Comparative Study of the Nonlinear Fractional Generalized Burger-Fisher Equations Using the Homotopy Perturbation Transform Method and New Iterative Transform Method

Abstract

The time-fractional generalized Burger–Fisher equation (TF-GBFE) is utilized in many physical applications and applied sciences, including nonlinear phenomena in plasma physics, gas dynamics, ocean engineering, fluid mechanics, and the simulation of financial mathematics. This mathematical expression explains the idea of dissipation and shows how advection and reaction systems can work together. We compare the homotopy perturbation transform method and the new iterative method in the current study. The suggested approaches are evaluated on nonlinear TF-GBFE. Two-dimensional (2D) and three-dimensional (3D) figures are displayed to show the dynamics and physical properties of some of the derived solutions. A comparison was made between the approximate and accurate solutions of the TF-GBFE. Simple tables are also given to compare the integer-order and fractional-order findings. It has been verified that the solution generated by the techniques given converges to the precise solution at an appropriate rate. In terms of absolute errors, the results obtained have been compared with those of alternative methods, including the Haar wavelet, OHAM, and q-HATM. The fundamental benefit of the offered approaches is the minimal amount of calculations required. In this research, we focus on managing the recurrence relation that yields the series solutions after a limited number of repetitions. The comparison table shows how well the methods work for different fractional orders, with results getting closer to precision as the fractional-order numbers get closer to integer values. The accuracy of the suggested techniques is greatly increased by obtaining numerical results in the form of a fast-convergent series. Maple is used to derive the approximate series solution’s behavior, which is graphically displayed for a number of fractional orders. The computational stability and versatility of the suggested approaches for examining a variety of phenomena in a broad range of physical science and engineering fields are highlighted in this work.

1. Introduction

A unique area of applied analysis, fractional calculus, deals with real or complex derivatives of any order. Therefore, we may generalise the formula from positive integer to any real-order differentiation because fractional-order calculus provides us with an understanding of differentiation and course integration. FC was founded by two mathematicians, Leibniz and L’Hospital, and its precise birthdate is thought to have been 30 September 1695. The extension of classical calculus known as fractional calculus is gaining popularity in various fields, including fluid mechanics, control theory, and signal processing. Although fractional calculus, particularly fractional differential equations, is a topic that is frequently addressed nowadays. In fact, the famous mathematician Abel first introduced the concept in 1823. Scientists from several fields have worked on fractional calculus for the past 200 years. Numerous definitions exist for fractional derivatives, but not all are frequently applied. The most commonly used operators are the Riemann-Liouville (R-L), Caputo-Fabrizio, Caputo, and conformable operators [1,2,3,4,5]. The R-L and Caputo fractional derivatives have singular kernels. The index law and other classical conditions were satisfied by this class of fractional differential operators. This singularity prevents the description of the entire physical structure of memory. The ability to simulate and analyse complex systems with intricate nonlinear processes and higher-order behaviours makes fractional derivatives better for modelling in some situations than integer-order derivatives. This has two primary reasons. First, the derivative operator does not have to be of an integer order; we can use any order. Second, non-integer-order derivatives are useful when the system has long-term memory, depending on the present and past situation. Non-classical differential equations are a generalisation of conventional differential equations with fractional derivatives. Fractional differential equations have been widely employed for their relaxation and oscillation models in physics, biology, engineering, chemistry, and other domains [6,7,8,9,10].

The study of fractional-order chaotic systems has gained popularity in recent years as chaotic theory research has advanced and gotten more refined. Specifically, the fractional-order chaotic systems’ complexity is correlated with both the system’s fractional order and parametric characteristics. The four-wing fractional chaotic system [11], fractional Lorenz hyperchaotic system [12], fractional Lorenz system [13], and other non-integer-order chaotic systems have been proposed by numerous scholars based on integer-order chaotic systems [14,15]. Many researchers have made some progress in the numerical finding of fractional chaotic systems, specifically in the discretisation of fractional chaotic systems, using the Adomian decomposition method (ADM) [11,12,13], the Adams-Bashforth-Moulton (ABM) algorithm [16,17], and the frequency-domain method (FDM) [18]. Using Laplace change, FDM employs a high-dimensional system that is nearly fractional-order, although the error is comparatively substantial [19]. Although ABM is the most widely employed technique, it operates slowly. In contrast, the ADM method is more accurate and uses less computing power than the ABM algorithm [11,12,13]. According to reference [20], complexity, the Lyapunov index, the bifurcation diagram, and other factors demonstrate that fractional-order chaotic systems exhibit more complicated chaotic behaviours than integer-order chaotic systems. The complexity of the system increases with decreasing system order [21].

The growing field of artificial intelligence provided new opportunities for estimating the parameters of dynamic systems. There have been attempts to include deep learning in dynamic system modelling, which uses neural networks to approximate unknown system dynamics, because of its powerful function approximation capabilities and end-to-end learning paradigm. ResNet is a highly important work that was proposed recently and has transformed deep learning with its deep residual learning, which allows networks to be trained deeper than previously utilised networks using shortcut connections [22,23,24]. Yan et al. [25] investigated how resilient neural ODEs were to input disturbances and offered strategies to improve their stability and dependability. Poli et al. [26] show how combining neural ODE with graph neural networks may efficiently simulate dynamic processes on networks, expanding the use of neural ODE to graph-structured data. Zhu et al. [27] worked on the neural ODE’s numerical integration feature. Their study focused on a crucial element influencing the effectiveness and performance of neural ODEs in a range of applications. Recently, Yong Yang et al. [28] proposed a novel approach to parameter estimation that combines physical prior knowledge with neural ordinary differential equations (neural ODEs). This method can achieve robust and accurate parameter estimation under high noise and small sample conditions, particularly for complex nonlinear dynamic systems. These papers demonstrate the advancements made and the areas that require further research to reach the full potential of neural ODEs in dynamic system modelling.

Our everyday lives are filled with randomness and uncertainty [29], such as estimating the number of points before a dice roll, forecasting the weather, and projecting a company’s stock price. These occurrences show how unpredictable and varied many facets of life are by nature. In recognition of this, scholars have long endeavoured to comprehend and measure uncertainty in a methodical manner. When only discrete observations are available, estimating unknown parameters is a critical problem in uncertain differential equations (UDE). Yao and Liu [30] first presented the method of moments, which was based on the Euler method. This approach was further expanded to parameter estimation for unknown delay differential equations by Liu and Jia [31]. A novel approach to estimating unknown parameters by creating a minimization optimization problem was presented by Wu [32]. In comparison to conventional UDE methods, this approach offers more parameter degrees of freedom, can better fit real data, and increases forecast accuracy. Jing Ning et al. [33] employ prediction-correction techniques to solve fractional UDEs and introduce rectangular and trapezoidal algorithms to numerically approximate the optimization problem in order to broaden its research on parameter estimation accuracy.

Fractional partial differential equations (FPDEs) have received a lot of attention in recent decades because of their rapidly expanding and wide-ranging applications in various scientific and technical fields, such as biology, chemistry, electrical engineering, medicine, and viscoelasticity. Further information on these and other uses can be found in earlier research [34,35,36,37]. In this perspective, including FPDE approximations and approximate solutions is crucial for perfectly describing the dynamics of basic physical processes. Given the previously discussed facts, mathematicians have created and applied a wide variety of approximate and analytical methods to solve several significant mathematical models related to real-world issues. Despite the difficulty of finding analytical or even close solutions to some nonlinear FPDEs and systems of FPDEs, mathematicians continue to work in this field [38,39,40,41]. Several approaches have been utilised for addressing FPDEs, including the optimal homotopy analysis method [42], the general residual power series method [43], the homotopy analysis method [44], the Galerkin finite element method [45], and the finite difference method [46].

Due to their ability to simulate tidal oscillations brought on by undersea landslides and tsunamis, nonlinear diffusion and convection equations are crucial to oceanography and fluid dynamics. Examples of nonlinear dispersive waves and travelling waves that frequently occur in oceanography, marine engineering, acoustics, and fluid dynamics include gravitational waves, surface water disturbances in shallow rivers and seas, ship bumps on water, and tsunami dispersion [47,48,49]. Burgers-type equations for various wave propagation processes have drawn a lot of interest in these fields. These models depict interface dynamics, non-equilibrium, and nonlinear turbulence in hydrodynamics and ocean sciences. This study aimed to better understand the mechanism dictated by nonlinear FPDEs by investigating TFGBFE. The response, dissipation mechanisms, and advection are combined in the nonlinear equation called the TF-GBFE. The Burgers and Fisher diffusion transfer qualities and reaction form properties are used in this nonlinear equation.

The TF-GBFE is an interesting fluid dynamic model that several researchers have studied to help examine various numerical techniques and the mathematical analysis of physical flows. Due to the inclusion of diffusion mechanisms, convection, and reaction, the TF-GBFE is highly nonlinear. It is called Burger–Fisher because it combines the diffusion and reaction qualities of Fisher’s equation with the convective and diffusion properties of Burger’s equation. The TF-GBFE is an important nonlinear diffusion equation in ocean engineering since it depicts the distant field of wave propagation in the ocean. Strong turbulent diffusion leads to the travelling waves due to convection, turbulent diffusion, and nonlinear radiation’s impact on the irregularity of sea surface temperature. The currents created by winds determine the travelling wavefronts’ speed and direction of movement [50]. According to [51], the TF-GBFE is responsible for the convection-diffusion model that may replicate underwater landslides, which could lead to the most dangerous tsunamis in the coastal area.

The TF-GBFE can be expressed as follows using the fractional order $\eta$ and any real constants $a,b$, and $\delta$:

$$\begin{array}{cc}\hfill & D_{\wp }^{\eta }\mathbb{P}(\delta ,\wp )-{\mathbb{P}}_{\delta \delta }(\delta ,\wp )+a{\mathbb{P}}^{\delta }(\delta ,\wp ){\mathbb{P}}_{\delta }(\delta ,\wp )+b\mathbb{P}(\delta ,\wp )({\mathbb{P}}^{\delta }(\delta ,\wp )-1)=0,0<\eta \le 1,\hfill \\ \hfill & 0\le \delta \le 1,\wp \ge 0\hfill \end{array}$$

(1)

here a, b, $\delta$ are positive parameters, having initial guess as:

$$\mathbb{P}(\delta ,0)={\left[{\displaystyle \frac{1}{2}}\left(1-tanh\left({\displaystyle \frac{a\delta }{2(1+\delta )}}\delta \right)\right)\right]}^{{\displaystyle \frac{1}{\delta }}}.$$

(2)

having an exact solution:

$$\mathbb{P}(\delta ,\wp )={\left[{\displaystyle \frac{1}{2}}\left(1-tanh\left({\displaystyle \frac{a\delta }{2(1+\delta )}}\left(\delta -\left({\displaystyle \frac{a^2+b{(1+\delta )}^2}{a(1+\delta )}}\right)\wp \right)\right)\right)\right]}^{{\displaystyle \frac{1}{\delta }}}.$$

(3)

Differential equations are usually solved with simple mathematical techniques such as the Fourier, Laplace, Sumudu, and Elzaki transforms. Khuri [52] uses the Laplace transform (LT) in combination with the Adomian decomposition approach to get the approximate solution of a class of ordinary differential equations that are nonlinear. Many scholars have used a variety of techniques in connection with the LT approach to solve partial differential equations in recent years. These include the homotopy perturbation transform method [53], laplace variational iteration strategy [54], the Laplace homotopy analysis method [55], and many more. The new iterative transform method (NITM) and the homotopy perturbation transform method (HPTM) are extended in this study to solve TF-GBFE. Guo created non-abelian extensions of Rota-Baxter algebras, which serve as the foundation for other decomposition techniques, such as the Adomian polynomials employed in NITM [56]. The Daftardar–Jafari and Adomian polynomials are combined in the NITM [57], which is a modified version of the Elzaki transform decomposition method. The offered method merges the New Iterative Method (NIM) and Elzaki transformation. Tarig ELzaki presented the Elzaki Transform (ET), a novel integral transform, in 2010. ET is a modified version of the Laplace and Sumudu transforms. It is important to note that some differential equations with variable coefficients may be difficult to solve using the Sumudu and Laplace transforms, but they can be quickly resolved with ET’s help. He introduced HPM in 1998 [58,59]. His polynomials are used to decompose the nonlinear terms after the differential equations are transformed into algebraic equations with the aid of the ET. The result of this method is taken to be in series form, which quickly converges to the precise answer after a small number of terms. This method can be used to solve nonlinear PDEs effectively. A higher level of accuracy was verified when the HPTM results were compared with the real solution to the issues. The methods offered accurate solutions to challenging issues, resulting in remarkable outcomes. The fractional problem findings acquired by applying the given approaches are also utilised to assess the issues from a fractional perspective. In summary, the original FPDE is transformed into its corresponding PDE by approximating the fractional Caputo derivative using the Elzaki transform method. The original FPDE may be solved quickly and easily by using the new iterative method and the homotopy perturbation method on the derived PDE. The current study is essential because it finds an approximate, fractional-order solution to the TF-GBFE equations by employing two relatively new and innovative methods. It also compares the accurate solution of the proposed models to fourth-order approximations for a range of values of the fractional derivative. The presentation of two novel strategies for TF-GBFE with minimal and progressive phases makes this work interesting. The outline of our work is given below: Important definitions of FC are given in Section 2. The main notion of NITM is presented in Section 3, and the basic concept of the HPTM approach is presented in Section 4. The convergence and uniqueness results were covered in Section 5. In Section 6, the suggested techniques with the validity of the error bound theorem are provided. Section 7 presents a numerical problem to demonstrate the significance of the approaches mentioned. A brief discussion of the core outcomes and conclusions is provided in Section 8.

2. Basic Definitions

Here, we give some important definitions related to the current study.

Definition 1. 

The Abel–Riemann non-integer derivative is as [60]

$$D^{\eta }\nu (\wp )=\left\{\begin{array}{c}{\displaystyle \frac{d^{\varpi }}{d{\wp }^{\varpi }}}\nu (\wp ),\eta =\varpi ,\hfill \\ {\displaystyle \frac{1}{\Gamma (\varpi -\eta )}}{\displaystyle \frac{d}{d{\wp }^{\varpi }}}{\int }_0^{\wp }{\displaystyle \frac{\nu (\wp )}{{(\wp -\delta )}^{\eta -\varpi +1}}}d\delta ,\varpi -1<\eta <\varpi ,\hfill \end{array}\right.$$

(4)

with $\varpi \in Z^{+}$, $\eta \in R^{+}$ and

$$D^{-\eta }\nu (\wp )={\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^{\wp }{(\wp -\delta )}^{\eta -1}\nu \left(\delta \right)d\delta ,0<\eta \le 1.$$

(5)

Definition 2. 

The Abel-Riemann non-integer integration operator is as [60]

$$J^{\eta }\nu (\wp )={\displaystyle \frac{1}{\Gamma \left(\eta \right)}}{\int }_0^{\wp }{(\wp -\delta )}^{\eta -1}\nu (\wp )d\wp ,\wp >0,\eta >0,$$

(6)

with given properties:

$$\begin{array}{c}\hfill J^{\eta }{\wp }^{\varpi }={\displaystyle \frac{\Gamma (\varpi +1)}{\Gamma (\varpi +\eta +1)}}{\wp }^{\varpi +\delta },\\ \hfill D^{\eta }{\wp }^{\varpi }={\displaystyle \frac{\Gamma (\varpi +1)}{\Gamma (\varpi -\eta +1)}}{\wp }^{\varpi -\wp }.\end{array}$$

Definition 3. 

The Caputo non-integer derivative is as [61]

$${}^{C}D^{\eta }\nu (\wp )=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma (\varpi -\eta )}}{\int }_0^{\wp }{\displaystyle \frac{{\nu }^{\varpi }\left(\delta \right)}{{(\wp -\delta )}^{\eta -\varpi +1}}}d\delta ,\varpi -1<\eta <\varpi ,\hfill \\ {\displaystyle \frac{d^{\varpi }}{d{\wp }^{\varpi }}}\nu (\wp ),\varpi =\eta ,\hfill \end{array}\right.$$

(7)

with given properties

• ${\varpi }_{\wp }^{\eta }{D}_{\wp }^{\eta }g(\wp )=g(\wp )-{\sum }_{k=0}^m{g}^k\left(0^{+}\right){\displaystyle \frac{{\wp }^k}{k!}},for\wp >0,\mathit{and}\varpi -1<\eta \le \varpi ,\varpi \in N.$
• $D_{\wp }^{\eta }{\varpi }_{\wp }^{\eta }g(\wp )=g(\wp ).$

Lemma 1. 

For $n<\eta \le 1$, $\wp \ge 0$ and $k\in R$, we have

$$\begin{array}{cc}\hfill & \left(a\right)D_{\wp }^{\eta }{\wp }^k={\displaystyle \frac{\Gamma (k+1)}{\Gamma (k+1-\eta )}}{\wp }^{k-\eta }.\hfill \\ \hfill & \left(b\right)D_{\wp }^{\eta }{I}_{\wp }^{\eta }\left[\mathbb{P}(\delta ,\wp )\right]=\mathbb{P}(\delta ,\wp ).\hfill \\ \hfill & \left(c\right)I_{\wp }^{\eta }{\wp }^k={\displaystyle \frac{\Gamma (k+1)}{\Gamma (k+1+\eta )}}{\wp }^{k+\eta }.\hfill \\ \hfill & \left(d\right)I_{\wp }^{\eta }{D}_{\wp }^{\eta }\left[\mathbb{P}(\delta ,\wp )\right]=\mathbb{P}(\delta ,\wp )-\sum _{i=0}^{n-1}{\partial }^i\mathbb{P}(\delta ,0){\displaystyle \frac{{\wp }^i}{i!}}.\hfill \end{array}$$

Definition 4. 

The ET of a function is as [62]

$$E\left[g(\wp )\right]=G\left(r\right)=r{\int }_0^{\infty }h(\wp )e^{{\displaystyle \frac{-\wp }{r}}}d\wp ,r>0.$$

(8)

Definition 5. 

The ET of Caputo operator is stated as [62]

$$\mathtt{E}[D_{\wp }^{\eta }g(\wp )]=s^{-\eta }\mathtt{E}[g(\wp )]-\sum _{k=0}^{\varpi -1}{s}^{2-\eta +k}{g}^{\left(k\right)}(0),where\varpi -1<\eta <\varpi .$$

3. General Procedure of NITM

Here, we illustrated the general analysis of the offered approach as given.

$$D_{\wp }^{\eta }\mathbb{P}(\delta ,\wp )+N\mathbb{P}(\delta ,\wp )+M\mathbb{P}(\delta ,\wp )=h(\delta ,\wp ),\wp >0,1<\eta \le 0,$$

(9)

with

$${\mathbb{P}}^k(\delta ,0)=f\left(\delta \right),$$

(10)

with $N,M$ indicates the linear and nonlinear terms.

By executing the ET to Equation (9), we may have

$$E\left[D_{\wp }^{\eta }\mathbb{P}(\delta ,\wp )\right]+E[N\mathbb{P}(\delta ,\wp )+M\mathbb{P}(\delta ,\wp )]=E\left[h(\delta ,\wp )\right].$$

(11)

In terms of differentiation property

$$E\left[\mathbb{P}(\delta ,\wp )\right]=\sum _{k=0}^m{s}^{2-\eta +k}{\mathbb{P}}^{\left(k\right)}(\delta ,0)+s^{\eta }E\left[h(\delta ,\wp )\right]-s^{\eta }E[N\mathbb{P}(\delta ,\wp )+M\mathbb{P}(\delta ,\wp )].$$

(12)

Now by executing the inverse ET to Equation (12),

$$\mathbb{P}(\delta ,\wp )=E^{-1}\left[\right\{\sum _{k=0}^m{s}^{2-\eta +k}{\mathbb{P}}^k(\delta ,0)+s^{\eta }E\left[h(\delta ,\wp )\right]\left\}\right]-E^{-1}\left[s^{\eta }E\right[N\mathbb{P}(\delta ,\wp )+M\mathbb{P}(\delta ,\wp )\left]\right].$$

(13)

By iterative process, we may have

$$\mathbb{P}(\delta ,\wp )=\sum _{m=0}^{\infty }{\mathbb{P}}_m(\delta ,\wp ),$$

(14)

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

(15)

the nonlinear term N is decomposed as

$$N\left(\sum _{m=0}^{\infty }{\mathbb{P}}_m(\delta ,\wp )\right)={\mathbb{P}}_0(\delta ,\wp )+N\left(\sum _{k=0}^m{\mathbb{P}}_k(\delta ,\wp )\right)-M\left(\sum _{k=0}^m{\mathbb{P}}_k(\delta ,\wp )\right).$$

(16)

On utilizing Equations (14)–(16) into Equation (13), we may have

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\infty }{\mathbb{P}}_m(\delta ,\wp )=E^{-1}\left[s^{\eta }\left(\sum _{k=0}^m{s}^{2-\delta +k}{\mathbb{P}}^k(\delta ,0)+E\left[h(\delta ,\wp )\right]\right)\right]\hfill \\ \hfill & -E^{-1}\left[s^{\eta }E\left[N\left(\sum _{k=0}^m{\mathbb{P}}_k(\delta ,\wp )\right)-M\left(\sum _{k=0}^m{\mathbb{P}}_k(\delta ,\wp )\right)\right]\right].\hfill \end{array}$$

(17)

By using iterative formula, we may have

$${\mathbb{P}}_0(\delta ,\wp )=E^{-1}\left[s^{\eta }\left(\sum _{k=0}^m{s}^{2-\delta +k}{\mathbb{P}}^k(\delta ,0)+s^{\eta }E\left(g(\delta ,\wp )\right)\right)\right],$$

(18)

$${\mathbb{P}}_1(\delta ,\wp )=-E^{-1}\left[s^{\eta }E[N\left[{\mathbb{P}}_0(\delta ,\wp )\right]+M\left[{\mathbb{P}}_0(\delta ,\wp )\right]\right],$$

(19)

$$\begin{array}{cc}\hfill & {\mathbb{P}}_{m+1}(\delta ,\wp )=-E^{-1}\left[s^{\eta }E\left[-N\left(\sum _{k=0}^m{\mathbb{P}}_k(\delta ,\wp )\right)-M\left(\sum _{k=0}^m{\mathbb{P}}_k(\delta ,\wp )\right)\right]\right],m\ge 1.\hfill \end{array}$$

(20)

Lastly, the approximate solution to Equation (9) is taken as

$$\mathbb{P}(\delta ,\wp )\cong {\mathbb{P}}_0(\delta ,\wp )+{\mathbb{P}}_1(\delta ,\wp )+{\mathbb{P}}_2(\delta ,\wp )+\cdots ,m=1,2,\cdots .$$

(21)

4. General Procedure of HPTM

Here, we illustrated the general analysis of the offered approach as given.

$$\begin{array}{cc}\hfill & D_{\wp }^{\eta }\mathbb{P}(\delta ,\wp )+M\mathbb{P}(\delta ,\wp )+N\mathbb{P}(\delta ,\wp )=h(\delta ,\wp ),\wp >0,0<\eta \le 1,\hfill \\ \hfill & \mathbb{P}(\delta ,0)=f\left(\delta \right).\hfill \end{array}$$

(22)

By executing the ET to Equation (22), we may have

$$\begin{array}{cc}\hfill & E[D_{\wp }^{\eta }\mathbb{P}(\delta ,\wp )+M\mathbb{P}(\delta ,\wp )+N\mathbb{P}(\delta ,\wp )]=E\left[h(\delta ,\wp )\right],\wp >0,0<\eta \le 1,\hfill \\ \hfill & \mathbb{P}(\delta ,\wp )=s^{2}g\left(\delta \right)+s^{\eta }E\left[h(\delta ,\wp )\right]-s^{\eta }E[M\mathbb{P}(\delta ,\wp )+N\mathbb{P}(\delta ,\wp )].\hfill \end{array}$$

(23)

Now by employing the inverse ET, we may have

$$\mathbb{P}(\delta ,\wp )=F(x,\wp )-E^{-1}\left[s^{\eta }E\{M\mathbb{P}(\delta ,\wp )+N\mathbb{P}(\delta ,\wp )\}\right],$$

(24)

where

$$F(\delta ,\wp )=E^{-1}\left[s^{2}g\left(\delta \right)+s^{\eta }E\left[h(\delta ,\wp )\right]\right]=g\left(\nu \right)+E^{-1}\left[s^{\eta }E\left[h(\delta ,\wp )\right]\right].$$

(25)

In terms of the HPM

$$\mathbb{P}(\delta ,\wp )=\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{P}}_k(\delta ,\wp ),$$

(26)

with $ϵ\in [0,1]$ indicates the perturbation parameter.

The nonlinear terms are decomposed as

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

(27)

with $H_n$ indicates He’s polynomials ${\mathbb{P}}_0,{\mathbb{P}}_1,{\mathbb{P}}_2,\dots ,{\mathbb{P}}_n,$ and is demonstrated as

$$H_n({\mathbb{P}}_0,{\mathbb{P}}_1,\dots ,{\mathbb{P}}_n)={\displaystyle \frac{1}{\eta (n+1)}}{D}_{ϵ}^k{\left[N\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{P}}_k\right)\right]}_{ϵ=0},$$

(28)

where $D_{ϵ}^k={\displaystyle \frac{{\partial }^k}{\partial {ϵ}^k}}.$

On utilizing Equations (26) and (27) into Equation (24), we may have

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathbb{P}}_k(\delta ,\wp )=F(\delta ,\wp )-ϵ\times \left[E^{-1}\left\{s^{\eta }E\{M\sum _{k=0}^{\infty }{p}^k{\mathbb{P}}_k(\delta ,\wp )+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left({\mathbb{P}}_k\right)\}\right\}\right].$$

(29)

On comparing the $ϵ$ coefficients, we may have

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

(30)

Lastly, the approximate solution to Equation (22) is taken as

$$\mathbb{P}(\delta ,\wp )=\underset{M\to \infty }{lim}\sum _{k=1}^M{\mathbb{P}}_k(\delta ,\wp ).$$

(31)

5. Convergence Analysis

The convergence analysis of the applied approaches are stated as below.

Theorem 1. 

Considering that ℘ is analytic in a neighborhood of $\mathbb{P}$ and $\left|\right|{\wp }^m\left({\mathbb{P}}_0\right)\left|\right|=sup\left\{\right||{\wp }^m\left({\mathbb{P}}_0\right)(b_0,b_0,\cdots b_n)/\left|\right|b_k\left|\right|\le 1,1\le k\le m\}\le l,$ for some real number $l>0$ and for every number m and ||${\mathbb{P}}_k\left|\right|\le M<{\displaystyle \frac{1}{e}},k=1,2,\cdots$ thus the series ${\sum }_{m=0}^{\infty }{\zeta }_m$ is convergent and also

$$\left|\right|{\zeta }_m\left|\right|\le l{M}^m{e}^{m-1}(e-1),m=1,2,\cdots .$$

Now to define boundedness of ||${\mathbb{P}}_k$||, for each k the conditions on ${\wp }^j\left({\mathbb{P}}_0\right)$ are assumed and is enough to assurance series convergence.

Theorem 2. 

If ℘ is $C^{\infty }$ and ||${\wp }^m\left({\mathbb{P}}_0\right)\left|\right|\le M\le e^{-1}$ for all m hence the series ${\sum }_{m=0}^{\infty }{\zeta }_m$ is convergent. These are the conditions for series ${\sum }_{j=0}^{\infty }{\mathbb{P}}_j$ to be convergent.

For proof check [63].

Theorem 3. 

Considering the precise solution of (22) is $\zeta (\delta ,\wp )$ and assume $\zeta (\delta ,\wp )$, ${\zeta }_n(\delta ,\wp )\in H$ and $\alpha \in (0,1)$, with H specifies the Hilbert space. The results achieved ${\sum }_{q=0}^{\infty }{\zeta }_q(\delta ,\wp )$ will converge $\zeta (\delta ,\wp )$ if ${\zeta }_q(\delta ,\wp )\le {\zeta }_{q-1}(\delta ,\wp )\forall q>A$, i.e., for each $\mathsf{\Omega }>0\exists A>0$, such that $\left|\right|{\zeta }_{q+n}(\delta ,\wp )\left|\right|\le \beta ,\forall m,n\in N.$

Proof. 

Considering a sequence of ${\sum }_{q=0}^{\infty }{\zeta }_q(\delta ,\wp ).$

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_0(\delta ,\wp )=& {\zeta }_0(\delta ,\wp ),\hfill \\ \hfill {\mathsf{\Omega }}_1(\delta ,\wp )=& {\zeta }_0(\delta ,\wp )+{\zeta }_1(\delta ,\wp ),\hfill \\ \hfill {\mathsf{\Omega }}_2(\delta ,\wp )=& {\zeta }_0(\delta ,\wp )+{\zeta }_1(\delta ,\wp )+{\zeta }_2(\delta ,\wp ),\hfill \\ \hfill {\mathsf{\Omega }}_3(\delta ,\wp )=& {\zeta }_0(\delta ,\wp )+{\zeta }_1(\delta ,\wp )+{\zeta }_2(\delta ,\wp )+{\zeta }_3(\delta ,\wp ),\hfill \\ \hfill & ⋮\hfill \\ \hfill {\mathsf{\Omega }}_q(\delta ,\wp )=& {\zeta }_0(\delta ,\wp )+{\zeta }_1(\delta ,\wp )+{\zeta }_2(\delta ,\wp )+\cdots +{\zeta }_q(\delta ,\wp ).\hfill \end{array}$$

(32)

We must specify that ${\mathsf{\Omega }}_q(\delta ,\wp )$ forms a “Cauchy sequence”. Additionally, let’s take

$$\begin{array}{cc}\hfill \left|\right|{\mathsf{\Omega }}_{q+1}(\delta ,\wp )-{\mathsf{\Omega }}_q(\delta ,\wp )\left|\right|& =\left|\right|{\zeta }_{q+1}(\delta ,\wp )\left|\right|\le \alpha \left|\right|{\zeta }_q(\delta ,\wp )\left|\right|\le {\alpha }^2\left|\right|{\zeta }_{q-1}(\delta ,\wp )\left|\right|\le {\alpha }^3\left|\right|{\zeta }_{q-2}(\delta ,\wp )\left|\right|\cdots \hfill \\ \hfill & \le {\alpha }_{q+1}\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|.\hfill \end{array}$$

(33)

For $q,n\in N$, we may have

$$\begin{array}{cc}\hfill \left|\right|{\mathsf{\Omega }}_q(\delta ,\wp )-{\mathsf{\Omega }}_n(\delta ,\wp )\left|\right|=& \left|\right|{\zeta }_{q+n}(\delta ,\wp )\left|\right|=\left|\right|{\mathsf{\Omega }}_q(\delta ,\wp )-{\mathsf{\Omega }}_{q-1}(\delta ,\wp )+({\mathsf{\Omega }}_{q-1}(\delta ,\wp )-{\mathsf{\Omega }}_{q-2}(\delta ,\wp ))\hfill \\ \hfill & +({\mathsf{\Omega }}_{q-2}(\delta ,\wp )-{\mathsf{\Omega }}_{q-3}(\delta ,\wp ))+\cdots +({\mathsf{\Omega }}_{n+1}(\delta ,\wp )-{\mathsf{\Omega }}_n(\delta ,\wp ))\left|\right|\hfill \\ \hfill \le & \left|\right|{\mathsf{\Omega }}_q(\delta ,\wp )-{\mathsf{\Omega }}_{q-1}(\delta ,\wp )\left|\right|+\left|\right|({\mathsf{\Omega }}_{q-1}(\delta ,\wp )-{\mathsf{\Omega }}_{q-2}(\delta ,\wp ))\left|\right|\hfill \\ \hfill & +\left|\right|({\mathsf{\Omega }}_{q-2}(\delta ,\wp )-{\mathsf{\Omega }}_{q-3}(\delta ,\wp ))\left|\right|+\cdots +\left|\right|({\mathsf{\Omega }}_{n+1}(\delta ,\wp )-{\mathsf{\Omega }}_n(\delta ,\wp ))\left|\right|\hfill \\ \hfill \le & {\alpha }^q\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|+{\alpha }^{q-1}\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|+\cdots +{\alpha }^{q+1}\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|\hfill \\ \hfill =& \left|\right|{\zeta }_0(\delta ,\wp )\left|\right|({\alpha }^q+{\alpha }^{q-1}+{\alpha }^{q+1})\hfill \\ \hfill =& \left|\right|{\zeta }_0(\delta ,\wp )\left|\right|{\displaystyle \frac{1-{\alpha }^{q-n}}{1-{\alpha }^{q+1}}}{\alpha }^{n+1}.\hfill \end{array}$$

(34)

As $0<\alpha <1$, and ${\zeta }_0(\delta ,\wp )$ are bound, so assume $\beta =1-\alpha /(1-{\alpha }_{q-n}){\alpha }^{n+1}\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|$, and we may have

$$\left|\right|{\zeta }_{q+n}(\delta ,\wp )\left|\right|\le \beta ,\forall q,n\in N.$$

(35)

Therefore, ${\left\{{\zeta }_q(\delta ,\wp )\right\}}_{q=0}^{\infty }$ forms a “Cauchy sequence” in H. It illustrate that the sequence ${\left\{{\zeta }_q(\delta ,\wp )\right\}}_{q=0}^{\infty }$ is a convergent sequence taking limit ${lim}_{q\to \infty }{\zeta }_q(\delta ,\wp )=\zeta (\delta ,\wp )$ for $\exists \zeta (\delta ,\wp )\in \mathcal{H}$ which verified the theorem. □

Theorem 4.

Considering that $\zeta (\delta ,\wp )$ reveals the acquired series solution and ${\sum }_{h=0}^k{\zeta }_h(\delta ,\wp )$ is finite. The relation that follows signifies the maximum absolute error, assuming $\alpha >0$ with $\left|\right|{\zeta }_{h+1}(\delta ,\wp )\left|\right|\le \left|\right|{\zeta }_h(\delta ,\wp )\left|\right|$.

$$\left|\right|\zeta (\delta ,\wp )-\sum _{h=0}^k{\zeta }_h(\delta ,\wp )\left|\right|<{\displaystyle \frac{{\alpha }^{k+1}}{1-\alpha }}\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|.$$

(36)

Proof. 

Assume ${\sum }_{h=0}^k{\zeta }_h(\delta ,\wp )$ is finite which indicates that ${\sum }_{h=0}^k{\zeta }_h(\delta ,\wp )<\infty$.

Considering

$$\begin{array}{cc}\hfill \left|\right|\zeta (\delta ,\wp )-\sum _{h=0}^k{\zeta }_h(\delta ,\wp )\left|\right|=& \left|\right|\sum _{h=k+1}^{\infty }{\zeta }_h(\delta ,\wp )\left|\right|\hfill \\ \hfill \le & \sum _{h=k+1}^{\infty }\left|\right|{\zeta }_h(\delta ,\wp )\left|\right|\hfill \\ \hfill \le & \sum _{h=k+1}^{\infty }{\alpha }^h\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|\hfill \\ \hfill \le & {\alpha }^{k+1}(1+\alpha +{\alpha }^2+\cdots )\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\alpha }^{k+1}}{1-\alpha }}\left|\right|{\zeta }_0(\delta ,\wp )\left|\right|.\hfill \end{array}$$

(37)

which verified the theorem. □

6. Error Estimation

In this section, we introduce error functions to analyse the accuracy and performance of the proposed methods. The absolute error $\left(E_n\right)$ in the nth-order approximation is as follows, assuming that $\mathbb{P}(\wp )$ is the accurate solution and ${\mathbb{P}}_n(\wp )$ is the nth-order approximation of $\mathbb{P}(\wp )$ obtained through the proposed methods.

$$E_n(\wp )=|\mathbb{P}(\wp )-{\mathbb{P}}_n(\wp )|,$$

(38)

then maximum absolute error $\left(M{E}_n\right)$ by

$$M{E}_n=ma{x}_{\wp \in [0,1]}|\mathbb{P}(\wp )-{\mathbb{P}}_n(\wp )|.$$

(39)

The authors of [64,65,66] discussed the error bound for their numerical approaches using well-known lemmas and theorems. We provide an upper bound on absolute error for the proposed approaches using the following theorem.

Theorem 5

(Error bound). Assume $\mathbb{P}(\wp )\in C^{(n+1)}[0,1]$ and ${\mathbb{P}}_n(\wp )={\sum }_{i=0}^n{c}_i{\wp }^i$ demonstrate the precise and nth-order achieved solution of (9) and (22), thus upper bound of the absolute error is as

$$\left|\right|\mathbb{P}(\wp )-{\mathbb{P}}_n{(\wp )\left|\right|}_{\infty }\le {\displaystyle \frac{M}{(n+1)!}}+ma{x}_{0\le i\le n}\left|a_i\right|,$$

(40)

where $M=ma{x}_{0\le t\le 1}\left|{\mathbb{P}}^{n+1}(\wp )\right|,a_i={\sum }_{i=0}^n\left({\displaystyle \frac{{\mathbb{P}}^i\left(0\right)}{i!}}-c_i\right).$

Proof. 

Assume $\mathbb{P}$ to be a continuous function on $[0,1]$; the upper bound of $\left|\mathbb{P}\right|$ is taken as

$${\left|\right|\mathbb{P}\left|\right|}_{\infty }=\underset{\wp \in [0,1]}{sup}\left|\mathbb{P}(\wp )\right|.$$

(41)

Employing norm property, we may have

$$\left|\right|\mathbb{P}(\wp )-{\mathbb{P}}_n{(\wp )\left|\right|}_{\infty }\le \left|\right|\mathbb{P}(\wp )-{\mathbb{P}}_n^{\ast }{(\wp )\left|\right|}_{\infty }+\left|\right|{\mathbb{P}}_n^{\ast }(\wp )-\mathbb{P}(\wp ){\left|\right|}_{\infty }$$

(42)

Since $\mathbb{P}(\wp )\in C^{(n+1)}[0,1]$ so by Taylor expansion, we get

$$\mathbb{P}(\wp )={\mathbb{P}}_n^{\ast }(\wp )+{\displaystyle \frac{{\mathbb{P}}^{n+1}\left({\wp }_0\right)}{(n+1)!}}{\wp }^{n+1},{\wp }_0\in (0,1),$$

(43)

with ${\mathbb{P}}_n^{\ast }(\wp )={\sum }_{i=0}^n{\displaystyle \frac{{\mathbb{P}}^i\left(0\right)}{i!}}{\wp }^i.$

From (43), we may have

$$\begin{array}{cc}\hfill & \left|\right|\mathbb{P}(\wp )-{\mathbb{P}}_n^{\ast }{(\wp )\left|\right|}_{\infty }=\underset{0\le \wp \le 1}{max}|{\displaystyle \frac{{\mathbb{P}}^{n+1}(\wp )}{(n+1)!}}{\wp }^{n+1}|\hfill \\ \hfill & \le {\displaystyle \frac{1}{(n+1)!}}\underset{0\le \wp \le 1}{max}\left|{\mathbb{P}}^{n+1}(\wp )\right|.\hfill \end{array}$$

(44)

Now we find the value of $\left|\right|\mathbb{P}(\wp )-{\mathbb{P}}_n^{\ast }(\wp ){\left|\right|}_{\infty }$.

Let $A=(a_0,a_1,\cdots ,a_n),T={(t_0,t_1,\cdots ,t_n)}^T,$

• $\mathrm{with} a_i={\displaystyle \frac{{\mathbb{P}}^i\left(0\right)}{i!}}-c_i,i=0,1,\cdots ,n \mathrm{thus}$

$$\begin{array}{cc}\hfill & \parallel {\mathbb{P}}_n^{\ast }(\wp )-{\mathbb{P}}_n(\wp )\parallel =\parallel \sum _{i=0}^n{\displaystyle \frac{{\mathbb{P}}^i\left(0\right)}{i!}}{\wp }^i-\sum _{i=0}^n{c}_i{\wp }^i\parallel =\parallel \sum _{i=0}^n\left({\displaystyle \frac{{\mathbb{P}}^i\left(0\right)}{i!}}-c_i\right){\wp }^i\parallel \hfill \\ \hfill & {\parallel A\parallel }_{\infty }.{\parallel T\parallel }_{\infty },\hfill \\ \hfill & \parallel {\mathbb{P}}_n^{\ast }(\wp )-{\mathbb{P}}_n{(\wp )\parallel =\parallel A\parallel }_{\infty }.{\parallel T\parallel }_{\infty }.\hfill \end{array}$$

(45)

From (44), (45) and (42) we may have

$$\begin{array}{cc}\hfill & \parallel \mathbb{P}(\wp )-{\mathbb{P}}_n(\wp )\parallel \le \le {\displaystyle \frac{1}{(n+1)!}}\underset{0\le \wp \le 1}{max}|{\mathbb{P}}^{n+1}(\wp )|+{\parallel A\parallel }_{\infty }.{\parallel T\parallel }_{\infty },\hfill \\ \hfill & \parallel \mathbb{P}(\wp )-{\mathbb{P}}_n(\wp )\parallel \le \le {\displaystyle \frac{M}{(n+1)!}}+\underset{0\le i\le n}{max}\left|a_i\right|.\hfill \end{array}$$

(46)

which complete the proof. □

7. Applications

7.1. Example

Consider the one-dimensional TF-GBFE

$${\displaystyle \frac{{\partial }^{\eta }\mathbb{P}(\delta ,\wp )}{\partial {\wp }^{\eta }}}={\displaystyle \frac{{\partial }^2\mathbb{P}}{\partial {\wp }^2}}-\varrho \mathbb{P}{\displaystyle \frac{\partial \mathbb{P}}{\partial \wp }}-\varrho {\mathbb{P}}^2+\varrho \mathbb{P},0<\eta \le 1,$$

(47)

with initial guess

$$\mathbb{P}(\delta ,0)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right).$$

(48)

By executing the ET to Equation (47), we may have

$$E\left[\mathbb{P}(\delta ,\wp )\right]=s^2\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right)\right)+s^{\eta }E\left[{\displaystyle \frac{{\partial }^2\mathbb{P}}{\partial {\wp }^2}}-\varrho \mathbb{P}{\displaystyle \frac{\partial \mathbb{P}}{\partial \wp }}-\varrho {\mathbb{P}}^2+\varrho \mathbb{P}\right],$$

(49)

Now by employing the inverse ET, we may have

$$\mathbb{P}(\delta ,\wp )={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right)+E^{-1}\left[s^{\eta }E\left[{\displaystyle \frac{{\partial }^2\mathbb{P}}{\partial {\wp }^2}}-\varrho \mathbb{P}{\displaystyle \frac{\partial \mathbb{P}}{\partial \wp }}-\varrho {\mathbb{P}}^2+\varrho \mathbb{P}\right]\right],$$

(50)

By NITM, we may have

$$\begin{array}{cc}\hfill & {\mathbb{P}}_0(\delta ,\wp )={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right),\hfill \\ \hfill & {\mathbb{P}}_1(\delta ,\wp )=E^{-1}\left[s^{\eta }E\left\{{\displaystyle \frac{{\partial }^2{\mathbb{P}}_0}{\partial {\wp }^2}}-\varrho {\mathbb{P}}_0{\displaystyle \frac{\partial {\mathbb{P}}_0}{\partial \wp }}-\varrho {\mathbb{P}}_0^2+\varrho {\mathbb{P}}_0\right\}\right]=\hfill \\ \hfill & {\displaystyle \frac{1}{16}}{\displaystyle \frac{\varrho (\varrho +4)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^2}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}},\hfill \end{array}$$

$$\begin{gathered} \begin{array}{cc}\hfill & {\mathbb{P}}_2(\delta ,\wp )=E^{-1}\left[s^{\eta }E\left\{{\displaystyle \frac{{\partial }^2({\mathbb{P}}_0+{\mathbb{P}}_1)}{\partial {\wp }^2}}-\varrho ({\mathbb{P}}_0+{\mathbb{P}}_1){\displaystyle \frac{\partial ({\mathbb{P}}_0+{\mathbb{P}}_1)}{\partial \wp }}-\varrho {({\mathbb{P}}_0+{\mathbb{P}}_1)}^2+\varrho ({\mathbb{P}}_0+{\mathbb{P}}_0)\right\}\right]-\hfill \\ \hfill & E^{-1}\left[s^{\eta }E\left\{{\displaystyle \frac{{\partial }^2{\mathbb{P}}_0}{\partial {\wp }^2}}-\varrho {\mathbb{P}}_0{\displaystyle \frac{\partial {\mathbb{P}}_0}{\partial \wp }}-\varrho {\mathbb{P}}_0^2+\varrho {\mathbb{P}}_0\right\}\right]={\displaystyle \frac{1}{64}}{\displaystyle \frac{{\varrho }^2{(\varrho +4)}^{2}sinh\left(\frac{1}{4}\varrho \delta \right)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^3}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}}-\hfill \\ \hfill & {\displaystyle \frac{1}{512}}{\displaystyle \frac{{\varrho }^3{(\varrho +4)}^2\left(-\varrho sinh\left(\frac{1}{4}\varrho \delta \right)+2cosh\left(\frac{1}{4}\varrho \delta \right)\right)\Gamma (2\eta +1)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^5\Gamma {(\eta +1)}^2}}{\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}, \\ ⋮\hfill \end{array} \end{gathered}$$

Lastly, the series solution is taken as

$$\begin{array}{cc}\hfill & \mathbb{P}(\delta ,\wp )={\mathbb{P}}_0(\delta ,\wp )+{\mathbb{P}}_1(\delta ,\wp )+{\mathbb{P}}_2(\delta ,\wp )+\cdots ,\hfill \\ \hfill & \mathbb{P}(\delta ,\wp )={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right)+{\displaystyle \frac{1}{16}}{\displaystyle \frac{\varrho (\varrho +4)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^2}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}}+{\displaystyle \frac{1}{64}}{\displaystyle \frac{{\varrho }^2{(\varrho +4)}^{2}sinh\left(\frac{1}{4}\varrho \delta \right)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^3}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}}-\hfill \\ \hfill & {\displaystyle \frac{1}{512}}{\displaystyle \frac{{\varrho }^3{(\varrho +4)}^2\left(-\varrho sinh\left(\frac{1}{4}\varrho \delta \right)+2cosh\left(\frac{1}{4}\varrho \delta \right)\right)\Gamma (2\eta +1)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^5\Gamma {(\eta +1)}^2}}{\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}+\cdots ,\hfill \end{array}$$

(51)

By employing the HPTM, we may have

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}^n(\delta ,\wp )={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right)+ϵ\left\{E^{-1}\right(s^{\eta }E[{\left(\sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}_n(\delta ,\wp )\right)}_{\delta \delta }-\left(\sum _{n=0}^{\infty }{ϵ}^n{H}_n^1(\delta ,\wp )\right)-\hfill \\ \hfill & \varrho \left(\sum _{n=0}^{\infty }{ϵ}^n{H}_n^2(\delta ,\wp )\right)+\varrho \left(\sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}_n(\delta ,\wp )\right)\left]\right)\},\hfill \end{array}$$

(52)

with He’s polynomials $H_k\left(\delta \right)$ represent the nonlinear terms and is stated as

$$\begin{array}{cc}\hfill & H_0^1\left(\delta \right)={\mathbb{P}}_0{\left({\mathbb{P}}_0\right)}_{\delta }\hfill \\ \hfill & H_1^1\left(\delta \right)={\mathbb{P}}_0{\left({\mathbb{P}}_1\right)}_{\delta }+{\mathbb{P}}_1{\left({\mathbb{P}}_0\right)}_{\delta }\hfill \\ \hfill & H_2^1\left(\delta \right)={\mathbb{P}}_2{\left({\mathbb{P}}_0\right)}_{\delta }+{\mathbb{P}}_1{\left({\mathbb{P}}_1\right)}_{\delta }+{\mathbb{P}}_0{\left({\mathbb{P}}_2\right)}_{\delta }\hfill \\ \hfill & ⋮\hfill \\ \hfill & H_0^2\left(\delta \right)={\left({\mathbb{P}}_0\right)}^2\hfill \\ \hfill & H_1^2\left(\delta \right)=2{\mathbb{P}}_0{\mathbb{P}}_1\hfill \\ \hfill & H_2^2\left(\delta \right)=2{\mathbb{P}}_0{\mathbb{P}}_1+{\left({\mathbb{P}}_0\right)}^2\hfill \end{array}$$

On comparing the $ϵ$ coefficients, we may have

$$\begin{array}{cc}\hfill {ϵ}^0:{\mathbb{P}}_0(\delta ,\wp )& ={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right),\hfill \\ \hfill {ϵ}^1:{\mathbb{P}}_1(\delta ,\wp )& ={\displaystyle \frac{1}{16}}{\displaystyle \frac{\varrho (\varrho +4)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^2}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}},\hfill \\ \hfill {ϵ}^2:{\mathbb{P}}_2(\delta ,\wp )& ={\displaystyle \frac{1}{64}}{\displaystyle \frac{{\varrho }^2{(\varrho +4)}^{2}sinh\left(\frac{1}{4}\varrho \delta \right)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^3}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}},\hfill \\ \hfill {ϵ}^3:{\mathbb{P}}_3(\delta ,\wp )& =-{\displaystyle \frac{{\varrho }^3{(\varrho +4)}^2}{512cosh{\left(\frac{1}{4}\varrho \delta \right)}^5}}(-2\varrho cosh{\left({\displaystyle \frac{1}{4}}\varrho \delta \right)}^3-8cosh{\left({\displaystyle \frac{1}{4}}\varrho \delta \right)}^3+2\varrho sinh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)+\hfill \\ \hfill & 3\varrho cosh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)+8cosh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)){\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}+{\displaystyle \frac{{\varrho }^3{(\varrho +4)}^2}{512cosh{\left(\frac{1}{4}\varrho \delta \right)}^5\Gamma {(\eta +1)}^2}}(\varrho sinh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)\hfill \\ \hfill & -2cosh\left({\displaystyle \frac{1}{4}}\varrho \delta \right))\Gamma (2\eta +1){\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}\hfill \\ \hfill & ⋮\hfill \end{array}$$

(53)

Lastly, the series solution is taken as

$$\mathbb{P}(\delta ,\wp )=\sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}_n(\delta ,\wp ).$$

(54)

Hence

$$\begin{array}{cc}\hfill & \mathbb{P}(\delta ,\wp )={\mathbb{P}}_0(\delta ,\wp )+{\mathbb{P}}_1(\delta ,\wp )+{\mathbb{P}}_2(\delta ,\wp )+\cdots ,\hfill \\ \hfill & \mathbb{P}(\delta ,\wp )={\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}tanh\left({\displaystyle \frac{\varrho \delta }{4}}\right)+{\displaystyle \frac{1}{16}}{\displaystyle \frac{\varrho (\varrho +4)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^2}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}}+{\displaystyle \frac{1}{64}}{\displaystyle \frac{{\varrho }^2{(\varrho +4)}^{2}sinh\left(\frac{1}{4}\varrho \delta \right)}{cosh{\left(\frac{1}{4}\varrho \delta \right)}^3}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}}-\hfill \\ \hfill & {\displaystyle \frac{{\varrho }^3{(\varrho +4)}^2}{512cosh{\left(\frac{1}{4}\varrho \delta \right)}^5}}(-2\varrho cosh{\left({\displaystyle \frac{1}{4}}\varrho \delta \right)}^3-8cosh{\left({\displaystyle \frac{1}{4}}\varrho \delta \right)}^3+2\varrho sinh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)+3\varrho cosh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)+\hfill \\ \hfill & 8cosh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)){\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}+{\displaystyle \frac{{\varrho }^3{(\varrho +4)}^2}{512cosh{\left(\frac{1}{4}\varrho \delta \right)}^5\Gamma {(\eta +1)}^2}}\left(\varrho sinh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)-2cosh\left({\displaystyle \frac{1}{4}}\varrho \delta \right)\right)\hfill \\ \hfill & \Gamma (2\eta +1){\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}+\cdots .\hfill \end{array}$$

(55)

At $\eta =1$, we get exact solution as

$$\mathbb{P}(\delta ,\wp )={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}tanh\left[{\displaystyle \frac{-\varrho }{4}}\left(\delta -\left({\displaystyle \frac{\varrho }{2}}+{\displaystyle \frac{2\varrho }{\varrho }}\right)\wp \right)\right]$$

(56)

This study’s graphical and tabular representations offer a thorough understanding of the precision and dependability of the NITM and HPTM when employed to the TF-GBFE. The HPTM approximation solution and the exact solution at $\eta =1$ are compared in Figure 1. The derived solution of NITM and the accurate solution at $\eta =1$ are compared in Figure 2. Excellent agreement can be seen in the graphical comparisons between the precise solution and the solutions from the suggested methods. The 3D depiction of the solutions derived by HPTM and NITM for the TF-GBFE at $\eta =0.6$ and $\eta =0.8$ is shown in Figure 3 and Figure 4. The 2D plot of the found solution for different fractional orders is shown in Figure 5 and Figure 6, along with a comparison of the generated and precise solution. Figure 5 and Figure 6 illustrate the impact of changing the parameter $\eta$ on the TF-GBFE solutions. The derived solutions are illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 for a range of $\eta$ values with $-5\le \delta \le 5$ and temporal variable $0\le \wp \le 0.1$.

Table 1. Behavior of the accurate as well as HPTM solution at several $\eta$ for $\mathbb{P}(\delta ,\wp )$.

$\mathsf{\delta }$	$\mathsf{\eta }=0.97$	$\mathsf{\eta }=0.98$	$\mathsf{\eta }=0.99$	$\mathsf{\eta }=1\left(\mathbf{HPTM}\right)$	$\mathsf{\eta }=1\left(\mathbf{Aacurate}\right)$
0.0	0.5002718980	0.5002646308	0.5002575411	0.5002506250	0.5002506250
0.1	0.5001468979	0.5001396308	0.5001325410	0.5001256250	0.5001256250
0.2	0.5000218980	0.5000146308	0.5000075411	0.5000006250	0.5000006250
0.3	0.4998968980	0.4998896309	0.4998825411	0.4998756251	0.4998756250
0.4	0.4997718981	0.4997646308	0.4997575411	0.4997506250	0.4997506250
0.5	0.4996468980	0.4996396309	0.4996325412	0.4996256251	0.4996256251
0.6	0.4995218981	0.4995146310	0.4995075413	0.4995006252	0.4995006252
0.7	0.4993968983	0.4993896312	0.4993825414	0.4993756253	0.4993756253
0.8	0.4992718985	0.4992646313	0.4992575416	0.4992506256	0.4992506256
0.9	0.4991468988	0.4991396317	0.4991325420	0.4991256259	0.4991256259
1.0	0.4990218993	0.4990146321	0.4990075424	0.4990006263	0.4990006263

and

Table 2. Behavior of the accurate as well as NITM solution at several $\eta$ for $\mathbb{P}(\delta ,\wp )$.

$\mathsf{\delta }$	$\mathsf{\eta }=0.97$	$\mathsf{\eta }=0.98$	$\mathsf{\eta }=0.99$	$\mathsf{\eta }=1\left(\mathbf{NITM}\right)$	$\mathsf{\eta }=1\left(\mathbf{Accurate}\right)$
0.0	0.5002718980	0.5002646308	0.5002575411	0.5002506250	0.5002506250
0.1	0.5001468979	0.5001396308	0.5001325410	0.5001256250	0.5001256250
0.2	0.5000218980	0.5000146308	0.5000075411	0.5000006250	0.5000006250
0.3	0.4998968980	0.4998896309	0.4998825411	0.4998756251	0.4998756250
0.4	0.4997718981	0.4997646308	0.4997575411	0.4997506250	0.4997506250
0.5	0.4996468980	0.4996396309	0.4996325412	0.4996256251	0.4996256251
0.6	0.4995218981	0.4995146310	0.4995075413	0.4995006252	0.4995006252
0.7	0.4993968983	0.4993896312	0.4993825414	0.4993756253	0.4993756253
0.8	0.4992718985	0.4992646313	0.4992575416	0.4992506256	0.4992506256
0.9	0.4991468988	0.4991396317	0.4991325420	0.4991256259	0.4991256259
1.0	0.4990218993	0.4990146321	0.4990075424	0.4990006263	0.4990006263

present the exact solution for the TF-GBFE while maintaining $\eta =0.97,0.98,0.98$ and 1, as well as the comparative numerical solutions derived by HPTM and the NITM.

Table 3. Comparative analysis of the analytical solution with Haar wavelet, OHAM, and q-HATM.

$\mathsf{\delta }$	℘	Haar Wavelet Error [67]	OHAM Error [67]	q-HATM Error [68]	HPTM Error	NITM Error
0.1	0.2	5.4804 $\times {10}^{-5}$	4.2290 $\times {10}^{-11}$	1.1102 $\times {10}^{-16}$	0	0
	0.4	2.3476 $\times {10}^{-5}$	8.4080 $\times {10}^{-10}$	8.8818 $\times {10}^{-16}$	0	0
	0.6	7.8526 $\times {10}^{-6}$	3.4030 $\times {10}^{-9}$	9.6589 $\times {10}^{-15}$	0	0
	0.8	3.9181 $\times {10}^{-5}$	8.7368 $\times {10}^{-9}$	4.7517 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
0.2	0.2	2.3553 $\times {10}^{-5}$	8.3330 $\times {10}^{-11}$	1.1102 $\times {10}^{-16}$	0	0
	0.4	7.7785 $\times {10}^{-6}$	3.3840 $\times {10}^{-10}$	4.4409 $\times {10}^{-16}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.6	3.9108 $\times {10}^{-5}$	2.2730 $\times {10}^{-9}$	2.7756 $\times {10}^{-15}$	0	0
	0.8	7.0440 $\times {10}^{-5}$	6.7268 $\times {10}^{-9}$	2.6090 $\times {10}^{-14}$	0	0
0.3	0.2	7.0426 $\times {10}^{-5}$	2.0890 $\times {10}^{-10}$	1.1102 $\times {10}^{-16}$	0	0
	0.4	3.9091 $\times {10}^{-5}$	1.6420 $\times {10}^{-10}$	1.7764 $\times {10}^{-15}$	0	0
	0.6	7.7594 $\times {10}^{-6}$	1.1420 $\times {10}^{-9}$	3.9968 $\times {10}^{-15}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.8	2.3578 $\times {10}^{-5}$	4.7168 $\times {10}^{-9}$	4.5519 $\times {10}^{-15}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
0.4	0.2	3.9169 $\times {10}^{-6}$	3.3460 $\times {10}^{-10}$	3.3307 $\times {10}^{-16}$	0	0
	0.4	7.8222 $\times {10}^{-5}$	6.6670 $\times {10}^{-10}$	3.3305 $\times {10}^{-15}$	1.0 × 10−10	1.0 × 10−10
	0.6	2.3516 $\times {10}^{-5}$	1.1300 $\times {10}^{-11}$	1.088 $\times {10}^{-14}$	0	0
	0.8	5.4870 $\times {10}^{-5}$	2.7068 $\times {10}^{-9}$	1.7097 $\times {10}^{-14}$	0	0
0.5	0.2	7.9054 $\times {10}^{-6}$	4.6020 $\times {10}^{-10}$	3.3307 $\times {10}^{-16}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.4	2.3463 $\times {10}^{-5}$	1.1692 $\times {10}^{-9}$	4.5519 $\times {10}^{-15}$	0	0
	0.6	5.4812 $\times {10}^{-5}$	1.1190 $\times {10}^{-9}$	1.7652 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.8	8.6199 $\times {10}^{-6}$	6.9680 $\times {10}^{-10}$	3.8636 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
0.6	0.2	5.4768 $\times {10}^{-5}$	5.8580 $\times {10}^{-10}$	4.9960 $\times {10}^{-16}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.4	2.3384 $\times {10}^{-5}$	1.6717 $\times {10}^{-9}$	5.9952 $\times {10}^{-15}$	0	0
	0.6	7.9731 $\times {10}^{-6}$	2.2490 $\times {10}^{-9}$	2.4536 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.8	3.9384 $\times {10}^{-5}$	1.3132 $\times {10}^{-9}$	6.0174 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
0.7	0.2	2.3489 $\times {10}^{-5}$	7.1150 $\times {10}^{-10}$	4.9960 $\times {10}^{-16}$	0	0
	0.4	7.9370 $\times {10}^{-6}$	2.1742 $\times {10}^{-9}$	7.2164 $\times {10}^{-15}$	0	0
	0.6	3.9317 $\times {10}^{-5}$	3.3810 $\times {10}^{-9}$	3.1419 $\times {10}^{-14}$	0	0
	0.8	7.0791 $\times {10}^{-5}$	3.3232 $\times {10}^{-9}$	8.1712 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
0.8	0.2	7.0337 $\times {10}^{-5}$	8.3710 $\times {10}^{-10}$	6.1062 $\times {10}^{-16}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.4	3.8884 $\times {10}^{-6}$	2.6767 $\times {10}^{-9}$	8.5487 $\times {10}^{-15}$	0	0
	0.6	7.4894 $\times {10}^{-5}$	4.5110 $\times {10}^{-9}$	3.8192 $\times {10}^{-14}$	0	0
	0.8	2.4026 $\times {10}^{-5}$	5.3332 $\times {10}^{-9}$	1.0325 $\times {10}^{-13}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
0.9	0.2	3.9031 $\times {10}^{-5}$	9.6270 $\times {10}^{-10}$	7.2164 $\times {10}^{-16}$	0	0
	0.4	7.5074 $\times {10}^{-6}$	3.1792 $\times {10}^{-9}$	9.9365 $\times {10}^{-15}$	0	0
	0.6	2.3923 $\times {10}^{-5}$	5.6420 $\times {10}^{-9}$	4.4964 $\times {10}^{-14}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
	0.8	5.5543 $\times {10}^{-5}$	7.3432 $\times {10}^{-9}$	1.2479 $\times {10}^{-13}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$
1.0	0.2	8.5852 $\times {10}^{-5}$	1.0883 $\times {10}^{-9}$	7.7716 $\times {10}^{-16}$	0	0
	0.4	5.4286 $\times {10}^{-5}$	3.6817 $\times {10}^{-9}$	1.1269 $\times {10}^{-14}$	0	0
	0.6	2.2833 $\times {10}^{-5}$	6.7720 $\times {10}^{-9}$	5.1736 $\times {10}^{-14}$	0	0
	0.8	8.8514 $\times {10}^{-6}$	9.3532 $\times {10}^{-9}$	1.4622 $\times {10}^{-13}$	1.0 $\times {10}^{-10}$	1.0 $\times {10}^{-10}$

provides the absolute differences derived from Haar wavelet, OHAM, q-HATM, HPTM, and NITM. This suggests that the solutions generated by NITM and HPTM are more suitable than those obtained using q-HATM, OHAM, and Haar wavelet. It is also clear from Table 1, Table 2 and Table 3 that the solution obtained by HPTM after the fourth iteration is obtained by NITM after the third iteration. Hence, the solutions obtained by NITM are more suitable and efficient than those obtained by HPTM. Overall, the tabular and graphical data support the finding that both approaches work well, while NITM provides better accuracy in particular fractional contexts.

7.2. Example

Consider the one-dimensional TF-GBFE

$${\displaystyle \frac{{\partial }^{\eta }\mathbb{P}(\delta ,\wp )}{\partial {\wp }^{\eta }}}={\displaystyle \frac{{\partial }^2\mathbb{P}}{\partial {\wp }^2}}+{\mathbb{P}}^2{\displaystyle \frac{\partial \mathbb{P}}{\partial \wp }}-{\mathbb{P}}^3+\mathbb{P},0<\eta \le 1,$$

(57)

with initial guess

$$\mathbb{P}(\delta ,0)=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)}.$$

(58)

By executing the ET to Equation (57), we may have

$$E\left[\mathbb{P}(\delta ,\wp )\right]=s^2\left(\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)}\right)+s^{\eta }E\left[{\displaystyle \frac{{\partial }^2\mathbb{P}}{\partial {\wp }^2}}+{\mathbb{P}}^2{\displaystyle \frac{\partial \mathbb{P}}{\partial \wp }}-{\mathbb{P}}^3+\mathbb{P}\right],$$

(59)

Now by employing the inverse ET, we may have

$$\mathbb{P}(\delta ,\wp )=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)}+E^{-1}\left[s^{\eta }E\left[{\displaystyle \frac{{\partial }^2\mathbb{P}}{\partial {\wp }^2}}+{\mathbb{P}}^2{\displaystyle \frac{\partial \mathbb{P}}{\partial \wp }}-{\mathbb{P}}^3+\mathbb{P}\right]\right],$$

(60)

By NITM, we may have

$$\begin{array}{cc}\hfill & {\mathbb{P}}_0(\delta ,\wp )=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)},\hfill \\ \hfill & {\mathbb{P}}_1(\delta ,\wp )=E^{-1}\left[s^{\eta }E\left\{{\displaystyle \frac{{\partial }^2{\mathbb{P}}_0}{\partial {\wp }^2}}+{\mathbb{P}}_0^2{\displaystyle \frac{\partial {\mathbb{P}}_0}{\partial \wp }}-{\mathbb{P}}_0^3+{\mathbb{P}}_0\right\}\right]=\hfill \\ \hfill & {\displaystyle \frac{5}{18}}{\displaystyle \frac{\sqrt{2}}{cosh{\left(\frac{\delta }{3}\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}},\hfill \end{array}$$

$$\begin{gathered} \begin{array}{cc}\hfill & {\mathbb{P}}_2(\delta ,\wp )=E^{-1}\left[s^{\eta }E\left\{{\displaystyle \frac{{\partial }^2({\mathbb{P}}_0+{\mathbb{P}}_1)}{\partial {\wp }^2}}+{({\mathbb{P}}_0+{\mathbb{P}}_1)}^2{\displaystyle \frac{\partial ({\mathbb{P}}_0+{\mathbb{P}}_1)}{\partial \wp }}-{({\mathbb{P}}_0+{\mathbb{P}}_1)}^3+({\mathbb{P}}_0+{\mathbb{P}}_1)\right\}\right]-\hfill \\ \hfill & E^{-1}\left[s^{\eta }E\left\{{\displaystyle \frac{{\partial }^2{\mathbb{P}}_0}{\partial {\wp }^2}}+{\mathbb{P}}_0^2{\displaystyle \frac{\partial {\mathbb{P}}_0}{\partial \wp }}-{\mathbb{P}}_0^3+{\mathbb{P}}_0\right\}\right]=\hfill \\ \hfill & -{\displaystyle \frac{25}{162}}{\displaystyle \frac{\sqrt{2}\left(8sinh\left(\frac{\delta }{3}\right)cosh{\left(\frac{\delta }{3}\right)}^2+8cosh{\left(\frac{\delta }{3}\right)}^3-3sinh\left(\frac{\delta }{3}\right)-7cosh\left(\frac{\delta }{3}\right)\right)}{cosh{\left(\frac{\delta }{3}\right)}^3{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}}\hfill \\ \hfill & -{\displaystyle \frac{25}{1944}}{\displaystyle \frac{\sqrt{2}\left(52sinh\left(\frac{\delta }{3}\right)cosh{\left(\frac{\delta }{3}\right)}^2+52cosh{\left(\frac{\delta }{3}\right)}^3-7sinh\left(\frac{\delta }{3}\right)-33cosh\left(\frac{\delta }{3}\right)\right)\Gamma (2q+1)}{cosh{\left(\frac{\delta }{3}\right)}^5{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}\Gamma {(\eta +1)}^2}}{\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}\hfill \\ \hfill & -{\displaystyle \frac{125}{17496}}{\displaystyle \frac{\sqrt{2}\left(10cosh{\left(\frac{\delta }{3}\right)}^2+10cosh\left(\frac{\delta }{3}\right)sinh\left(\frac{\delta }{3}\right)-3\right)\Gamma (3q+1)}{cosh{\left(\frac{\delta }{3}\right)}^6{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}\Gamma {(\eta +1)}^3}}{\displaystyle \frac{{\wp }^{4\eta }}{\Gamma (4\eta +1)}} \\ ⋮\hfill \end{array} \end{gathered}$$

Lastly, the series solution is taken as

$$\begin{array}{cc}\hfill & \mathbb{P}(\delta ,\wp )={\mathbb{P}}_0(\delta ,\wp )+{\mathbb{P}}_1(\delta ,\wp )+{\mathbb{P}}_2(\delta ,\wp )+\cdots ,\hfill \\ \hfill & \mathbb{P}(\delta ,\wp )=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)}+{\displaystyle \frac{5}{18}}{\displaystyle \frac{\sqrt{2}}{cosh{\left(\frac{\delta }{3}\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}}\hfill \\ \hfill & -{\displaystyle \frac{25}{162}}{\displaystyle \frac{\sqrt{2}\left(8sinh\left(\frac{\delta }{3}\right)cosh{\left(\frac{\delta }{3}\right)}^2+8cosh{\left(\frac{\delta }{3}\right)}^3-3sinh\left(\frac{\delta }{3}\right)-7cosh\left(\frac{\delta }{3}\right)\right)}{cosh{\left(\frac{\delta }{3}\right)}^3{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}}\hfill \\ \hfill & -{\displaystyle \frac{25}{1944}}{\displaystyle \frac{\sqrt{2}\left(52sinh\left(\frac{\delta }{3}\right)cosh{\left(\frac{\delta }{3}\right)}^2+52cosh{\left(\frac{\delta }{3}\right)}^3-7sinh\left(\frac{\delta }{3}\right)-33cosh\left(\frac{\delta }{3}\right)\right)\Gamma (2\eta +1)}{cosh{\left(\frac{\delta }{3}\right)}^5{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}\Gamma {(\eta +1)}^2}}{\displaystyle \frac{{\wp }^{3\eta }}{\Gamma (3\eta +1)}}\hfill \\ \hfill & -{\displaystyle \frac{125}{17496}}{\displaystyle \frac{\sqrt{2}\left(10cosh{\left(\frac{\delta }{3}\right)}^2+10cosh\left(\frac{\delta }{3}\right)sinh\left(\frac{\delta }{3}\right)-3\right)\Gamma (3\eta +1)}{cosh{\left(\frac{\delta }{3}\right)}^6{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}\Gamma {(\eta +1)}^3}}{\displaystyle \frac{{\wp }^{4\eta }}{\Gamma (4\eta +1)}}+\cdots ,\hfill \end{array}$$

(61)

By employing the HPTM, we may have

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}^n(\delta ,\wp )=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)}+ϵ\left\{E^{-1}\right(s^{\eta }E[{\left(\sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}_n(\delta ,\wp )\right)}_{\delta \delta }+\left(\sum _{n=0}^{\infty }{ϵ}^n{H}_n^1(\delta ,\wp )\right)-\hfill \\ \hfill & \left(\sum _{n=0}^{\infty }{ϵ}^n{H}_n^2(\delta ,\wp )\right)+\left(\sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}_n(\delta ,\wp )\right)\left]\right)\},\hfill \end{array}$$

(62)

with He’s polynomials $H_k\left(\delta \right)$ represent the nonlinear terms and is stated as

$$\begin{array}{cc}\hfill & H_0^1\left(\delta \right)={\mathbb{P}}_0^2{\left({\mathbb{P}}_0\right)}_{\delta }\hfill \\ \hfill & H_1^1\left(\delta \right)={\mathbb{P}}_0^2{\left({\mathbb{P}}_1\right)}_{\delta }+2{\mathbb{P}}_0{\mathbb{P}}_1{\left({\mathbb{P}}_0\right)}_{\delta }\hfill \\ \hfill & ⋮\hfill \\ \hfill & H_0^2\left(\delta \right)={\left({\mathbb{P}}_0\right)}^3\hfill \\ \hfill & H_1^2\left(\delta \right)=3{\mathbb{P}}_0^2{\mathbb{P}}_1\hfill \end{array}$$

On comparing the $ϵ$ coefficients, we may have

$$\begin{array}{cc}\hfill {ϵ}^0:{\mathbb{P}}_0(\delta ,\wp )& =\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)},\hfill \\ \hfill {ϵ}^1:{\mathbb{P}}_1(\delta ,\wp )& ={\displaystyle \frac{5}{18}}{\displaystyle \frac{\sqrt{2}}{cosh{\left(\frac{\delta }{3}\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}},\hfill \\ \hfill {ϵ}^2:{\mathbb{P}}_2(\delta ,\wp )& =-{\displaystyle \frac{25}{162}}{\displaystyle \frac{\sqrt{2}\left(8sinh\left(\frac{\delta }{3}\right)cosh{\left(\frac{\delta }{3}\right)}^2+8cosh{\left(\frac{\delta }{3}\right)}^3-3sinh\left(\frac{\delta }{3}\right)-7cosh\left(\frac{\delta }{3}\right)\right)}{cosh{\left(\frac{\delta }{3}\right)}^3{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}},\hfill \\ \hfill & ⋮\hfill \end{array}$$

(63)

Lastly, the series solution is taken as

$$\mathbb{P}(\delta ,\wp )=\sum _{n=0}^{\infty }{ϵ}^n{\mathbb{P}}_n(\delta ,\wp ).$$

(64)

Hence

$$\begin{array}{cc}\hfill & \mathbb{P}(\delta ,\wp )={\mathbb{P}}_0(\delta ,\wp )+{\mathbb{P}}_1(\delta ,\wp )+{\mathbb{P}}_2(\delta ,\wp )+\cdots ,\hfill \\ \hfill & \mathbb{P}(\delta ,\wp )=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}\right)\right)}+{\displaystyle \frac{5}{18}}{\displaystyle \frac{\sqrt{2}}{cosh{\left(\frac{\delta }{3}\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{\eta }}{\Gamma (\eta +1)}}+\hfill \\ \hfill & -{\displaystyle \frac{25}{162}}{\displaystyle \frac{\sqrt{2}\left(8sinh\left(\frac{\delta }{3}\right)cosh{\left(\frac{\delta }{3}\right)}^2+8cosh{\left(\frac{\delta }{3}\right)}^3-3sinh\left(\frac{\delta }{3}\right)-7cosh\left(\frac{\delta }{3}\right)\right)}{cosh{\left(\frac{\delta }{3}\right)}^3{\left(cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)\right)}^2\sqrt{\frac{cosh\left(\frac{\delta }{3}\right)+sinh\left(\frac{\delta }{3}\right)}{cosh\left(\frac{\delta }{3}\right)}}}}{\displaystyle \frac{{\wp }^{2\eta }}{\Gamma (2\eta +1)}}+\cdots .\hfill \end{array}$$

(65)

At $\eta =1$, we get exact solution as

$$\mathbb{P}(\delta ,\wp )=\sqrt{{\displaystyle \frac{1}{2}}\left(1+tanh\left({\displaystyle \frac{\delta }{3}}+{\displaystyle \frac{10\wp }{9}}\right)\right)}$$

(66)

The HPTM approximation solution and the accurate solution at $\eta =1$ are compared in Figure 7. The derived solution of NITM and the accurate solution at $\eta =1$ are compared in Figure 8. Excellent agreement can be seen in the graphical comparisons between the precise solution and the solutions from the suggested methods. The 3D depiction of the solutions derived by HPTM and NITM for the TF-GBFE at $\eta =0.6$ and $\eta =0.8$ is shown in Figure 9 and Figure 10. The 2D plot of the found solution for different fractional orders is shown in Figure 11 and Figure 12, along with a comparison of the generated and precise solution. Figure 11 and Figure 12 illustrate the impact of changing the parameter $\eta$ on the TF-GBFE solutions. The derived solutions are illustrated in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 for a range of $\eta$ values with $-5\le \delta \le 5$ and temporal variable $0\le \wp \le 0.1$.

Table 4. Behavior of the accurate as well as HPTM solution at several of $\eta$ for $\mathbb{P}(\delta ,\wp )$.

$\mathsf{\delta }$	$\mathsf{\eta }=0.97$	$\mathsf{\eta }=0.98$	$\mathsf{\eta }=0.99$	$\mathsf{\eta }=1\left(\mathbf{HPTM}\right)$	$\mathsf{\eta }=1\left(\mathbf{Accurate}\right)$
0.0	0.7116581992	0.7114365594	0.7112253981	0.7110242400	0.7110241395
0.1	0.7232619661	0.7230443269	0.7228369684	0.7226394249	0.7226393305
0.2	0.7346476339	0.7344342783	0.7342309925	0.7340373213	0.7340372335
0.3	0.7458007792	0.7455919690	0.7453930058	0.7452034453	0.7452033650
0.4	0.7567081326	0.7565041070	0.7563096947	0.7561244628	0.7561243895
0.5	0.7673576479	0.7671586222	0.7669689664	0.7667882594	0.7667881945
0.6	0.7777385630	0.7775447277	0.7773600105	0.7771840022	0.7771839460
0.7	0.7878414402	0.7876529605	0.7874733396	0.7873021809	0.7873021325
0.8	0.7976581905	0.7974752057	0.7973008145	0.7971346329	0.7971345930
0.9	0.8071820859	0.8070047096	0.8068356568	0.8066745564	0.8066745250
1.0	0.8164077556	0.8162360756	0.8160724457	0.8159165074	0.8159164840

and

Table 5. Behavior of the accurate as well as NITM solution at several $\eta$ for $\mathbb{P}(\delta ,\wp )$.

$\mathsf{\delta }$	$\mathsf{\eta }=0.97$	$\mathsf{\eta }=0.98$	$\mathsf{\eta }=0.99$	$\mathsf{\eta }=1\left(\mathbf{NITM}\right)$	$\mathsf{\eta }=1\left(\mathbf{Accurate}\right)$
0.0	0.7116580097	0.7114363988	0.7112252619	0.7110241247	0.7110241395
0.1	0.7232617778	0.7230441673	0.7228368331	0.7226393103	0.7226393305
0.2	0.7346474475	0.7344341203	0.7342308586	0.7340372079	0.7340372335
0.3	0.7458005955	0.7455918133	0.7453928739	0.7452033335	0.7452033650
0.4	0.7567079523	0.7565039542	0.7563095652	0.7561243531	0.7561243895
0.5	0.7673574716	0.7671584728	0.7669688398	0.7667881521	0.7667881945
0.6	0.7777383914	0.7775445823	0.7773598872	0.7771838978	0.7771839460
0.7	0.7878412738	0.7876528195	0.7874732201	0.7873020797	0.7873021325
0.8	0.7976580298	0.7974750695	0.7973006991	0.7971345351	0.7971345930
0.9	0.8071819313	0.8070045785	0.8068355457	0.8066744623	0.8066745250
1.0	0.8164076073	0.8162359500	0.8160723393	0.8159164172	0.8159164840

present the exact solution for the TF-GBFE while maintaining $\eta =0.97,0.98,0.98$ and 1, as well as the comparative numerical solutions derived by HPTM and the NITM.

8. Conclusions

In this work, we offer the HPTM and NITM for nonlinear TF-GBFE solutions. The derivative is considered in the Caputo sense. The solution derived from the methods presented shows that our findings strongly match the precise solution. We have compared the solutions we found with some of the solutions provided in the literature to demonstrate the effectiveness of the existing methodologies. A comparison with the other three approaches validates the accuracy and convergence of our offered strategies. Finding the solution to fractional problems is made more straightforward by the hybrid techniques that have been offered. The results acquired are plotted in their graphical form. Graphs and tables are used to show a very strong correlation between the actual and suggested technique solutions. The fractional solutions are used to illustrate the behaviour of different dynamics of the specified physical phenomena. The aforementioned issues can be seen in tabular and graphical form with Maple’s assistance.

In conclusion, the suggested methods were found to be quite precise, efficient, and simple to use. It provides a practical tool for solving nonlinear fractional equations in a variety of academic fields and creates new opportunities for fractional differential equation research. Thus, we can say that the methods presented are sufficiently consistent and applicable to the analysis of a wide range of fractional-order nonlinear mathematical models that aid in explaining the behaviour of complex, highly nonlinear phenomena in significant scientific and engineering domains.

Future Work

In future work, these techniques are expected to be considered for fractional problems in the sense of Atangana–Baleanu derivatives and other partial differential equations employing fractional calculus and fractal theory. We expect that these methods will be used in the future to swiftly and efficiently tackle other fractional differential problems in scientific domains.