0,0,2)}-6{\mathbb{F}}_0^{(0,0,4)}+{\mathbb{F}}_0^{(0,0,6)}+12{\mathbb{F}}_0^{(0,2,0)}\\ -12{\mathbb{F}}_0^{(0,2,2)}+3{\mathbb{F}}_0^{(0,2,4)}-6{\mathbb{F}}_0^{(0,4,0)}+3{\mathbb{F}}_0^{(0,4,2)}\\ +{\mathbb{F}}_0^{(0,6,0)}+12{\mathbb{F}}_0^{(2,0,0)}-12{\mathbb{F}}_0^{(2,0,2)}+3{\mathbb{F}}_0^{(2,0,4)}\\ -12{\mathbb{F}}_0^{(2,2,0)}+6{\mathbb{F}}_0^{(2,2,2)}+3{\mathbb{F}}_0^{(2,4,0)}-6{\mathbb{F}}_0^{(4,0,0)}\\ +3{\mathbb{F}}_0^{(4,0,2)}+3{\mathbb{F}}_0^{(4,2,0)}+{\mathbb{F}}_0^{(6,0,0)}-8{\mathbb{F}}_0\end{array}\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(3\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_4& =\left(\begin{array}{c}-32{\mathbb{F}}_0^{(0,0,2)}+24{\mathbb{F}}_0^{(0,0,4)}-8{\mathbb{F}}_0^{(0,0,6)}+{\mathbb{F}}_0^{(0,0,8)}\\ -32{\mathbb{F}}_0^{(0,2,0)}+48{\mathbb{F}}_0^{(0,2,2)}-24{\mathbb{F}}_0^{(0,2,4)}+4{\mathbb{F}}_0^{(0,2,6)}\\ +24{\mathbb{F}}_0^{(0,4,0)}-24{\mathbb{F}}_0^{(0,4,2)}+6{\mathbb{F}}_0^{(0,4,4)}-8{\mathbb{F}}_0^{(0,6,0)}\\ +4{\mathbb{F}}_0^{(0,6,2)}+{\mathbb{F}}_0^{(0,8,0)}-32{\mathbb{F}}_0^{(2,0,0)}+48{\mathbb{F}}_0^{(2,0,2)}\\ -24{\mathbb{F}}_0^{(2,0,4)}+4{\mathbb{F}}_0^{(2,0,6)}+48{\mathbb{F}}_0^{(2,2,0)}-48{\mathbb{F}}_0^{(2,2,2)}\\ +12{\mathbb{F}}_0^{(2,2,4)}-24{\mathbb{F}}_0^{(2,4,0)}+12{\mathbb{F}}_0^{(2,4,2)}+4{\mathbb{F}}_0^{(2,6,0)}\\ +24{\mathbb{F}}_0^{(4,0,0)}-24{\mathbb{F}}_0^{(4,0,2)}+6{\mathbb{F}}_0^{(4,0,4)}-24{\mathbb{F}}_0^{(4,2,0)}\\ +12{\mathbb{F}}_0^{(4,2,2)}+6{\mathbb{F}}_0^{(4,4,0)}-8{\mathbb{F}}_0^{(6,0,0)}+4{\mathbb{F}}_0^{(6,0,2)}\\ +4{\mathbb{F}}_0^{(6,2,0)}+{\mathbb{F}}_0^{(8,0,0)}+16{\mathbb{F}}_0\end{array}\right).{\displaystyle \frac{1}{\mathrm{\Gamma }\left(4\sigma +1\right)}},\hfill \end{array}$$

(81)

where ${\mathbb{F}}_i^{\left(2,0,0,0\right)}\equiv {\partial }_{\xi }^2{\mathbb{F}}_i$, ${\mathbb{F}}_i^{\left(0,2,0,0\right)}\equiv {\partial }_{\zeta }^2\mathbb{F}$, ${\mathbb{F}}_i^{\left(0,0,3,0\right)}\equiv {\partial }_{\eta }^2\mathbb{F}$, and so on.

Step (7)

Using the value of the IC ${\mathbb{F}}_0$ given in Equation (72) in the system (81), we finally get the explicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, $\cdots ,$ as follows:

$${\mathbb{F}}_i={\left(-5\right)}^i\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{1}{\mathrm{\Gamma }(i\sigma +1)}},\forall i=1,2,3,\dots .$$

(82)

Step (8)

By inserting the obtained values of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, ⋯, into the Ansatz (74), we ultimately obtain an analytical approximate solution to Equation (71) as follows:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1{\tau }^{\sigma }+{\mathbb{F}}_2{\tau }^{2\sigma }+{\mathbb{F}}_3{\tau }^{3\sigma }+\cdots \hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)\left(\sum _{i=0}^{\infty }{\left(-5\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)\hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)E_{\sigma }\left(-5{\tau }^{\sigma }\right).\hfill \end{array}$$

(83)

where $E_{\sigma }\left(-5{\tau }^{\sigma }\right)$ indicates the Mittag-Leffler function.

4.2.2. Analyzing Example (II) via the NITM

To analyze Equation (71) using the NITM, we can encapsulate this procedure in the following succinct steps:

Step (1)

By implementing the ET to Equation (71), we have

$$\left(E\left[\mathbb{F}\right]-s^2{\mathbb{F}}_0\right)=s^{\sigma }E\left({\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}-2\mathbb{F}\right)$$

(84)

Step (2)

On implementing the inverse ET to Equation (84), we obtain

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}-2\mathbb{F}\right]\right].$$

(85)

Step (3)

Note that Equation (71) is linear and according to the recurrence relation (25), we get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $$\begin{array}{c}{\mathbb{F}}_0=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\partial }_{{\tau }^k}^k\mathbb{F}(\xi ,\zeta ,\eta ,\tau )\right|}_{\tau =0}\right]=E^{-1}\left[s^2\mathbb{F}(\xi ,\zeta ,\eta ,0)\right]=\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right).\end{array}$$

  (86)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0\right)\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0+{\partial }_{\eta }^2{\mathbb{F}}_0-2{\mathbb{F}}_0\right]\right]\hfill \\ \hfill & =-5\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (87)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_1\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_1+{\partial }_{\zeta }^2{\mathbb{F}}_1+{\partial }_{\eta }^2{\mathbb{F}}_1-2{\mathbb{F}}_1\right]\right]\hfill \\ \hfill & =5^2\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (88)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_2\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_2+{\partial }_{\zeta }^2{\mathbb{F}}_2+{\partial }_{\eta }^2{\mathbb{F}}_2-2{\mathbb{F}}_2\right]\right]\hfill \\ \hfill & =-5^3\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.\hfill \end{array}$$

  (89)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_3\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_3+{\partial }_{\zeta }^2{\mathbb{F}}_3+{\partial }_{\eta }^2{\mathbb{F}}_3-2{\mathbb{F}}_3\right]\right]\hfill \\ \hfill & =5^4\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}},\hfill \end{array}$$

  (90)

  and so on.

Step (4)

At the end, the solution in the convergence series form is taken the following form

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots \hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)\left(\sum _{i=0}^{\infty }{\left(-5\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)\hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)E_{\sigma }\left(-5{\tau }^{\sigma }\right).\hfill \end{array}$$

(91)

4.2.3. Analyzing Example (II) via the HPTM

To examine Equation (71) using the HPTM, we can summarize this process in the following concise steps:

Step (1)

By applying the ET to Equation (71), we obtain

$$E\left[\mathbb{F}\right]=s^2{\mathbb{F}}_0+s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}-2\mathbb{F}\right].$$

(92)

Step (2)

On applying the inverse ET to Equation (92), we have

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}-2\mathbb{F}\right]\right].$$

(93)

Step (3)

Note that Equation (71) is linear, thus, relation (33) according to Equation (93) can be written as follows:

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵE^{-1}\left[s^{\sigma }E\left[M\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)\right]\right],$$

(94)

which leads to

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵ\left(E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\zeta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\eta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)-2\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right]\right]\right).$$

(95)

Step (4)

By comparing the coefficients of $ϵ$ in Equation (95), we finally get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $${\mathbb{F}}_0=\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right).$$

  (96)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0+{\partial }_{\eta }^2{\mathbb{F}}_0-2{\mathbb{F}}_0\right]\right]\hfill \\ \hfill & =-5\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}},\hfill \end{array}$$

  (97)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_1+{\partial }_{\zeta }^2{\mathbb{F}}_1+{\partial }_{\eta }^2{\mathbb{F}}_1-2{\mathbb{F}}_1\right]\right]\hfill \\ \hfill & =5^2\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (98)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_2+{\partial }_{\zeta }^2{\mathbb{F}}_2+{\partial }_{\eta }^2{\mathbb{F}}_2-2{\mathbb{F}}_2\right]\right]\hfill \\ \hfill & =-5^3\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.\hfill \end{array}$$

  (99)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_3+{\partial }_{\zeta }^2{\mathbb{F}}_3+{\partial }_{\eta }^2{\mathbb{F}}_3-2{\mathbb{F}}_3\right]\right]\hfill \\ \hfill & =5^4\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right){\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}}.\hfill \end{array}$$

  (100)

Step (5)

Thus, the series convergent solution takes the following final form:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots \hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)\left(\sum _{i=0}^{\infty }{\left(-5\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)\hfill \\ \hfill & =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)E_{\sigma }\left(-5{\tau }^{\sigma }\right).\hfill \end{array}$$

(101)

Equations (83), (91), and (101) unequivocally demonstrate that all derived approximations by the Tantawy technique, the NITM, and the HPTM are entirely identical. Therefore, we will examine one of these approximations to understand the dynamics of the various phenomena described by problem (71). Figure 3 illustrates the influence of fractionality $\sigma$ on the profile of the approximate solution (83). Furthermore, a comparison between the approximation (83) and the exact solution (73) for $\sigma =1$ is presented in Figure 4 to assess the effectiveness and stability of the applied methods. Figure 4 illustrates that the concordance between the derived approximations and the exact solution increases with an increase in the value of fractionality $\sigma$, achieving complete agreement at $\sigma =1$. This confirms the high accuracy of the generated approximations and the effectiveness of the utilized methods. In addition, a numerical analysis is performed to estimate the absolute error $E_n=\left|\mathrm{Exact}\mathrm{Sol}.\left(73\right)-\mathrm{Approx}.\left(83\right)\right|$ for the generated approximations (83) as presented in

Table 3. The absolute error $E_n$ to the approximation (83) is estimated at $\left(\sigma ,\zeta ,\eta ,\tau \right)=\left(1,1,1,0.1\right)$.

$\mathit{\xi }$	$\mathbf{Exact}\mathbf{Sol}.$ (73)	${\mathbf{Approx.}}_{\mathit{\sigma }=1}$ (83)	${\mathit{E}}_{\mathit{n}}$
−10	0.23364	0.233732	0.925165 ×${10}^{-4}$
−8	−0.424898	−0.425066	1.68251 ×${10}^{-4}$
−6	0.12	0.120048	0.475176 ×${10}^{-4}$
−4	0.325023	0.325151	1.28702 ×${10}^{-4}$
−2	−0.390514	−0.390669	1.54636 ×${10}^{-4}$
0	0	0	0
2	0.390514	0.390669	1.54636 ×${10}^{-4}$
4	−0.325023	−0.325151	1.28702 ×${10}^{-4}$
6	−0.12	−0.120048	0.475176 ×${10}^{-4}$
8	0.424898	0.425066	1.68251 ×${10}^{-4}$
10	−0.23364	−0.233732	0.925165 ×${10}^{-4}$

. Furthermore, to guarantee the convergence of the generated approximate solution utilizing the proposed approaches, the absolute error of these approximations is computed at various orders, as illustrated in

Table 4. The absolute error $E_n$ for the different order approximations to the generated approximation (83) is estimated at $\left(\sigma ,\zeta ,\eta ,\tau \right)=\left(1,1,1,0.1\right)$.

$\mathit{\xi }$	$4^{\mathbf{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${10}^{\mathbf{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${20}^{\mathbf{th}}$-Order ${\mathit{E}}_{\mathit{n}}$
−10	0.925165 ×${10}^{-4}$	4.52294 ×${10}^{-12}$	0.
−8	1.68251 ×${10}^{-4}$	8.22564 ×${10}^{-12}$	0.
−6	0.475176 ×${10}^{-4}$	2.32306 ×${10}^{-12}$	0.
−4	1.28702 ×${10}^{-4}$	6.29208 ×${10}^{-12}$	0.
−2	1.54636 ×${10}^{-4}$	7.55984 ×${10}^{-12}$	0.
0	0	0	0
2	1.54636 ×${10}^{-4}$	7.55984 ×${10}^{-12}$	0.
4	1.28702 ×${10}^{-4}$	6.29208 ×${10}^{-12}$	0.
6	0.475176 ×${10}^{-4}$	2.32306 ×${10}^{-12}$	0.
8	1.68251 ×${10}^{-4}$	8.22564 ×${10}^{-12}$	0.
10	0.925165 ×${10}^{-4}$	4.52294 ×${10}^{-12}$	0.

. Additionally, the maximum residual error for all derived approximations across the whole study domain $\mathsf{\Omega }$ is computed using the following relationship:

$$\begin{array}{ll}\mathbb{F}& =\sin \left(\xi \right)\sin \left(\zeta \right)\sin \left(\eta \right)\left(1+{\displaystyle \sum _{i=1}^{30}}{\left(-5\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right),\\ \mathbb{R}& ={\displaystyle \underset{\mathsf{\Omega }}{max}}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}-{\partial }_{\zeta }^2\mathbb{F}-{\partial }_{\eta }^2\mathbb{F}+2\mathbb{F}\right|=5.23903\times {10}^{-12},\end{array}$$

where $\mathsf{\Omega }\equiv \left(\xi ,\zeta ,\eta ,\tau \right)\in \left[-5,5\right]\times \left[0,1\right]\times \left[0,1\right]\times \left[0,1\right]$ and $\sigma =1$. Note here that $\mathbb{R}=5.23903\times {10}^{-12}$ at $\left(\xi ,\zeta ,\eta ,\tau \right)=\left(1.1636,0.926666,0.557893,0.991053\right)$.

We can conclude that the proposed techniques enable the computation of higher-order approximations yield more precise approximations, which, hence, validates the efficacy of the existing methods.

4.3. Example (III)

Assume the following time-fractional three-dimensional heat flow inhomogenous problem:

$$D_{\tau }^{\sigma }\mathbb{F}\equiv {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}={\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}+\sin \left(\eta \right),$$

(102)

with the IC

$$\mathbb{F}\left(\xi ,\zeta ,\eta ,0\right)\equiv {\mathbb{F}}_0=\sin \left(\xi +\zeta \right)+\sin \left(\eta \right),$$

(103)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)$ and $0<\sigma \le 1$.

For $\sigma =1$, the exact solution to problem (102) reads

$$\mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)=e^{-2\tau }\sin (\xi +\zeta )+\sin \left(\eta \right).$$

(104)

This problem can be analyzed and solved using the suggested methods, as shown below.

4.3.1. Analyzing Example (III) via the Tantawy Technique

Step (1)

The analytical approximation to problem (102) is assumed to have the following convergent series form:

$$\mathbb{F}={\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma },$$

(105)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)$, ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi ,\zeta ,\eta \right)$ ∀$i=1,2,3,\cdots$ are unknown functions in the independent variable $\left(\xi ,\zeta ,\eta \right)$ and ${\mathbb{F}}_0\equiv {\mathbb{F}}_0\left(\xi ,\zeta ,\eta ,{\tau }_0\right)$.

Step (2)

By inserting the Ansatz (105) into problem (102), we get

$$\begin{array}{cc}\hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+{\partial }_{\zeta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & +{\partial }_{\eta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+\sin \left(\eta \right),\hfill \end{array}$$

(106)

and for $m^{th}$-order (say $m=3$), Equation (106) reduces to the following form:

$$\begin{array}{cc}\hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+{\partial }_{\zeta }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & +{\partial }_{\eta }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+\sin \left(\eta \right).\hfill \end{array}$$

(107)

Step (3)

Using the following Mathematica command for the CFD in Equation (76),

$$D_{\tau }^{\sigma }\mathbb{F}\equiv \mathrm{CaputoD}\left[\left({\mathbb{F}}_0\left[\xi ,\zeta ,\eta \right]+\sum _{i=1}^m{\mathbb{F}}_i\left[\xi ,\zeta ,\eta \right]{\tau }^{i\sigma }\right),\left\{\tau ,\sigma \right\}\right].$$

(108)

Step (4)

After inserting the Mathematica command (108) into Equation (107) and collecting the various coefficients of ${\tau }^{i\sigma }$, we get

$$\sum _{i=0}^m{Q}_i{\tau }^{i\sigma }=Q_0+Q_1{\tau }^{\sigma }+Q_2{\tau }^{2\sigma }+Q_3{\tau }^{3\sigma }+\cdots =0,$$

(109)

with

$$\begin{array}{cc}\hfill \mathrm{Coeff}.\left({\tau }^{0\sigma }\right)& :Q_0=g_1{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\zeta }^2{\mathbb{F}}_0-{\partial }_{\eta }^2{\mathbb{F}}_0+\sin \left(\eta \right),\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{1\sigma }\right)& :Q_1=g_2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\zeta }^2{\mathbb{F}}_1-{\partial }_{\eta }^2{\mathbb{F}}_1\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{2\sigma }\right)& :Q_2=g_3{\mathbb{F}}_3-{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\zeta }^2{\mathbb{F}}_2-{\partial }_{\eta }^2{\mathbb{F}}_2\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{3\sigma }\right)& :Q_3=g_4{\mathbb{F}}_4-{\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\zeta }^2{\mathbb{F}}_3-{\partial }_{\eta }^2{\mathbb{F}}_3\hfill \\ \hfill & ⋮,\hfill \end{array}$$

(110)

where $g_i$∀$i=1,2,3,\dots$ are defined in Equation (48).

Step (5)

Equating the coefficients $Q_i$ to zero, we obtain a system of DEs involving ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯:

$$\begin{array}{cc}\hfill g_1{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\zeta }^2{\mathbb{F}}_0-{\partial }_{\eta }^2{\mathbb{F}}_0+\sin \left(\eta \right)& =0,\hfill \\ \hfill g_2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\zeta }^2{\mathbb{F}}_1-{\partial }_{\eta }^2{\mathbb{F}}_1& =0,\hfill \\ \hfill g_3{\mathbb{F}}_3-{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\zeta }^2{\mathbb{F}}_2-{\partial }_{\eta }^2{\mathbb{F}}_2& =0,\hfill \\ \hfill g_4{\mathbb{F}}_4-{\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\zeta }^2{\mathbb{F}}_3-{\partial }_{\eta }^2{\mathbb{F}}_3& =0,\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(111)

Step (6)

By solving system (111) simultaneously, we can finally determine the implicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯ as functions of the IC ${\mathbb{F}}_0$:

$$\begin{array}{cc}\hfill {\mathbb{F}}_1& =\left({\mathbb{F}}_0^{(2,0,0)}+{\mathbb{F}}_0^{(0,2,0)}+{\mathbb{F}}_0^{(0,0,2)}+\sin \left(\eta \right)\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_2& =\left(\begin{array}{c}-4{\mathbb{F}}_0^{(0,0,2)}+{\mathbb{F}}_0^{(0,0,4)}-4{\mathbb{F}}_0^{(0,2,0)}+2{\mathbb{F}}_0^{(0,2,2)}\\ +{\mathbb{F}}_0^{(0,4,0)}-4{\mathbb{F}}_0^{(2,0,0)}+2{\mathbb{F}}_0^{(2,0,2)}+2{\mathbb{F}}_0^{(2,2,0)}+{\mathbb{F}}_0^{(4,0,0)}\end{array}\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(2\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_3& =\left(\begin{array}{c}12{\mathbb{F}}_0^{(0,0,2)}-6{\mathbb{F}}_0^{(0,0,4)}+{\mathbb{F}}_0^{(0,0,6)}+12{\mathbb{F}}_0^{(0,2,0)}\\ -12{\mathbb{F}}_0^{(0,2,2)}+3{\mathbb{F}}_0^{(0,2,4)}-6{\mathbb{F}}_0^{(0,4,0)}+3{\mathbb{F}}_0^{(0,4,2)}\\ +{\mathbb{F}}_0^{(0,6,0)}+12{\mathbb{F}}_0^{(2,0,0)}-12{\mathbb{F}}_0^{(2,0,2)}+3{\mathbb{F}}_0^{(2,0,4)}\\ -12{\mathbb{F}}_0^{(2,2,0)}+6{\mathbb{F}}_0^{(2,2,2)}+3{\mathbb{F}}_0^{(2,4,0)}-6{\mathbb{F}}_0^{(4,0,0)}\\ +3{\mathbb{F}}_0^{(4,0,2)}+3{\mathbb{F}}_0^{(4,2,0)}+{\mathbb{F}}_0^{(6,0,0)}\end{array}\right){\displaystyle \frac{1}{\mathrm{\Gamma }\left(3\sigma +1\right)}},\hfill \\ \hfill {\mathbb{F}}_4& =\left(\begin{array}{c}-32{\mathbb{F}}_0^{(0,0,2)}+24{\mathbb{F}}_0^{(0,0,4)}-8{\mathbb{F}}_0^{(0,0,6)}+{\mathbb{F}}_0^{(0,0,8)}\\ -32{\mathbb{F}}_0^{(0,2,0)}+48{\mathbb{F}}_0^{(0,2,2)}-24{\mathbb{F}}_0^{(0,2,4)}+4{\mathbb{F}}_0^{(0,2,6)}\\ +24{\mathbb{F}}_0^{(0,4,0)}-24{\mathbb{F}}_0^{(0,4,2)}+6{\mathbb{F}}_0^{(0,4,4)}-8{\mathbb{F}}_0^{(0,6,0)}\\ +4{\mathbb{F}}_0^{(0,6,2)}+{\mathbb{F}}_0^{(0,8,0)}-32{\mathbb{F}}_0^{(2,0,0)}+48{\mathbb{F}}_0^{(2,0,2)}\\ -24{\mathbb{F}}_0^{(2,0,4)}+4{\mathbb{F}}_0^{(2,0,6)}+48{\mathbb{F}}_0^{(2,2,0)}-48{\mathbb{F}}_0^{(2,2,2)}\\ +12{\mathbb{F}}_0^{(2,2,4)}-24{\mathbb{F}}_0^{(2,4,0)}+12{\mathbb{F}}_0^{(2,4,2)}+4{\mathbb{F}}_0^{(2,6,0)}\\ +24{\mathbb{F}}_0^{(4,0,0)}-24{\mathbb{F}}_0^{(4,0,2)}+6{\mathbb{F}}_0^{(4,0,4)}-24{\mathbb{F}}_0^{(4,2,0)}\\ +12{\mathbb{F}}_0^{(4,2,2)}+6{\mathbb{F}}_0^{(4,4,0)}-8{\mathbb{F}}_0^{(6,0,0)}+4{\mathbb{F}}_0^{(6,0,2)}\\ +4{\mathbb{F}}_0^{(6,2,0)}+{\mathbb{F}}_0^{(8,0,0)}\end{array}\right).{\displaystyle \frac{1}{\mathrm{\Gamma }\left(4\sigma +1\right)}},\hfill \end{array}$$

(112)

where ${\mathbb{F}}_i^{\left(2,0,0,0\right)}\equiv {\partial }_{\xi }^2{\mathbb{F}}_i$, ${\mathbb{F}}_i^{\left(0,2,0,0\right)}\equiv {\partial }_{\zeta }^2\mathbb{F}$, ${\mathbb{F}}_i^{\left(0,0,3,0\right)}\equiv {\partial }_{\eta }^2\mathbb{F}$, and so on.

Step (7)

Using the value of the IC ${\mathbb{F}}_0$ given in Equation (103) in the system (112), we finally get the explicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, $\cdots ,$ as follows:

$${\mathbb{F}}_i={\left(-2\right)}^i\sin (\xi +\zeta ){\displaystyle \frac{1}{\mathrm{\Gamma }(i\sigma +1)}},\forall i=1,2,3,\cdots .$$

(113)

Step (8)

By inserting the obtained values of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, ⋯, into the Ansatz (105), we ultimately obtain an analytical approximate solution to Equation (102) as follows:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1{\tau }^{\sigma }+{\mathbb{F}}_2{\tau }^{2\sigma }+{\mathbb{F}}_3{\tau }^{3\sigma }+\cdots \hfill \\ \hfill & =\sin \left(\eta \right)+\sin (\xi +\zeta )\left(\sum _{i=0}^{\infty }{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)\hfill \\ \hfill & =\sin \left(\eta \right)+\sin (\xi +\zeta )E_{\sigma }\left(-2{\tau }^{\sigma }\right),\hfill \end{array}$$

(114)

where $E_{\sigma }\left(-2{\tau }^{\sigma }\right)$ indicates the Mittag-Leffler function.

4.3.2. Analyzing Example (III) via the NITM

To analyze Equation (102) using the NITM, we can encapsulate this procedure in the following succinct steps:

Step (1)

By applying the ET to Equation (102), we have

$$\left(E\left[\mathbb{F}\right]-s^2{\mathbb{F}}_0\right)=s^{\sigma }E\left({\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}+\sin \left(\eta \right)\right)$$

(115)

Step (2)

Applying the inverse ET to Equation (115), we obtain

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}+\sin \left(\eta \right)\right]\right].$$

(116)

Step (3)

Note that Equation (102) is linear and according to the recurrence relation (25), we get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $$\begin{array}{c}{\mathbb{F}}_0=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\partial }_{{\tau }^k}^k\mathbb{F}(\xi ,\zeta ,\eta ,\tau )\right|}_{\tau =0}\right]=E^{-1}\left[s^2\mathbb{F}(\xi ,\zeta ,\eta ,0)\right]=\sin (\xi +\zeta )+\sin \left(\eta \right).\end{array}$$

  (117)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0\right)+h\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0+{\partial }_{\eta }^2{\mathbb{F}}_0+\sin \left(\eta \right)\right]\right]\hfill \\ \hfill & =-2\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (118)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_1\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_1+{\partial }_{\zeta }^2{\mathbb{F}}_1+{\partial }_{\eta }^2{\mathbb{F}}_1\right]\right]\hfill \\ \hfill & =2^2\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (119)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_2\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_2+{\partial }_{\zeta }^2{\mathbb{F}}_2+{\partial }_{\eta }^2{\mathbb{F}}_2\right]\right]\hfill \\ \hfill & =-2^3\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.\hfill \end{array}$$

  (120)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E\left[\left[M\left({\mathbb{F}}_3\right)\right]\right]\right]=E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_3+{\partial }_{\zeta }^2{\mathbb{F}}_3+{\partial }_{\eta }^2{\mathbb{F}}_3\right]\right]\hfill \\ \hfill & =2^4\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}},\hfill \end{array}$$

  (121)

  and so on.

Step (4)

At the end, the solution in the convergence series form takes the following form:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots \hfill \\ \hfill & =\sin \left(\eta \right)+\sin (\xi +\zeta )\left(\sum _{i=0}^m{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)\hfill \\ \hfill & =\sin \left(\eta \right)+\sin (\xi +\zeta )E_{\sigma }\left(-2{\tau }^{\sigma }\right).\hfill \end{array}$$

(122)

4.3.3. Analyzing Example (III) via the HPTM

To analyze Equation (102) using the HPTM, we can summarize this process in the following brief steps:

Step (1)

Applying the ET to Equation (102) yields

$$E\left[\mathbb{F}\right]=s^2{\mathbb{F}}_0+s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}+\sin \left(\eta \right)\right].$$

(123)

Step (2)

On applying the inverse ET to Equation (123), we have

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\zeta }^2\mathbb{F}+{\partial }_{\eta }^2\mathbb{F}+\sin \left(\eta \right)\right]\right].$$

(124)

Step (3)

Note that Equation (102) is linear; thus, relation (33) according to Equation (124) can be written as follows:

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵE^{-1}\left[s^{\sigma }E\left[M\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)-h\right]\right],$$

(125)

which leads to

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵ\left(E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\zeta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\eta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+\sin \left(\eta \right)\right]\right]\right).$$

(126)

Step (4)

By comparing the coefficients of $ϵ$ in Equation (126), we finally get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $${\mathbb{F}}_0=\sin (\xi +\zeta )+\sin \left(\eta \right).$$

  (127)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0+{\partial }_{\zeta }^2{\mathbb{F}}_0+{\partial }_{\eta }^2{\mathbb{F}}_0+\sin \left(\eta \right)\right]\right]\hfill \\ \hfill & =-2\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}},\hfill \end{array}$$

  (128)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_1+{\partial }_{\zeta }^2{\mathbb{F}}_1+{\partial }_{\eta }^2{\mathbb{F}}_1\right]\right]\hfill \\ \hfill & =2^2\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (129)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_2+{\partial }_{\zeta }^2{\mathbb{F}}_2+{\partial }_{\eta }^2{\mathbb{F}}_2\right]\right]\hfill \\ \hfill & =-2^3\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.\hfill \end{array}$$

  (130)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_3+{\partial }_{\zeta }^2{\mathbb{F}}_3+{\partial }_{\eta }^2{\mathbb{F}}_3\right]\right]\hfill \\ \hfill & =2^4\sin (\xi +\zeta ){\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}}.\hfill \end{array}$$

  (131)

Step (5)

Thus, the series convergent solution takes the following final form:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\zeta ,\eta ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots \hfill \\ \hfill & =\sin \left(\eta \right)+\sin (\xi +\zeta )\left(\sum _{i=0}^{\infty }{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)\hfill \\ \hfill & =\sin \left(\eta \right)+\sin (\xi +\zeta )E_{\sigma }\left(-2{\tau }^{\sigma }\right).\hfill \end{array}$$

(132)

One can see that the generated approximations (114), (122), and (132) to problem (102), using the three proposed methods, show perfect agreement. Figure 5 illustrates an analysis of one of these approximations and how the fractional parameter $\sigma$ influences the behavior of the phenomenon described by this problem. It is also noted that when the fractional parameter value is close to one, there is almost complete agreement between the derived approximations and the exact solution (104) at $\sigma =1$. This confirms the high accuracy of the generated approximations and also enhances the efficiency of the approaches employed in this investigation. We also performed a graphical and numerical comparison between the derived approximations and the exact solution for the integer case, as shown in Figure 6 and

Table 5. The absolute error $E_n$ to the approximation (114) is estimated at $\left(\sigma ,\zeta ,\eta ,\tau \right)=\left(1,1,1,0.1\right)$.

$\mathit{\xi }$	$\mathbf{Exact}\mathbf{Sol}.$ (104)	${\mathbf{Approx.}}_{\mathit{\sigma }=1}$ (114)	${\mathit{E}}_{\mathit{n}}$
−10	0.504057	0.504056	1.06337 ×${10}^{-6}$
−8	0.303576	0.303574	1.69519 ×${10}^{-6}$
−6	1.62657	1.62657	2.47427 ×${10}^{-6}$
−4	0.725932	0.725931	0.364126 ×${10}^{-6}$
−2	0.152533	0.152531	2.17121 ×${10}^{-6}$
0	1.53041	1.53041	2.17121 ×${10}^{-6}$
2	0.95701	0.957011	0.364126 ×${10}^{-6}$
4	0.0563702	0.0563677	2.47427 ×${10}^{-6}$
6	1.37937	1.37937	1.69519 ×${10}^{-6}$
8	1.17889	1.17889	1.06337 ×${10}^{-6}$
10	0.0227482	0.0227457	2.58023 ×${10}^{-6}$

, respectively. The comparison results show complete agreement between all the generated approximations (114) and the exact solution (104) at $\sigma =1$. Furthermore, the numerical analysis presented in Table 5 demonstrates the high precision of the derived approximations by calculating their absolute error $E_n=\left|\mathrm{Exact}\mathrm{Sol}.\left(104\right)-\mathrm{Approx}.\left(114\right)\right|$ compared to the exact solution (104) at $\sigma =1$. Additionally, to ensure the convergence of the generated approximate solution using the proposed approaches, the absolute error of these approximations is estimated at different orders, as demonstrated in

Table 6. The absolute error $E_n$ for the different order approximations to the generated approximation (114) is estimated at $\left(\sigma ,\zeta ,\eta ,\tau \right)=\left(1,1,1,0.1\right)$.

$\mathit{\xi }$	$4^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${10}^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${20}^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$
−10	1.06337 ×${10}^{-6}$	2.22045 ×${10}^{-16}$	0.
−8	1.69519 ×${10}^{-6}$	3.33067 ×${10}^{-16}$	0.
−6	2.47427 ×${10}^{-6}$	5.55112 ×${10}^{-16}$	0.
−4	0.364126 ×${10}^{-6}$	0.832667 ×${10}^{-16}$	0.
−2	2.17121 ×${10}^{-6}$	4.44089 ×${10}^{-16}$	0.
0	2.17121 ×${10}^{-6}$	4.44089 ×${10}^{-16}$	0.
2	0.364126 ×${10}^{-6}$	0.832667 ×${10}^{-16}$	0.
4	2.47427 ×${10}^{-6}$	5.55112 ×${10}^{-16}$	0.
6	1.69519 ×${10}^{-6}$	3.33067 ×${10}^{-16}$	0.
8	1.06337 ×${10}^{-6}$	2.22045 ×${10}^{-16}$	0.
10	2.58023 ×${10}^{-6}$	5.55112 ×${10}^{-16}$	1.11022 ×${10}^{-16}$

. In addition, the maximum residual error for all derived approximations across the whole study domain $\mathsf{\Omega }$ is computed using the following relationship:

$$\begin{array}{ll}\mathbb{F}& =\sin \left(\eta \right)+\sin (\xi +\zeta )\left(1+{\displaystyle \sum _{i=1}^{20}}{\left(-2\right)}^i{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right),\\ \mathbb{R}& ={\displaystyle \underset{\mathsf{\Omega }}{max}}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}-{\partial }_{\zeta }^2\mathbb{F}-{\partial }_{\eta }^2\mathbb{F}-\sin \left(\eta \right)\right|=8.42659\times {10}^{-13},\end{array}$$

where $\mathsf{\Omega }\equiv \left(\xi ,\zeta ,\eta ,\tau \right)\in \left[-5,5\right]\times \left[0,1\right]\times \left[0,1\right]\times \left[0,1\right]$ and $\sigma =1$. Note here that $\mathbb{R}=8.42659\times {10}^{-13}$ at $\left(\xi ,\zeta ,\eta ,\tau \right)=\left(0.469392,0.997728,0.999987,0.999131\right)$.

One can conclude that the proposed methods facilitate the computation of higher-order approximations, resulting in increased precision and verifying the effectiveness of the used methods.

4.4. Example (IV)

Assume the following time-fractional three-dimensional heat flow nonlinear problem [72,73]:

$$D_{\tau }^{\sigma }\mathbb{F}\equiv {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}={\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F},$$

(133)

with the IC

$$\mathbb{F}\left(\xi ,0\right)\equiv {\mathbb{F}}_0=e^{\xi },$$

(134)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\tau \right)$ and $0<\sigma \le 1$.

For $\sigma =1$, the exact solution to problem (133) reads

$$\mathbb{F}\left(\xi ,\tau \right)=e^{\xi +\tau }.$$

(135)

This problem can be analyzed and solved using the suggested methods, as shown below.

4.4.1. Analyzing Example (IV) via the Tantawy Technique

Step (1)

The analytical approximation to problem (133) is assumed to have the following convergent series form:

$$\mathbb{F}={\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma },$$

(136)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\tau \right)$, ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi \right)$∀ $i=1,2,3,\cdots$ are unknown functions in the independent variable $\xi$ and ${\mathbb{F}}_0\equiv {\mathbb{F}}_0\left(\xi ,{\tau }_0\right)$.

Step (2)

By inserting the Ansatz (136) into problem (133), we get

$$\begin{array}{cc}\hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & +\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & -{\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)}^2+\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right),\hfill \end{array}$$

(137)

and for $m^{th}$-order (say $m=3$), Equation (137) reduces to the following form:

$$\begin{array}{cc}\hfill {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}& ={\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F},\hfill \\ \hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & +\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)\hfill \\ \hfill & -{\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)}^2+\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right).\hfill \end{array}$$

(138)

Step (3)

Using the following Mathematica command for the CFD in Equation (138),

$$D_{\tau }^{\sigma }\mathbb{F}\equiv \mathrm{CaputoD}\left[\left({\mathbb{F}}_0\left[\xi \right]+\sum _{i=1}^m{\mathbb{F}}_i\left[\xi \right]{\tau }^{i\sigma }\right),\left\{\tau ,\sigma \right\}\right].$$

(139)

Step (4)

After inserting the Mathematica command (139) into Equation (138) and collecting the various coefficients of ${\tau }^{i\sigma }$, we get

$$\sum _{i=0}^m{Q}_i{\tau }^{i\sigma }=Q_0+Q_1{\tau }^{\sigma }+Q_2{\tau }^{2\sigma }+Q_3{\tau }^{3\sigma }+\cdots =0,$$

(140)

with

$$\begin{array}{cc}\hfill \mathrm{Coeff}.\left({\tau }^{0\sigma }\right)& :Q_0=g_1{\mathbb{F}}_1+{\partial }_{\xi }{\mathbb{F}}_0-{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_0+{\mathbb{F}}_0^2-{\mathbb{F}}_0,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{1\sigma }\right)& :Q_1=g_2{\mathbb{F}}_2+{\partial }_{\xi }{\mathbb{F}}_1-{\mathbb{F}}_1{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\xi }^2{\mathbb{F}}_1-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_1-{\mathbb{F}}_1+2{\mathbb{F}}_0{\mathbb{F}}_1,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{2\sigma }\right)& :Q_2=g_3{\mathbb{F}}_3+{\partial }_{\xi }{\mathbb{F}}_2-{\mathbb{F}}_2{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_1{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_2\hfill \\ \hfill & -{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_2+{\mathbb{F}}_1^2-{\mathbb{F}}_2+2{\mathbb{F}}_0{\mathbb{F}}_2,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{3\sigma }\right)& :Q_3=g_4{\mathbb{F}}_4+{\partial }_{\xi }{\mathbb{F}}_3-{\mathbb{F}}_3{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_2{\partial }_{\xi }^2{\mathbb{F}}_1-{\mathbb{F}}_1{\partial }_{\xi }^2{\mathbb{F}}_2\hfill \\ \hfill & -{\partial }_{\xi }^2{\mathbb{F}}_3-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_3+2{\mathbb{F}}_1{\mathbb{F}}_2-{\mathbb{F}}_3+2{\mathbb{F}}_0{\mathbb{F}}_3,\hfill \\ \hfill & ⋮,\hfill \end{array}$$

(141)

where $g_i$∀$i=1,2,3,\dots$ are defined in Equation (48).

Step (5)

Equating the coefficients $Q_i$ to zero, we obtain a system of DEs involving ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, …:

$$\begin{array}{cc}\hfill g_1{\mathbb{F}}_1+{\partial }_{\xi }{\mathbb{F}}_0-{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_0+{\mathbb{F}}_0^2-{\mathbb{F}}_0& =0,\hfill \\ \hfill g_2{\mathbb{F}}_2+{\partial }_{\xi }{\mathbb{F}}_1-{\mathbb{F}}_1{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\xi }^2{\mathbb{F}}_1-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_1-{\mathbb{F}}_1+2{\mathbb{F}}_0{\mathbb{F}}_1& =0,\hfill \\ \hfill g_3{\mathbb{F}}_3+{\partial }_{\xi }{\mathbb{F}}_2-{\mathbb{F}}_2{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_1{\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\xi }^2{\mathbb{F}}_2-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_2+{\mathbb{F}}_1^2-{\mathbb{F}}_2+2{\mathbb{F}}_0{\mathbb{F}}_2& =0,\hfill \\ \hfill g_4{\mathbb{F}}_4+{\partial }_{\xi }{\mathbb{F}}_3-{\mathbb{F}}_3{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_2{\partial }_{\xi }^2{\mathbb{F}}_1-{\mathbb{F}}_1{\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\xi }^2{\mathbb{F}}_3-{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_3+2{\mathbb{F}}_1{\mathbb{F}}_2-{\mathbb{F}}_3+2{\mathbb{F}}_0{\mathbb{F}}_3& =0,\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(142)

Step (6)

Using the value of the IC ${\mathbb{F}}_0$ given in Equation (134) in the system (142) and solving the obtained system simultaneously, we finally get the explicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, $\cdots ,$ as follows:

$${\mathbb{F}}_i={\displaystyle \frac{e^{\xi }}{\mathrm{\Gamma }(i\sigma +1)}},\forall i=1,2,3,\cdots .$$

Step (7)

Inserting the obtained values of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ${\mathbb{F}}_4$, ⋯, into the Ansatz (136), the following analytical approximate solution to Equation (133) is obtained:

$$\mathbb{F}\left(\xi ,\tau \right)={\mathbb{F}}_0+{\mathbb{F}}_1{\tau }^{\sigma }+{\mathbb{F}}_2{\tau }^{2\sigma }+{\mathbb{F}}_3{\tau }^{3\sigma }+\cdots =e^{\xi }\sum _{i=0}^{\infty }{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}=e^{\xi }{E}_{\sigma }\left({\tau }^{\sigma }\right),$$

(143)

where $E_{\sigma }\left({\tau }^{\sigma }\right)$ indicates the Mittag-Leffler.

4.4.2. Analyzing Example (IV) via the NITM

To solve Equation (133) using the NITM, we can encapsulate this procedure in the following brief steps:

Step (1)

Applying the ET to Equation (133) yields

$$\left(E\left[\mathbb{F}\right]-s^2{\mathbb{F}}_0\right)=s^{\sigma }E\left({\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F}\right).$$

(144)

Step (2)

By applying the inverse ET to Equation (144), we obtain

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F}\right]\right].$$

(145)

Step (3)

According to the recurrence relation (25), the following first fourth-order approximations are obtained:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $$\begin{array}{c}{\mathbb{F}}_0=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\partial }_{{\tau }^k}^k\mathbb{F}(\xi ,\tau )\right|}_{\tau =0}\right]=E^{-1}\left[s^2\mathbb{F}(\xi ,0)\right]=e^{\xi }.\end{array}$$

  (146)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0\right)+M\left({\mathbb{F}}_0\right)\right]\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\xi }{\mathbb{F}}_0+{\mathbb{F}}_0\right)+\left({\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_0^2\right)\right]\right]\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (147)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-N\left({\mathbb{F}}_0\right)\right]\right]+E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-M\left({\mathbb{F}}_0\right)\right]\right]\hfill \\ \hfill & =\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)+\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\xi }{\mathbb{F}}_0+{\mathbb{F}}_0\right]\right]\end{array}\right\}\hfill \\ \hfill & +\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[\left({\mathbb{F}}_0+{\mathbb{F}}_1\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-{\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)}^2\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[{\mathbb{F}}_0{\partial }_{\xi }^2{\mathbb{F}}_0-{\mathbb{F}}_0^2\right]\right]\end{array}\right\}\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (148)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)-N\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)\right]\right]\hfill \\ \hfill & +E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)-M\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)\right]\right]\hfill \\ \hfill & =\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)+\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)+\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)\right]\right]\end{array}\right\}\hfill \\ \hfill & +\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)-{\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)}^2\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[\left({\mathbb{F}}_0+{\mathbb{F}}_1\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-{\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)}^2\right]\right]\end{array}\right\}\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (149)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)-N\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)\right]\right]\hfill \\ \hfill & +E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)-M\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)\right]\right]\hfill \\ \hfill & =\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)+\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)-{\partial }_{\xi }\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)+\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)\right]\right]\end{array}\right\}\hfill \\ \hfill & +\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)-{\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3\right)}^2\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right){\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)-{\left({\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2\right)}^2\right]\right]\end{array}\right\}\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (150)

  and so on.

Step (4)

At the end, the solution in the convergence series form is taken the following form:

$$\mathbb{F}\left(\xi ,\tau \right)={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots =e^{\xi }\sum _{i=0}^{\infty }{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}=e^{\xi }{E}_{\sigma }\left({\tau }^{\sigma }\right).$$

(151)

4.4.3. Analyzing Example (IV) via the HPTM

To analyze Equation (133) using the HPTM, the following brief steps are considered:

Step (1)

Applying the ET to Equation (133) implies

$$E\left[\mathbb{F}\right]=s^2{\mathbb{F}}_0+s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F}\right].$$

(152)

Step (2)

By applying the inverse ET to Equation (152), we have

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F}\right]\right].$$

(153)

Step (3)

The relation (33) according to Equation (153) can be written as follows:

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵE^{-1}\left[s^{\sigma }E\left[M\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{F}}_k\right)\right]\right],$$

(154)

which leads to

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵ\left(E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\zeta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+{\partial }_{\eta }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)\right)+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{F}}_k\right)\right]\right]\right).$$

(155)

Step (4)

By comparing the coefficients of $ϵ$ in Equation (155), we finally get the following first fourth-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $${\mathbb{F}}_0=e^{\xi }.$$

  (156)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_0\right)+H_0\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_0-{\partial }_{\xi }{\mathbb{F}}_0+{\mathbb{F}}_0\right)+H_0\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (157)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_1\right)+H_1\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_1-{\partial }_{\xi }{\mathbb{F}}_1+{\mathbb{F}}_1\right)+H_1\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (158)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_2\right)+H_2\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_2-{\partial }_{\xi }{\mathbb{F}}_2+{\mathbb{F}}_2\right)+H_2\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}.\hfill \end{array}$$

  (159)
• Fourth-order approximation ${\mathbb{F}}_4$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_4& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_3\right)+H_3\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_3-{\partial }_{\xi }{\mathbb{F}}_3+{\mathbb{F}}_3\right)+H_3\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & =e^{\xi }{\displaystyle \frac{{\tau }^{4\sigma }}{\mathrm{\Gamma }(4\sigma +1)}}.\hfill \end{array}$$

  (160)

Step (5)

Thus, the series convergent solution takes the following final form:

$$\mathbb{F}\left(\xi ,\tau \right)={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+{\mathbb{F}}_4+\cdots =e^{\xi }\left(\sum _{i=0}^{\infty }{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right)=e^{\xi }{E}_{\sigma }\left({\tau }^{\sigma }\right).$$

(161)

We can observe that all derived approximations (143), (151), and (161) to the problem (133) using the proposed methods are also completely identical, although this problem is strongly nonlinear. Due to the problem’s nonlinearity, we expected some differences between the derived approximations by the proposed methods, but it turned out that all the approximations are identical. Perhaps the reason for this identity is that the initial condition is not affected by differentiation. Figure 7 displays the graphical analyses of the obtained approximations at various values for the fractionality $\sigma$. It is clear that these approximations are very sensitive to any change in the fractionality value, as illustrated in Figure 7. Figure 8 shows the almost perfect harmony between the exact solution (135) and the obtained approximations (143) at $\sigma =1$, which proves the high precision of the generated approximations. Furthermore, the analysis results demonstrate the high accuracy of the derived approximations and the efficiency of the proposed methods by calculating the absolute error $L_{\infty }=\left|\mathrm{Exact}\mathrm{Sol}.\left(135\right)-\mathrm{Approx}.\left(143\right)\right|$ of these approximations, as shown in

Table 7. The absolute error $E_n$ to the approximation (143) is estimated at $\sigma =1\&\tau =0.1$.

$\mathit{\xi }$	$\mathbf{Exact}\mathbf{Sol}.$ (135)	${\mathbf{Approx.}}_{\mathit{\sigma }=1}$ (143)	${\mathit{E}}_{\mathit{n}}$
0	1.10517	1.10517	0.847423 ×${10}^{-7}$
0.2	1.34986	1.34986	1.03504 ×${10}^{-7}$
0.4	1.64872	1.64872	1.26421 ×${10}^{-7}$
0.6	2.01375	2.01375	1.54411 ×${10}^{-7}$
0.8	2.4596	2.4596	1.88597 ×${10}^{-7}$
1	3.00417	3.00417	2.30353 ×${10}^{-7}$
1.2	3.6693	3.6693	2.81354 ×${10}^{-7}$
1.4	4.48169	4.48169	3.43647 ×${10}^{-7}$
1.6	5.47395	5.47395	4.19731 ×${10}^{-7}$
1.8	6.68589	6.68589	5.12661 ×${10}^{-7}$
2	8.16617	8.16617	6.26166 ×${10}^{-7}$

. Moreover, to ensure the convergence of the derived approximation using the proposed techniques, the absolute error of these approximations is computed at different orders, as demonstrated in

Table 8. The absolute error $E_n$ for the different order approximations to the generated approximation (143) is estimated at $\sigma =1\&\tau =0.1$.

$\mathit{\xi }$	$4^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${10}^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$	${20}^{\mathit{th}}/{30}^{\mathit{th}}$-Order ${\mathit{E}}_{\mathit{n}}$
0	0.847423 ×${10}^{-7}$	8.32667 ×${10}^{-17}$	8.32667 ×${10}^{-17}$
0.2	1.03504 ×${10}^{-7}$	5.55112 ×${10}^{-17}$	5.55112 ×${10}^{-17}$
0.4	1.26421 ×${10}^{-7}$	0.	0.
0.6	1.54411 ×${10}^{-7}$	5.55112 ×${10}^{-17}$	5.55112 ×${10}^{-17}$
0.8	1.88597 ×${10}^{-7}$	11.1022 ×${10}^{-17}$	11.1022 ×${10}^{-17}$
1	2.30353 ×${10}^{-7}$	44.4089 ×${10}^{-17}$	44.4089 ×${10}^{-17}$
1.2	2.81354 ×${10}^{-7}$	38.8578 ×${10}^{-17}$	38.8578 ×${10}^{-17}$
1.4	3.43647 ×${10}^{-7}$	38.8578 ×${10}^{-17}$	38.8578 ×${10}^{-17}$
1.6	4.19731 ×${10}^{-7}$	88.8178 ×${10}^{-17}$	88.8178 ×${10}^{-17}$
1.8	5.12661 ×${10}^{-7}$	33.3067 ×${10}^{-17}$	33.3067 ×${10}^{-17}$
2	6.26166 ×${10}^{-7}$	44.4089 ×${10}^{-17}$	44.4089 ×${10}^{-17}$

. Furthermore, the maximum residual error for all derived approximations across the whole study domain $\mathsf{\Omega }$ is computed using the following relationship:

$$\begin{array}{ll}\mathbb{F}& =e^{\xi }\left(1+{\displaystyle \sum _{i=1}^{30}}{\displaystyle \frac{{\tau }^{i\sigma }}{\mathrm{\Gamma }(i\sigma +1)}}\right),\\ \mathbb{R}& ={\displaystyle \underset{\mathsf{\Omega }}{max}}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}+{\partial }_{\xi }\mathbb{F}-\mathbb{F}{\partial }_{\xi }^2\mathbb{F}+{\mathbb{F}}^2-\mathbb{F}\right|=1.13687\times {10}^{-13},\end{array}$$

where $\mathsf{\Omega }\equiv \left(\xi ,\tau \right)\in \left[-5,5\right]\times \left[0,1\right]$ and $\sigma =1$. Note here that $\mathbb{R}=1.13687\times {10}^{-13}$ at $\left(\xi ,\tau \right)=\left(4.90429,0.853881\right)$.

It is observed that the proposed techniques facilitate the computation of higher-order approximations, resulting in increased precision and verifying the effectiveness of the used approaches.

4.5. Example (V)

Assume the following time-fractional the Fisher-type equation (nonlinear diffusion equation) [74]:

$$D_{\tau }^{\sigma }\mathbb{F}\equiv {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}={\partial }_{\xi }^2\mathbb{F}+\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right),$$

(162)

with the IC

$$\mathbb{F}\left(\xi ,0\right)\equiv {\mathbb{F}}_0={\displaystyle \frac{1}{1+e^{\frac{-1}{\sqrt{2}}\xi }}},$$

(163)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\tau \right)$ and $0<\sigma \le 1$.

For $\sigma =1$, the exact solution to problem (162) is given by

$$\mathbb{F}\left(\xi ,\tau \right)={\displaystyle \frac{1}{1+e^{\frac{-\xi }{\sqrt{2}}-\left(\frac{1}{2}-a\right)\tau }}}.$$

(164)

This problem can be solved via the proposed techniques, as seen below.

4.5.1. Analyzing Example (V) via the Tantawy Technique

Step (1)

The analytical approximate solution to problem (162) is assumed to have the following convergent series form:

$$\mathbb{F}={\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma },$$

(165)

where $\mathbb{F}\equiv \mathbb{F}\left(\xi ,\tau \right)$, ${\mathbb{F}}_i\equiv {\mathbb{F}}_i\left(\xi \right)$∀ $i=1,2,3,\cdots$ are unknown functions in the independent variable $\xi$ and ${\mathbb{F}}_0\equiv {\mathbb{F}}_0\left(\xi ,{\tau }_0\right)$.

Step (2)

By substituting the Ansatz (165) into problem (162), we have

$$\begin{array}{cc}\hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)+\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)\times \hfill \\ \hfill & \left(1-{\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }\right)\left({\mathbb{F}}_0+\sum _{i=1}^{\infty }{\mathbb{F}}_i{\tau }^{i\sigma }-a\right)\hfill \end{array}$$

(166)

and for $m^{th}$-order (say $m=3$), Equation (166) reduces to the following form

$$\begin{array}{cc}\hfill {\partial }_{{\tau }^{\sigma }}^{\sigma }\mathbb{F}& ={\partial }_{\xi }^2\mathbb{F}-{\partial }_{\xi }\mathbb{F}+\mathbb{F}{\partial }_{\xi }^2\mathbb{F}-{\mathbb{F}}^2+\mathbb{F},\hfill \\ \hfill D_{\tau }^{\sigma }\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)& ={\partial }_{\xi }^2\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)+\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)\times \hfill \\ \hfill & \left(1-{\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }\right)\left({\mathbb{F}}_0+\sum _{i=1}^m{\mathbb{F}}_i{\tau }^{i\sigma }-a\right).\hfill \end{array}$$

(167)

Step (3)

Employing the subsequent Mathematica command for the CFD in Equation (167),

$$D_{\tau }^{\sigma }\mathbb{F}\equiv \mathrm{CaputoD}\left[\left({\mathbb{F}}_0\left[\xi \right]+\sum _{i=1}^m{\mathbb{F}}_i\left[\xi \right]{\tau }^{i\sigma }\right),\left\{\tau ,\sigma \right\}\right].$$

(168)

Step (4)

Upon putting the Mathematica command (168) into Equation (167) and aggregating the distinct coefficients of ${\tau }^{i\sigma }$, we obtain

$$\sum _{i=0}^m{Q}_i{\tau }^{i\sigma }=Q_0+Q_1{\tau }^{\sigma }+Q_2{\tau }^{2\sigma }+Q_3{\tau }^{3\sigma }+\cdots =0,$$

(169)

with

$$\begin{array}{cc}\hfill \mathrm{Coeff}.\left({\tau }^{0\sigma }\right)& :Q_0=g_1{\mathbb{F}}_1-{\mathbb{F}}_0^{″}+a{\mathbb{F}}_0-{\mathbb{F}}_0^2-a{\mathbb{F}}_0^2+{\mathbb{F}}_0^3,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{1\sigma }\right)& :Q_1=g_2{\mathbb{F}}_2-{\mathbb{F}}_1^{″}+a{\mathbb{F}}_1-2{\mathbb{F}}_0{\mathbb{F}}_1-2a{\mathbb{F}}_0{\mathbb{F}}_1+3{\mathbb{F}}_0^2{\mathbb{F}}_1,\hfill \\ \hfill \mathrm{Coeff}.\left({\tau }^{2\sigma }\right)& :Q_2=g_3{\mathbb{F}}_3-{\mathbb{F}}_2^{″}-{\mathbb{F}}_1^2-a{\mathbb{F}}_1^2+3{\mathbb{F}}_0{\mathbb{F}}_1^2+a{\mathbb{F}}_2-2{\mathbb{F}}_0{\mathbb{F}}_2\hfill \\ \hfill & -2a{\mathbb{F}}_0{\mathbb{F}}_2+3{\mathbb{F}}_0^2{\mathbb{F}}_2\hfill \\ \hfill & ⋮,\hfill \end{array}$$

(170)

where $g_i$∀$i=1,2,3,\dots$ are defined in Equation (48).

Step (5)

Equating the coefficients $Q_i$ to zero, we obtain a system of DEs involving ${\mathbb{F}}_0$, ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯:

$$\begin{array}{cc}\hfill \mathrm{\Gamma }\left(\sigma +1\right){\mathbb{F}}_1-{\mathbb{F}}_0^{″}+a{\mathbb{F}}_0-{\mathbb{F}}_0^2-a{\mathbb{F}}_0^2+{\mathbb{F}}_0^3& =0,\hfill \\ \hfill {\displaystyle \frac{\mathrm{\Gamma }\left(2\sigma +1\right)}{\mathrm{\Gamma }\left(\sigma +1\right)}}{\mathbb{F}}_2-{\mathbb{F}}_1^{″}+a{\mathbb{F}}_1-2{\mathbb{F}}_0{\mathbb{F}}_1-2a{\mathbb{F}}_0{\mathbb{F}}_1+3{\mathbb{F}}_0^2{\mathbb{F}}_1& =0,\hfill \\ \hfill {\displaystyle \frac{\mathrm{\Gamma }\left(3\sigma +1\right)}{\mathrm{\Gamma }\left(2\sigma +1\right)}}{\mathbb{F}}_3-{\mathbb{F}}_2^{″}-{\mathbb{F}}_1^2-a{\mathbb{F}}_1^2+3{\mathbb{F}}_0{\mathbb{F}}_1^2+a{\mathbb{F}}_2-2{\mathbb{F}}_0{\mathbb{F}}_2-2a{\mathbb{F}}_0{\mathbb{F}}_2+3{\mathbb{F}}_0^2{\mathbb{F}}_2& =0,\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(171)

Step (6)

Using the value of the IC ${\mathbb{F}}_0$ given in Equation (163) in the system (171) and solving the obtained system simultaneously, we finally get the explicit value of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, $\cdots ,$ as follows:

$$\begin{array}{cc}\hfill {\mathbb{F}}_1& ={\displaystyle \frac{\left(1-2a\right)e^{\frac{1}{\sqrt{2}}\xi }}{2{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^2}}{\displaystyle \frac{1}{\mathrm{\Gamma }(\sigma +1)}},\hfill \\ \hfill {\mathbb{F}}_2& =-{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^3}}{\displaystyle \frac{1}{\mathrm{\Gamma }(2\sigma +1)}},\hfill \\ \hfill {\mathbb{F}}_3& =-{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)\left[1-6e^{\frac{1}{\sqrt{2}}\xi }-e^{\sqrt{2}\xi }+2a\left(-1+e^{\sqrt{2}\xi }\right)\right]}{8{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{1}{\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & +{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{e^{\sqrt{2}\xi }}\left(1+a-2e^{\frac{1}{\sqrt{2}}\xi }+a{e}^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{\mathrm{\Gamma }(2\sigma +1)}{\mathrm{\Gamma }{(\sigma +1)}^2\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(172)

Step (7)

Inserting the obtained values of ${\mathbb{F}}_1$, ${\mathbb{F}}_2$, ${\mathbb{F}}_3$, ⋯ into the Ansatz (165), the following analytical approximate solution to Equation (162) is obtained:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1{\tau }^{\sigma }+{\mathbb{F}}_2{\tau }^{2\sigma }+{\mathbb{F}}_3{\tau }^{3\sigma }+\cdots \hfill \\ \hfill & ={\displaystyle \frac{1}{1+e^{\frac{-1}{\sqrt{2}}\xi }}}+{\displaystyle \frac{\left(1-2a\right)e^{\frac{1}{\sqrt{2}}\xi }}{2{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^2}}{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}-{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^3}}{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}\hfill \\ \hfill & -{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)\left[1-6e^{\frac{1}{\sqrt{2}}\xi }-e^{\sqrt{2}\xi }+2a\left(-1+e^{\sqrt{2}\xi }\right)\right]}{8{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & +{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{e^{\sqrt{2}\xi }}\left(1+a-2e^{\frac{1}{\sqrt{2}}\xi }+a{e}^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{\mathrm{\Gamma }(2\sigma +1){\tau }^{3\sigma }}{\mathrm{\Gamma }{(\sigma +1)}^2\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(173)

4.5.2. Analyzing Example (V) via the NITM

To solve Equation (162) using the NITM, we can encapsulate this procedure in the following brief steps:

Step (1)

By applying the ET to Equation (162), we get

$$\left(E\left[\mathbb{F}\right]-s^2{\mathbb{F}}_0\right)=s^{\sigma }E\left({\partial }_{\xi }^2\mathbb{F}+\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right)\right).$$

(174)

Step (2)

By applying the inverse ET to Equation (174), we obtain

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left({\partial }_{\xi }^2\mathbb{F}+\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right)\right)\right].$$

(175)

Step (3)

According to the recurrence relation (25), the following first $2^{rd}$-order approximations are obtained:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $$\begin{array}{c}{\mathbb{F}}_0=E^{-1}\left[\sum _{k=0}^{m-1}{s}^{2+k}{\left.{\mathbb{F}}^{\left(k\right)}(\xi ,\tau )\right|}_{\tau =0}\right]=E^{-1}\left[s^2\mathbb{F}(\xi ,0)\right]={\displaystyle \frac{1}{1+e^{\frac{-1}{\sqrt{2}}\xi }}}.\end{array}$$

  (176)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0\right)+M\left({\mathbb{F}}_0\right)\right]\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_0-a{\mathbb{F}}_0\right)+\left({\mathbb{F}}_0^2+a{\mathbb{F}}_0^2-{\mathbb{F}}_0^3\right)\right]\right]\hfill \\ \hfill & ={\displaystyle \frac{\left(1-2a\right)e^{\frac{1}{\sqrt{2}}\xi }}{2{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^2}}{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (177)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E\left[N\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-N\left({\mathbb{F}}_0\right)\right]\right]+E^{-1}\left[s^{\sigma }E\left[M\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-M\left({\mathbb{F}}_0\right)\right]\right]\hfill \\ \hfill & =\left\{E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)-a\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)\right]\right]-E^{-1}\left[s^{\sigma }E\left[{\partial }_{\xi }^2{\mathbb{F}}_0-a{\mathbb{F}}_0\right]\right]\right\}\hfill \\ \hfill & +\left\{\begin{array}{c}{E}^{-1}\left[s^{\sigma }E\left[{\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)}^2+a{\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)}^2-{\left({\mathbb{F}}_0+{\mathbb{F}}_1\right)}^3\right]\right]\\ -E^{-1}\left[s^{\sigma }E\left[{\left({\mathbb{F}}_0\right)}^2+a{\left({\mathbb{F}}_0\right)}^2-{\left({\mathbb{F}}_0\right)}^3\right]\right]\end{array}\right\}\hfill \\ \hfill & ={\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }}{8{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^6}}{\tau }^{2\sigma }\left\{\begin{array}{c}-{\displaystyle \frac{2\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right){\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^3}{\mathrm{\Gamma }(2\sigma +1)}}+\left(-1+2a\right)e^{\sqrt{2}\xi }{\displaystyle \frac{{\tau }^{2\sigma }\mathrm{\Gamma }(3\sigma +1)}{\mathrm{\Gamma }{(\sigma +1)}^3\mathrm{\Gamma }(4\sigma +1)}}\\ +2e^{{\displaystyle \frac{1}{\sqrt{2}}}\xi }\left(1+e^{{\displaystyle \frac{1}{\sqrt{2}}}\xi }\right)\left(1+a+\left(a-2\right)e^{{\displaystyle \frac{1}{\sqrt{2}}}\xi }\right){\displaystyle \frac{{\tau }^{\sigma }\mathrm{\Gamma }(2\sigma +1)}{\mathrm{\Gamma }{(\sigma +1)}^2\mathrm{\Gamma }(3\sigma +1)}}\end{array}\right\},\hfill \end{array}$$

  (178)

  and so on.

Step (4)

At the end, the solution in the convergence series form up to $2^{rd}$-order approximations reads

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+\cdots \hfill \\ \hfill & ={\displaystyle \frac{1}{1+e^{\frac{-1}{\sqrt{2}}\xi }}}+{\displaystyle \frac{\left(1-2a\right)e^{\frac{1}{\sqrt{2}}\xi }}{2{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^2}}{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}\hfill \\ \hfill & +{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }}{8{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^6}}{\tau }^{2\sigma }\left\{\begin{array}{c}-{\displaystyle \frac{2\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right){\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^3}{\mathrm{\Gamma }(2\sigma +1)}}+\left(-1+2a\right)e^{\sqrt{2}\xi }{\displaystyle \frac{{\tau }^{2\sigma }\mathrm{\Gamma }(3\sigma +1)}{\mathrm{\Gamma }{(\sigma +1)}^3\mathrm{\Gamma }(4\sigma +1)}}\\ +2e^{{\displaystyle \frac{1}{\sqrt{2}}}\xi }\left(1+e^{{\displaystyle \frac{1}{\sqrt{2}}}\xi }\right)\left(1+a+\left(a-2\right)e^{{\displaystyle \frac{1}{\sqrt{2}}}\xi }\right){\displaystyle \frac{{\tau }^{\sigma }\mathrm{\Gamma }(2\sigma +1)}{\mathrm{\Gamma }{(\sigma +1)}^2\mathrm{\Gamma }(3\sigma +1)}}\end{array}\right\}\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(179)

4.5.3. Analyzing Example (V) via the HPTM

To analyze Equation (162) using the HPTM, the following brief steps are considered:

Step (1)

Applying the ET to Equation (162) implies

$$E\left[\mathbb{F}\right]=s^2{\mathbb{F}}_0+s^{\sigma }E\left[{\partial }_{\xi }^2\mathbb{F}+\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right)\right].$$

(180)

Step (2)

By applying the inverse ET to Equation (180), we have

$$\mathbb{F}={\mathbb{F}}_0+E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2\mathbb{F}-a\mathbb{F}\right)+\left({\mathbb{F}}^2+a{\mathbb{F}}^2-{\mathbb{F}}^3\right)\right]\right].$$

(181)

Step (3)

The relation (33) according to Equation (181) can be written as follows:

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵE^{-1}\left[s^{\sigma }E\left[M\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{F}}_k\right)\right]\right],$$

(182)

which leads to

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k={\mathbb{F}}_0+ϵ\left(E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)-a\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{F}}_k\right)+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{F}}_k\right)\right]\right]\right).$$

(183)

Step (4)

By comparing the coefficients of $ϵ$ in Equation (183), we finally get the following first $2^{rd}$-order approximations:

• Zeroth-order approximation ${\mathbb{F}}_0$:

  $${\mathbb{F}}_0={\displaystyle \frac{1}{1+e^{\frac{-1}{\sqrt{2}}\xi }}}.$$

  (184)
• First-order approximation ${\mathbb{F}}_1$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_1& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_0\right)+H_0\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_0-a{\mathbb{F}}_0\right)+H_0\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & ={\displaystyle \frac{\left(1-2a\right)e^{\frac{1}{\sqrt{2}}\xi }}{2{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^2}}{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}.\hfill \end{array}$$

  (185)
• Second-order approximation ${\mathbb{F}}_2$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_2& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_1\right)+H_1\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_1-a{\mathbb{F}}_1\right)+H_1\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & =-{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^3}}{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}.\hfill \end{array}$$

  (186)
• Third-order approximation ${\mathbb{F}}_3$:

  $$\begin{array}{cc}\hfill {\mathbb{F}}_3& =E^{-1}\left[s^{\sigma }E(M\left({\mathbb{F}}_2\right)+H_2\left(\mathbb{F}\right))\right]\hfill \\ \hfill & =E^{-1}\left[s^{\sigma }E\left[\left({\partial }_{\xi }^2{\mathbb{F}}_2-a{\mathbb{F}}_2\right)+H_2\left(\mathbb{F}\right)\right]\right]\hfill \\ \hfill & =-{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)\left[1-6e^{\frac{1}{\sqrt{2}}\xi }-e^{\sqrt{2}\xi }+2a\left(-1+e^{\sqrt{2}\xi }\right)\right]}{8{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & +{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{e^{\sqrt{2}\xi }}\left(1+a-2e^{\frac{1}{\sqrt{2}}\xi }+a{e}^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{\mathrm{\Gamma }(2\sigma +1){\tau }^{3\sigma }}{\mathrm{\Gamma }{(\sigma +1)}^2\mathrm{\Gamma }(3\sigma +1)}},\hfill \end{array}$$

  (187)

  and so on.

Step (5)

Thus, the series convergent solution takes the following final form:

$$\begin{array}{cc}\hfill \mathbb{F}\left(\xi ,\tau \right)& ={\mathbb{F}}_0+{\mathbb{F}}_1+{\mathbb{F}}_2+{\mathbb{F}}_3+\cdots \hfill \\ \hfill & ={\displaystyle \frac{1}{1+e^{\frac{-1}{\sqrt{2}}\xi }}}+{\displaystyle \frac{\left(1-2a\right)e^{\frac{1}{\sqrt{2}}\xi }}{2{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^2}}{\displaystyle \frac{{\tau }^{\sigma }}{\mathrm{\Gamma }(\sigma +1)}}-{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^3}}{\displaystyle \frac{{\tau }^{2\sigma }}{\mathrm{\Gamma }(2\sigma +1)}}\hfill \\ \hfill & -{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{\frac{1}{\sqrt{2}}\xi }\left(-1+e^{\frac{1}{\sqrt{2}}\xi }\right)\left[1-6e^{\frac{1}{\sqrt{2}}\xi }-e^{\sqrt{2}\xi }+2a\left(-1+e^{\sqrt{2}\xi }\right)\right]}{8{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{{\tau }^{3\sigma }}{\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & +{\displaystyle \frac{{\left(1-2a\right)}^2{e}^{e^{\sqrt{2}\xi }}\left(1+a-2e^{\frac{1}{\sqrt{2}}\xi }+a{e}^{\frac{1}{\sqrt{2}}\xi }\right)}{4{\left(1+e^{\frac{1}{\sqrt{2}}\xi }\right)}^5}}{\displaystyle \frac{\mathrm{\Gamma }(2\sigma +1){\tau }^{3\sigma }}{\mathrm{\Gamma }{(\sigma +1)}^2\mathrm{\Gamma }(3\sigma +1)}}\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(188)

In this example, it is clear that the derived approximations (173) and (188) using the Tantawy technique and the HPTM are completely identical. However, these approximations do not align with the approximation (179) derived by the NITM. The impact of fractional-order parameter $\sigma$ on the shock wave profile is examined according to the approximation (173) as illustrated in Figure 9. Moreover, the comparison between the generated approximations (173) and (179) is presented in Figure 10. Figure 10 demonstrates the perfect harmony between the exact solution (164) and the derived approximations (173) and (179) at $\sigma =1$, which confirms the high accuracy of the generated approximations. Furthermore, the absolute error for the derived approximations (173) and (179) is estimated as shown in

Table 9. The absolute error $E_n$ to the approximations (173) and (179) is estimated at $\sigma =1\&\tau =0.1$.

$\mathit{\xi }$	$\mathbf{Exact}\mathbf{Sol}.$ (164)	${\mathbf{NITM}}_{\mathit{\sigma }=1}$ (179)	2nd ${\mathbf{Tantawy}}_{\mathit{\sigma }=1}$ (173)	${\mathit{E}}_{\mathit{n}}$ to (179)	${\mathit{E}}_{\mathit{n}}$ to (173)
−7	0.00717645	0.00717645	0.00717645	0.813417 ×${10}^{-8}$	0.896627 ×${10}^{-8}$
−6	0.0144481	0.0144481	0.0144481	1.38663 ×${10}^{-8}$	1.71358 ×${10}^{-8}$
−5	0.0288735	0.0288734	0.0288734	1.84853 ×${10}^{-8}$	3.07471 ×${10}^{-8}$
−4	0.0568705	0.0568705	0.0568704	0.625998 ×${10}^{-8}$	4.81592 ×${10}^{-8}$
−3	0.108969	0.108969	0.108969	6.47895 ×${10}^{-8}$	5.42794 ×${10}^{-8}$
−2	0.198736	0.198736	0.198736	22.342 ×${10}^{-8}$	1.11902 ×${10}^{-8}$
−1	0.334677	0.334677	0.334677	29.6863 ×${10}^{-8}$	9.7287 ×${10}^{-8}$
0	0.505	0.505	Indeterminate	0.313167 ×${10}^{-8}$	Indeterminate
1	0.67417	0.67417	0.67417	36.9186 ×${10}^{-8}$	9.56308 ×${10}^{-8}$
2	0.807557	0.807557	0.807557	38.0494 ×${10}^{-8}$	1.23241 ×${10}^{-8}$
3	0.894855	0.894855	0.894855	22.2572 ×${10}^{-8}$	5.44266 ×${10}^{-8}$
4	0.945237	0.945237	0.945237	10.4696 ×${10}^{-8}$	4.79297 ×${10}^{-8}$
5	0.972227	0.972227	0.972227	4.68234 ×${10}^{-8}$	3.05153 ×${10}^{-8}$
6	0.986111	0.986111	0.986111	2.12959 ×${10}^{-8}$	1.69851 ×${10}^{-8}$
7	0.993103	0.993103	0.993103	0.997465 ×${10}^{-8}$	0.888214 ×${10}^{-8}$

. It is crucial to recognize that both the Tantawy technique and HPTM provide straightforward approximations up to the third order and beyond with low computational costs. In contrast, NITM requires a high computational cost to generate higher-order approximations. Thus, in this investigation, we derived the second-order approximation using the NITM, whereas the third-order approximations are derived using both the Tantawy technique and the HPTM. The numerical findings indicate the high accuracy of the generated approximations and the efficiency of the proposed approaches by calculating the absolute errors $E_n=\left|\mathrm{Exact}\mathrm{Sol}.\left(164\right)-\mathrm{Approx}.\left(173\right)\right|$ and $E_n=\left|\mathrm{Exact}\mathrm{Sol}.\left(164\right)-\mathrm{Approx}.\left(179\right)\right|$, as presented in Table 5. Moreover, the absolute error $E_n=\left|\mathrm{Exact}\mathrm{Sol}.\left(164\right)-\mathrm{Approx}.\left(173\right)\right|$ for the $2^{nd}$- and 3^rd-order approximations is estimated as shown in

Table 10. The absolute error $E_n$ for the different order approximations to the generated approximation (173) is compared to the absolute error $E_n$ of $2^{nd}$-order approximation (179) at $\sigma =1\&\tau =0.1$.

$\mathit{\xi }$	${\mathit{E}}_{\mathit{n}}$: $2^{\mathit{nd}}$-Order (179)	${\mathit{E}}_{\mathit{n}}$: $2^{\mathit{nd}}$-Order (173)	${\mathit{E}}_{\mathit{n}}$: $3^{\mathit{rd}}$-Order (173)
−7	0.813417 ×${10}^{-8}$	0.896627 ×${10}^{-8}$	4.22165 ×${10}^{-11}$
−6	1.38663 ×${10}^{-8}$	1.71358 ×${10}^{-8}$	7.55295 ×${10}^{-11}$
−5	1.84853 ×${10}^{-8}$	3.07471 ×${10}^{-8}$	11.6105 ×${10}^{-11}$
−4	0.625998 ×${10}^{-8}$	4.81592 ×${10}^{-8}$	11.4394 ×${10}^{-11}$
−3	6.47895 ×${10}^{-8}$	5.42794 ×${10}^{-8}$	7.55969 ×${10}^{-11}$
−2	22.342 ×${10}^{-8}$	1.11902 ×${10}^{-8}$	57.0119 ×${10}^{-11}$
−1	29.6863 ×${10}^{-8}$	9.7287 ×${10}^{-8}$	82.6723 ×${10}^{-11}$
0	0.313167 ×${10}^{-8}$	Indeterminate	0.666643 ×${10}^{-11}$
1	36.9186 ×${10}^{-8}$	9.56308 ×${10}^{-8}$	82.9496 ×${10}^{-11}$
2	38.0494 ×${10}^{-8}$	1.23241 ×${10}^{-8}$	56.383 ×${10}^{-11}$
3	22.2572 ×${10}^{-8}$	5.44266 ×${10}^{-8}$	7.16657 ×${10}^{-11}$
4	10.4696 ×${10}^{-8}$	4.79297 ×${10}^{-8}$	11.5089 ×${10}^{-11}$
5	4.68234 ×${10}^{-8}$	3.05153 ×${10}^{-8}$	11.5715 ×${10}^{-11}$
6	2.12959 ×${10}^{-8}$	1.69851 ×${10}^{-8}$	7.50794 ×${10}^{-11}$
7	0.997465 ×${10}^{-8}$	0.888214 ×${10}^{-8}$	4.19199 ×${10}^{-11}$

. Furthermore, the maximum residual error for all derived approximations across the whole study domain $\mathsf{\Omega }$ is computed using the following relationship:

$$\begin{array}{cc}\hfill {\mathbb{R}}_{2^{nd}—\mathrm{NITM}\left(179\right)}& =\underset{\mathsf{\Omega }}{max}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}-\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right)\right|=0.00174796\mathrm{at}\left(\xi ,\tau \right)=\left(1.30491,1.\right),\hfill \\ \hfill {\mathbb{R}}_{2^{nd}—\mathrm{Tan}\mathrm{tawy}\left(173\right)}& =\underset{\mathsf{\Omega }}{max}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}-\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right)\right|=0.000626514\mathrm{at}\left(\xi ,\tau \right)=\left(-0.0622468,1.\right),\hfill \\ \hfill {\mathbb{R}}_{3^{rd}—\mathrm{Tan}\mathrm{tawy}\left(173\right)}& =\underset{\mathsf{\Omega }}{max}\left|D_{\tau }^{\sigma }\mathbb{F}-{\partial }_{\xi }^2\mathbb{F}-\mathbb{F}\left(1-\mathbb{F}\right)\left(\mathbb{F}-a\right)\right|=0.0000464475\mathrm{at}\left(\xi ,\tau \right)=\left(1.02815,1.\right),\hfill \end{array}$$

where $\mathsf{\Omega }\equiv \left(\xi ,\tau \right)\in \left[-7,7\right]\times \left[0,1\right]$ and $\left(\sigma ,a\right)=\left(1,0.3\right)$.

The obtained results indicate that increasing the order of derived approximations can result in more accurate and stable approximations, which enhances the proposed methods’ efficiency and confirms their convergence approximations.

5. Conclusions

In this investigation, three distinct and highly accurate approaches have been employed, namely, the Tantawy technique, the new iterative transform method (NITM), and the homotopy perturbation transform method (HPTM), to study and analyze multidimensional fractional heat flow problems and derive highly accurate and more stable analytical approximations for the suggested problems. The Tantawy technique is a versatile approach that effectively resolves any fractional partial differential equations (FPDEs) without issues researchers can face during implementation. This novel technique is more effective than comparable methods as it is devoid of difficulties, irrespective of the problem’s complexity. This technique assumes the solution to the FPDE in the form of a convergent series, and by calculating the coefficients of this series, we can generate highly accurate approximations without any challenges that researchers may face. One of the essential features of the Tantawy technique is that it does not require any other means to facilitate the calculations, unlike its counterparts from different methods, which may require applying some transforms to simplify the calculations, such as the NITM and the HPTM. In hybrid approaches (NITM and HPTM), the Elzaki transform (ET) has been utilized in the analytical process to solve and analyze fractional differential equations, as directly solving equations with fractional order is complex. During our numerical analysis, we analyzed five test problems using the proposed methods to demonstrate the usefulness and efficiency of these suggested approaches. The application of the three proposed methods has generated numerous analytical approximations for the four suggested problems, and it was found that all derived approximations for each problem are completely identical. This may be explained by the fact that the first three problems are linear fractional differential equations (FDEs); thus, if the FDE is linear, all derived approximations via the three offered methods will be completely identical. Nevertheless, for the fourth problem, despite being a nonlinear FDE, all generated approximations via the three proposed methods are completely identical. This may be ascribed to the initial condition that is not affected by differential operators. In the fifth example, which represents a strong nonlinear fractional differential equation, the derived analytical approximations using the Tantawy technique and the HPTM are completely identical, but these approximations differ from the approximation derived by the NITM. Also, we conducted a graphical and numerical analysis of the derived approximations for the fifth proposed problems to comprehend the dynamics of the phenomena represented by these approximations and the impact of fractionality on their behavior. Moreover, we estimated both the absolute and residual errors for all generated approximations to assess their accuracy and stability and validate the efficacy of the proposed techniques. It was found that the generated approximations exhibit good accuracy and stability, with potential for additional accuracy enhancement through an increase in the order of the approximations, which, in turn, enhances the efficiency of the proposed methods.

Future Work

The techniques employed in this study, especially the Tantawy technique, are straightforward and applicable to analyzing a diverse range of physical and engineering challenges. This technique enables the generation of very accurate approximations with low computational cost without the need for decomposition, perturbation, linearization, or the use of transforms to aid calculations, unlike previous methods. This technique is anticipated to be widely utilized by researchers due to its efficacy and simplicity in application to all fractional partial differential equations. This approach can be utilized to investigate nonlinear phenomena that arise in various plasma models, specifically by analyzing fractional damped Kawahara-type equations [75,76,77] and fractional nonplanar Kawahara-type equations [78,79] to enhance comprehension of the characteristics and dynamics of fractional solitary waves and cnoidal waves described by this family. This technique can be readily used to study fractional nonlinear Schrödinger-type equations [80,81,82] to investigate the characteristics of rogue waves and breathers, as well as to understand their propagation dynamics.