High-Precision and Stability-Preserving Approximations to the Time-Fractional Harry Dym Model Using the Tantawy Technique

Abstract

Fractional differential equations provide a flexible framework for describing evolutionary processes in complex media, where nonlocality and memory effects play central roles, and classical integer-order models are frequently inadequate to capture these behaviors. In this work, we revisit the time-fractional Harry Dym (HD) evolution equation in the Caputo sense and construct high-precision analytical approximations using the recently developed Tantawy technique (TT). The method generates a rapidly convergent fractional-power series in time without resorting to perturbative assumptions, auxiliary decomposition polynomials, linearization procedures, or integral transforms, and it remains computationally economical even at high approximation orders. Closed, compact expressions are derived up to the fifth-order approximation and can be systematically extended, yielding excellent agreement with the known exact solution of the classical/integer HD model and with approximations obtained via the new iterative method. A detailed error analysis is carried out by computing absolute and maximum residual errors over the entire computational domain, demonstrating the accuracy, stability, and robustness of the TT for the HD-type fractional nonlinear evolution equation. From a physical perspective, the proposed framework offers a reliable tool for modeling nonlinear wave structures in dispersive media with significant memory and, more generally, for treating a broad class of fractional nonlinear wave equations arising in physics and engineering.

1. Introduction

Fractional calculus (FC) is the study of the properties of integrals and derivatives of non-integer orders [1]. Fractional calculus has recently gained significant prominence and attracted considerable interest from senior researchers due to its efficacy in precisely modeling many physical, chemical, biological, medical, and engineering problems [2,3,4]. Using FC to study different partial differential equations (PDEs) helped researchers understand how complex behaviors work in areas like fluid dynamics [5], electronic circuits [6], and nonlinear waves in a plasma [7,8,9,10], among other fields in physics. The power and importance of fractional calculus lie in its ability to uncover some mysterious behaviors that traditional calculus cannot detect, such as the effect of locality and memory. Furthermore, in many cases, the theoretical results may not be consistent with the scientific results or space observations. One of the reasons may be that the problem is approached from the perspective of traditional fractional calculus. Therefore, after applying fractional calculus, the results obtained often fill the gap, or at least pave the way for researchers to uncover other causes to bridge the gap between theoretical and scientific studies. Numerous pioneering books [11,12,13] in the field of fractional calculus deal with the subject in detail from various aspects, in addition to multiple applications [14,15]. However, after that, various studies and various applications have followed, through modeling many different and real multidisciplinary problems, such as those in chemical engineering [16], biological phenomena [17], bioengineering [18], engineering problems [19], and modeling nonlinear waves in various plasma models [20,21,22,23]. The efficacy of fractional differential equations in modeling diverse real-world problems often yields results that align with the experimental findings. From this perspective, numerous academics have effectively modeled real-world issues by deriving analytical and numerical approximations for evolutionary equations in their fractional representations. This motivates the development and application of novel approaches, such as the Tantawy technique (TT) [20,21,22,23], which aims to overcome the drawbacks of some existing methods by generating highly analytical approximate solutions and higher-order approximations without incurring computational costs.

Numerous researchers have devoted their efforts to developing highly accurate approximations of various fractional differential equations to simulate the natural phenomena they describe. Many analytical and numerical methods have effectively developed analytical and reliable approximations, including the homotopy perturbation method (HPM) [24,25,26], variational iteration method (VIM) [7,8,9], Adomian decomposition method [27,28,29], differential transform method (DTM) [30], new iterative method (NIM) [31,32,33], residual power series method (RPSM) [34,35,36], finite difference method (FDM) [37,38] and many others. Notwithstanding the efficacy of these methods in analyzing diverse differential equations (DEs), there may exist certain restrictions, limitations, or challenges during their implementation, particularly for physical researchers. From this perspective, we will discuss one of the most accurate and straightforward techniques for analyzing various types of FDEs without any restrictions or challenges that researchers might encounter, known as the TT [39,40,41].

In the specific context of nonlinear dispersive waves, the Harry Dym (HD) equation occupies a distinguished position as a prototype integrable model with highly nonlinear dispersion and a rich solution structure, including compactons, cuspons, and other nonclassical waveforms in appropriate settings. The standard form of the integer HD model reads [42]

$${\partial }_t\Psi -{\Psi }^3{\partial }_x^3\Psi =0,$$

(1)

and this model admits the exact solution

$${\Psi }_{\mathrm{exact}}\equiv \Psi ={\left[a-{\displaystyle \frac{3}{2}}\sqrt{b}\left(x+bt\right)\right]}^{{\displaystyle \frac{2}{3}}},$$

(2)

where the parameters “$a\ne 0$” and “$b\ne 0$” and $\Psi \equiv \Psi \left(x,t\right)$. The constant “a” essentially sets a characteristic scale for the amplitude or density profile of the wave. Meanwhile, the constant “$b\ne 0$” governs the propagation speed and direction of the traveling-wave structure.

The first study on the HD model was conducted by Kruskal and Moser in 1974, examining its appearance and behavior in various physical environments [43]. In its integer-order form, the HD model (1) has been connected to inverse scattering theory, to the Korteweg–de Vries (KdV) hierarchy, and to several problems in hydrodynamics and quantum physics, and has therefore attracted extensive analytical and numerical attention [42,43,44]. Many investigations have been conducted on the HD model (1) to analyze and solve it. For instance, Mokhtari [42] solved the HD model (1) using several techniques, including He’s variational iteration method (VIM), the Adomian decomposition method (ADM), and the direct integration method (DIM), to derive the exact traveling wave solutions. Also, the RPSM was applied to analyze model (1) and obtain a highly accurate approximate analytical solution [45]. Moreover, the Darboux transformation was employed to generate soliton solutions to the (2 + 1)-dimensional HD model [44]. Nevertheless, many physical environments in which HD-type dynamics arise, such as complex fluids, disordered media and plasmas with anomalous transport exhibit clear signatures of nonlocality and memory that the classical model cannot capture. This observation has stimulated a growing body of work on fractional variants of HD-like systems, typically formulated using Caputo or related operators, which finally led to the following Caputo time-fractional HD (FHD) model:

$$D_t^p\Psi ={\Psi }^3{\partial }_x^3\Psi , \forall 0<p<1,$$

(3)

having the initial condition (IC):

$$\Psi \left(x,t_0\right)=f\left(x\right),$$

(4)

where $D_t^p$ indicates the time Caputo fractional derivative operator (CFDO) of order $p>0$ to the function $\Psi \equiv \Psi \left(x,t\right)$.

Why pick Caputo over Riemann–Liouville (RL) or Caputo–Fabrizio (CF)? The Caputo fractional derivative emerges as the preferred choice for describing physical systems, especially those involving nonlinear wave dynamics, owing to its seamless compatibility with experimentally measurable initial states (the IC $\Psi \left(x,0\right)$) [46,47]. In contrast to the Riemann–Liouville (RL) operator, which imposes initial conditions of fractional order that defy straightforward physical interpretation, the Caputo formulation accommodates familiar integer-order specifications, such as displacement and velocity profiles, essential for wave problems where hereditary memory effects coexist with conventional boundary constraints [46,48]. Moreover, under the Caputo operator, constant functions vanish, faithfully reflecting the persistence of steady states in nature, a property conspicuously absent in the RL framework. When juxtaposed with the CF derivative, whose exponential kernel delivers the non-singular mitigation of memory influences for gentler decay profiles, the Caputo operator’s power-law kernel more faithfully embodies the protracted, algebraically decaying correlations prevalent in dissipative wave propagation as evidenced in fractional extensions of the KdV hierarchy [46]. Mathematically, for orders $1<\alpha <2$, the Caputo definition takes the concrete form

$${}^{C}D_t^p\Psi (x,t)={\displaystyle \frac{1}{\Gamma (2-p)}}{\int }_0^t{(t-\tau )}^{1-p} {\partial }_{\tau }^2\Psi (x,\tau ) d\tau ,$$

(5)

guaranteeing that emergent solutions honor the intrinsic dissipation of physical waves without the inadvertent regularization imposed by Caputo–Fabrizio’s kernel [49]. Consequently, within the domain of nonlinear wave mechanics, the Caputo derivative affords unmatched fidelity to empirical reality [46].

This fractional setup gets at the heart of the equation’s physical evolution, capturing those memory effects and nonlocal traits that appear consistently in real-world physical systems but are inadequately described by classical integer-order derivatives. By applying the tools of fractional calculus, we aim to uncover new characteristics of the HD model that may have been overlooked in the standard formulation, thereby contributing to the broader understanding of nonlinear phenomena in mathematical physics. Numerous analytical and numerical methods were applied to analyze the FHD model (3), such as the fractional reduced differential transform method (RDTM) [50], ADM [51,52], family of HPM with different transforms [52,53,54,55], the RPSM [56,57], NIM [57], a Lie symmetry technique [58], the finite difference approach [59] and many other methods. However, it is observed that the FHD model (3) has not yet been solved using the TT [20,21,22,23]. Therefore, the subject of our study will be to analyze this equation using the TT and to generate fast, low-computational-cost, high-degree approximations. This technique addresses all limitations and challenges that researchers may encounter when solving fractional differential equations, unlike alternative methods that can pose difficulties, particularly for physicists lacking expertise in complex mathematical techniques. This is the primary motivation for this study, which focuses on applying the TT to a wide range of physical and engineering problems that were previously difficult to analyze due to numerous constraints faced by researchers. We also emphasize that this new technique imposes no overly strict limitations on its application and is a practical, straightforward method for addressing any fractional problem. Moreover, it eliminates the need for decomposition, perturbation, linearization, and other constraints commonly found in similar methods. Now, due to the adaptability of TT and its ability to solve practical issues, it has become an indispensable tool for studying and modeling complex physical and engineering problems. Thus, this study aims to apply this unique technique to understand better the FHD model (3) behavior.

2. Anatomy of the FHD Model Using the TT

In this section, we outline, in a transparent way, how the TT is used to construct rapidly convergent, high-order approximations for the FHD model (3) in the Caputo sense [13,14,15]. We consider the nonlinear model [55].

2.1. Fractional Model and Caputo Operator

We consider the following FHD model in the Caputo sense [55]:

$$D_t^p\Psi \left(x,t\right)-{\Psi }^3\left(x,t\right) {\partial }_x^3\Psi \left(x,t\right)=0, x\in \mathbb{R} \& 0<p<1,$$

(6)

supplemented by the physically meaningful IC

$$\Psi (x,0)\equiv {\Psi }_0\left(x\right)={\left[a-{\displaystyle \frac{3}{2}}x\sqrt{b}\right]}^{{\displaystyle \frac{2}{3}}},$$

(7)

where a and b are non-zero constants.

For $p=1$, the fractional model (6) reduces to the standard HD model (1), and the corresponding exact solution reads

$$\Psi \left(x,t\right)={\left[a-{\displaystyle \frac{3}{2}}\sqrt{b}\left(x+bt\right)\right]}^{{\displaystyle \frac{2}{3}}},$$

(8)

which will later be used as a reference for the integer-order limit.

Generally, the time derivative operator “$D_t^p$” denotes the Caputo fractional derivative of order p ∀$n-1<p<n\in \mathbb{N}$ of a sufficiently smooth function $\Psi \equiv \Psi \left(x,t\right)$ with respect to t, defined by [31]:

$${}^{C}D_t^p\Psi (x,t)={\displaystyle \frac{1}{\Gamma (n-p)}}{\int }_0^t{(t-\tau )}^{n-p-1} {\partial }_{\tau }^n\Psi (x,\tau ) d\tau ,$$

(9)

where $\Gamma \left(·\right)$ indicates the standard Gamma function.

For model (6), the Caputo fractional derivative of order $p\in (0,1]$ of a sufficiently smooth function $\Psi \equiv \Psi \left(x,t\right)$ with respect to t, reads [31]:

$${}^{C}D_t^p\Psi (x,t)={\displaystyle \frac{1}{\Gamma (1-p)}}{\int }_0^t{(t-\tau )}^{-p} {\partial }_{\tau }\Psi (x,\tau ) d\tau .$$

(10)

This definition has two crucial points for our analysis: First, the derivative acts on the time derivative ${\partial }_{\tau }\Psi$ and involves a weakly singular kernel ${(t-\tau )}^{-p}$, which encodes the memory of the past evolution in a power-law fashion. Second, in contrast with the RL derivative, the Caputo operator allows the initial condition to be imposed directly in terms of the function itself, $\Psi (x,0)={\Psi }_0\left(x\right)$, without requiring fractional-order initial data.

Two simple properties of the definition (10) are considered [31]:

$$\begin{array}{cc}\hfill {}^{C}D_t^{p}c& =0, c\in \mathbb{R},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {}^{C}D_t^p{t}^{\alpha }& =\left\{\begin{array}{c}{\displaystyle \frac{\Gamma (\alpha +1)}{\Gamma (\alpha +1-p)}} t^{\alpha -p}, \forall \alpha >0,\\ 0, \forall \alpha =0,\end{array}\right.\hfill \end{array}$$

(12)

Physically, the choice of the Caputo derivative in model (6) is motivated by two complementary reasons. On the one hand, specifying $\Psi (x,0)$ in IC (7) corresponds to fixing an initial wave profile (for instance, a density or velocity field in a plasma), which is directly observable. This is consistent with Caputo, but it would lead to fractional initial conditions for the RL version of the derivative, which do not have a clear experimental interpretation. On the other hand, the kernel ${(t-\tau )}^{-p}$ in Equation (10) captures long-range memory in time through a power-law tail, a feature that is frequently reported in anomalous transport and dispersive processes, whereas kernels of the exponential type, such as those used in Caputo–Fabrizio-type derivatives, tend to model short-range memory and introduce additional time scales that are not needed in the present setting.

2.2. Outline of the TT

In operational terms, the TT may be viewed as a fractional-power series procedure carefully adapted to the structure of the Caputo operator and the nonlinearities in the governing equation. The central idea is to represent the solution $\Psi (x,t)$ as a series in powers of $t^p$ with spatially dependent coefficients ${\Psi }_k\equiv {\Psi }_k\left(x\right)$∀$k=0,1,2,3,\dots$, and then to substitute this expansion into the FHD model (6) together with the prescribed initial data (initial condition) ${\Psi }_0$. By matching coefficients of like powers of $t^p$, one obtains a hierarchy of ordinary differential equations in x for the unknown coefficient functions ${\Psi }_k$∀$k=1,2,3,\dots ,$ (note here that at $k=0$, the IC is known ${\Psi }_0$), which can be solved successively without introducing any perturbations, linearization, or decomposition polynomials, and does not need to apply any transformation to facilitate the solution. This recursive construction yields closed expressions for the coefficients up to any desired order and makes transparent how the fractional order p modifies the temporal scaling of the wave profile. In this way, the TT provides a direct, algorithmic route from the fractional PDE to explicit approximate solutions while keeping the algebraic manipulations at a level that remains tractable and physically interpretable.

The algorithm for this technique can be summarized as follows for analyzing the FHD model (6).

Step (1)

Consider the fractional power series ansatz: In this step, the solution $\Psi \equiv \Psi \left(x,t\right)$ is considered a series in fractional powers of t, accordingly to the following Tantawy ansatz:

$$\Psi \left(x,t\right)=\sum _{k=0}^{\infty }{\Psi }_k t^{kp}, \forall 0<p<1,$$

(13)

with ${\Psi }_0\left(x\right)\equiv \Psi (x,0)$ so that the ansatz (13) automatically satisfies the IC (7) by construction and ${\Psi }_k\equiv {\Psi }_k\left(x\right)$, ∀$k=0,1,2,\dots$. This choice reflects the fact that, under Caputo differentiation, powers $t^{kp}$ are mapped to other powers $t^{(k-1)p}$ according to Equation (12). In other words, the fractional power series structure is preserved by the Caputo operator, which makes the ansatz (13) a natural and efficient ansatz for developing the solution.

The physical meaning of the ansatz (13) is straightforward: the function ${\Psi }_0$ encodes the initial profile, while the higher-order terms ${\Psi }_k{t}^{kp}$ describe successive fractional-time corrections driven by the nonlinearity and dispersion in Equation (6). The fractional parameter p controls how strongly the past influences the present evolution: when $p\to 1$, the series tends to the usual Taylor expansion; when $p<1$, the same expansion is stretched in time, reflecting slower dynamics with memory.

Step (2)

Truncation and residual construction: For an $M^{th}$-order approximation, the ansatz (13) is truncated as follows

$${\Psi }^{\left(M\right)}(x,t)=\sum _{k=0}^M{\Psi }_k{t}^{kp}={\Psi }_0+\sum _{k=1}^M{\Psi }_k t^{kp}, \forall k=1,2,\dots .$$

(14)

We insert relation (14) into Equation (6) in order to construct the truncated residual as follows:

$$\begin{array}{cc}\hfill {\mathcal{R}}^{\left(M\right)}(x,t)& =D_t^p{\Psi }^{\left(M\right)}(x,t)-{\left({\Psi }^{\left(M\right)}(x,t)\right)}^3 {\partial }_x^3{\Psi }^{\left(M\right)}(x,t)\hfill \\ \hfill & =D_t^p\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)-{\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)}^3{\partial }_x^3\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)\hfill \\ \hfill & =I_1-I_2\times I_3.\hfill \end{array}$$

(15)

At this stage, we are simply translating the differential equation into an identity that must hold term-by-term in the fractional power series. The rationale for this truncation is that we are interested in a finite-order approximation m that already offers a high level of accuracy, while keeping the algebraic expressions manageable. In the present work, we go up to $m=5$, which turns out to be a good compromise between compactness and precision.

Step (3)

Application of the Caputo operator to the Tantawy ansatz: Using the two definitions (11) and (12), we compute the Caputo derivative term is explicitly as follows:

$$I_1=D_t^p\Psi (x,t)=D_t^p\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)=\sum _{k=1}^M{\Psi }_k{D}_t^p{t}^{kp}=\sum _{k=1}^M{\Psi }_k L_{k-1}\left(p\right) t^{(k-1)p},$$

(16)

with

$$L_{k-1}\left(p\right)={\displaystyle \frac{\Gamma (kp+1)}{\Gamma \left(\right(k-1)p+1)}}, k=1,2,\dots .$$

Notice that the sum in Equation (16) starts from $k=1$ because the Caputo derivative of the constant term ${\Psi }_0$ vanishes, as in condition (10). This is another reason why Caputo is particularly convenient here: the IC enters only through ${\Psi }_0$, and it does not generate additional singular contributions in the derivative.

It is helpful to move the summation index in relation (16) for later use. Thus, set $k=n+1$, and n can take any number from $0\to \infty$, but in our case we took only the $M^{th}$-order approximation. Thus, the relation (14) can be written as follows:

$$\begin{array}{cc}\hfill I_1& =D_t^p\Psi (x,t)=D_t^p\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)=\sum _{k=1}^M{\Psi }_k{D}_t^p{t}^{kp}\hfill \\ \hfill & =\sum _{n=0}^M{\Psi }_{n+1}{D}_t^p{t}^{\left(n+1\right)p}=\sum _{n=0}^M{\Psi }_{n+1}{L}_n\left(p\right) t^{np},\hfill \end{array}$$

(17)

with

$$L_n\left(p\right)={\displaystyle \frac{\Gamma (\left(n+1\right)p+1)}{\Gamma (np+1)}}, \forall n=0,1,2,\dots .$$

Step (4)

Expansion of the nonlinear term: The second term in Equation (15) involves the cubic nonlinearity $I_2={\left({\sum }_{k=0}^M{\Psi }_k{t}^{kp}\right)}^3$, and the third spatial derivative $I_3={\partial }_x^3\left({\sum }_{k=0}^M{\Psi }_k{t}^{kp}\right)$, which can be expanded as follows:

(i)

The third spatial derivative $I_3$ can expand as

$$I_3={\partial }_x^3\Psi (x,t)={\partial }_x^3\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)=\sum _{k=0}^M{\Psi }_k^{\left(3\right)}{t}^{kp}.$$

(18)

(ii)

The cubic nonlinearity $I_2$ can expand as

$$I_2={\Psi }^3(x,t) ={\left(\sum _{k=0}^M{\Psi }_k{t}^{kp}\right)}^3.$$

(19)

Executing the cubic product explicitly, each term in the expansion derives from a triple product

$${\Psi }_i t^{ip}\times {\Psi }_j t^{jp}\times {\Psi }_{\ell } t^{\ell p}={\Psi }_i{\Psi }_j{\Psi }_{\ell } t^{(i+j+\ell )p}.$$

Summing over all triples $(i,j,\ell )$ with non-negative integer entries leads to

$$I_2={\Psi }^3(x,t) =\sum _{n=0}^M{A}_n\left(x\right) t^{np},$$

(20)

where the coefficient $A_n\left(x\right)$ collects all contributions with the same time exponent $np$. Thus, the expression of $A_n\left(x\right)$ reads

$$A_n\left(x\right)=\sum _{\begin{array}{c}i,j,\ell \ge 0\\ i+j+\ell =n\end{array}}{\Psi }_i {\Psi }_j {\Psi }_{\ell }, \forall n=0,1,2,\dots .$$

(21)

Finally, we have

$$I_2\times I_3={\Psi }^3(x,t) {\partial }_x^3\Psi (x,t)=\left(\sum _{n=0}^M{A}_n\left(x\right) t^{np}\right)\left(\sum _{m=0}^M{\Psi }_m^{\left(3\right)} t^{mp}\right).$$

(22)

The coefficient of $t^{qp}$ in this product arises from all pairs $(n,m)$ with $n+m=q$, since

$$A_n\left(x\right) t^{np} \times {\Psi }_m^{\left(3\right)} t^{mp}=A_n\left(x\right) {\Psi }_m^{\left(3\right)}{t}^{(n+m)p}.$$

Therefore, we may write the cubic nonlinear term $I_2\times I_3$ as follows:

$$I_2\times I_3={\Psi }^3(x,t) {\partial }_x^3\Psi (x,t)=\sum _{q=0}^m{B}_q\left(x\right) t^{qp},$$

(23)

with

$$B_q\left(x\right)=\sum _{\begin{array}{c}n,m\ge 0\\ n+m=q\end{array}}{A}_n\left(x\right) {\Psi }_m^{\left(3\right)}, \forall q=0,1,2,\dots .$$

(24)

Step (5)

Finding the coefficients of equal powers of time: Now, by substituting the obtained series representations (17) and (23) into the governing Equation (6), we obtain

$$\sum _{n=0}^M{\Psi }_{n+1}{L}_n\left(p\right) t^{np}=\sum _{q=0}^{\infty }{B}_q\left(x\right) t^{qp}.$$

(25)

The two series on the left and on the right are written in terms of the same basis functions $t^{np}$, with $n=0,1,2,\dots$. For two such series to be equal for all t in an interval, it is necessary and sufficient that the coefficients of each power $t^{np}$ coincide. Therefore, for every integer $n\ge 0$, we have

$${\Psi }_{n+1}{L}_n\left(p\right) =B_n\left(x\right).$$

(26)

This relation is the basic recurrence that links the coefficient ${\Psi }_{n+1}$ to the previously determined coefficients through $B_n\left(x\right)$.

Step (6)

Explicit recursive formula and the first few coefficients: Solving Equation (26) in ${\Psi }_{n+1}$, the recurrence relation for generating the $\left(n+1\right)$-coefficient ${\Psi }_{n+1}$ can be written in the following beauty formula:

$${\Psi }_{n+1}={\displaystyle \frac{1}{L_n\left(p\right)}}{B}_n\left(x\right)={\displaystyle \frac{\Gamma (np+1)}{\Gamma (\left(n+1\right)p+1)}}{B}_n\left(x\right), \forall n=0,1,2,\dots ,$$

(27)

with

$$B_n\left(x\right)=\sum _{j=0}^n{\Psi }_j^{\left(3\right)}\left[\sum _{\begin{array}{c}{i}_1+i_2+i_3=n-j\\ i_1,i_2,i_3\ge 0\end{array}}{\Psi }_{i_1}{\Psi }_{i_2}{\Psi }_{i_3}\right],$$

(28)

where ${\Psi }_j^{\left(3\right)}\equiv {\Psi }_j^{\left(3\right)}\left(x\right)$, ${\Psi }_{i_1}\equiv {\Psi }_{i_1}\left(x\right)$, ${\Psi }_{i_2}\equiv {\Psi }_{i_2}\left(x\right)$, and ${\Psi }_{i_3}\equiv {\Psi }_{i_3}\left(x\right).$

To see how this works in practice, we begin calculating the first few approximations using this relationship.

(i)

Determination of ${\Psi }_1$: For $n=0$, relation (27) becomes

$${\Psi }_1\left(x\right)={\displaystyle \frac{\Gamma \left(1\right)}{\Gamma (p+1)}} B_0\left(x\right)={\displaystyle \frac{1}{\Gamma (p+1)}} B_0\left(x\right),$$

(29)

To compute $B_0\left(x\right)$, we use relation (24) with $q=0$. The only way to have $n+m=0$ with $n,m\ge 0$ is to take $n=0$ and $m=0$, so, we have

$$B_0\left(x\right)=A_0\left(x\right) {\Psi }_0^{\left(3\right)}.$$

(30)

From relation (21), the only admissible triple $(i,j,\ell )$ with $i+j+\ell =0$ is $(0,0,0)$, and hence, we obtain

$$A_0\left(x\right)={\Psi }_0 {\Psi }_0{\Psi }_0={\Psi }_0^3.$$

(31)

Combining Equations (30) and (31), we obtain

$$B_0\left(x\right)={\Psi }_0^3 {\Psi }_0^{\left(3\right)}.$$

(32)

We can get the same result given in Equation (32) directly by using the relation (28). In

Table 1. Combinatorial sums for each j and computation of $B_0\left(x\right)$ coefficient of $t^0$.

j	Constraint	Partitions of $\left(0-\mathit{j}\right)$	Sum
0	$i_1+i_2+i_3=0$	$\left(0,0,0\right)$	${\Psi }_0^3$

, the computation of the $B_0\left(x\right)$ coefficient of $t^0$ is estimated and then inserted into Equation (33)

$$B_0\left(x\right)=\sum _{j=0}^0{\Psi }_j^{\left(3\right)} \left(\sum _{\begin{array}{c}{i}_1+i_2+i_3=0-0\\ i_1,i_2,i_3\ge 0\end{array}}{\Psi }_{i_1}{\Psi }_{i_2}{\Psi }_{i_3}\right)={\Psi }_0^3{\Psi }_0^{\left(3\right)}.$$

(33)

Inserting relation (33) into Equation (29), we get the value of ${\Psi }_1$ as follows:

$${\Psi }_1={\displaystyle \frac{1}{\Gamma (p+1)}}{\Psi }_0^3 {\Psi }_0^{\left(3\right)}.$$

(34)

(ii)

Determination of ${\Psi }_2$: For $n=1$, relation (27) reads

$${\Psi }_2\left(x\right)={\displaystyle \frac{\Gamma (p+1)}{\Gamma (2p+1)}}{B}_1\left(x\right).$$

(35)

The coefficient $B_1\left(x\right)$ is obtained by setting $q=1$ in Equation (24):

$$B_1\left(x\right)=A_0\left(x\right) {\Psi }_1^{\left(3\right)}+A_1\left(x\right) {\Psi }_0^{\left(3\right)},$$

(36)

since the pairs $(n,m)$ with $n+m=1$ are $(0,1)$ and $(1,0)$. We already know $A_0\left(x\right)$ from Equation (31). Now, to find $A_1\left(x\right)$, we set $n=1$ in relation (21), which leads to

$$A_1\left(x\right)=\sum _{\begin{array}{c}i,j,\ell \ge 0\\ i+j+\ell =1\end{array}}{\Psi }_i{\Psi }_j{\Psi }_{\ell }.$$

(37)

The triples $(i,j,\ell )$ with sum 1 are permutations of $(1,0,0)$. There are three such permutations, so, we have

$$A_1\left(x\right)={\Psi }_1{\Psi }_0{\Psi }_0+{\Psi }_0{\Psi }_1{\Psi }_0+{\Psi }_0{\Psi }_0{\Psi }_1=3 {\Psi }_0^2 {\Psi }_1.$$

(38)

Consequently, we obtain

$$B_1\left(x\right)={\Psi }_0^3 {\Psi }_1^{\left(3\right)}+3 {\Psi }_0^2{\Psi }_1{\Psi }_0^{\left(3\right)}.$$

(39)

We can get the same result given in Equation (39) directly by using relation (28). In

Table 2. Combinatorial sums for each j and computation of $B_1\left(x\right)$ coefficient of $t^p$.

j	Constraint	Partitions of $\left(1-\mathit{j}\right)$	Sum
0	$i_1+i_2+i_3=1$	$3\times \left(1,0,0\right)$	$3{\Psi }_0^2{\Psi }_1$
1	$i_1+i_2+i_3=0$	$\left(0,0,0\right)$	${\Psi }_0^3$

, the computation of $B_1\left(x\right)$ coefficient of $t^p$ is estimated and then inserted into Equation (40)

$$\begin{array}{cc}\hfill B_1\left(x\right)& =\sum _{j=0}^1{\Psi }_j^{\left(3\right)} \left(\sum _{\begin{array}{c}{i}_1+i_2+i_3=1-j\\ i_1,i_2,i_3\ge 0\end{array}}{\Psi }_{i_1}{\Psi }_{i_2}{\Psi }_{i_3}\right)\hfill \\ \hfill & ={\Psi }_0^3 \left[3{\Psi }_0^2 {\Psi }_1\right]+{\Psi }_1^3\left[{\Psi }_0^3\right]=3{\Psi }_0^2 {\Psi }_1{\Psi }_0^{\left(3\right)}+{\Psi }_0^3{\Psi }_1^{\left(3\right)}.\hfill \end{array}$$

(40)

By substituting Equation (40) into Equation (35), we ultimately obtain

$${\Psi }_2={\displaystyle \frac{\Gamma (p+1)}{\Gamma (2p+1)}}\left[{\Psi }_0^3 {\Psi }_1^{\left(3\right)}+3 {\Psi }_0^2{\Psi }_1{\Psi }_0^{\left(3\right)}\right].$$

(41)

(iii)

Determination of ${\Psi }_3$: For $n=2$, relation (27) reads

$${\Psi }_3={\displaystyle \frac{\Gamma (2p+1)}{\Gamma (3p+1)}}{B}_2\left(x\right).$$

(42)

The coefficient $B_2\left(x\right)$ is obtained by setting $n=2$ in Equation (28):

$$\begin{array}{cc}\hfill B_2\left(x\right)& =\sum _{j=0}^2{\Psi }_j^{\left(3\right)}\left(\sum _{\begin{array}{c}{i}_1+i_2+i_3=2-j\\ i_1,i_2,i_3\ge 0\end{array}}{\Psi }_{i_1}{\Psi }_{i_2}{\Psi }_{i_3}\right)\hfill \\ \hfill & ={\Psi }_0^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0\right]+{\Psi }_1^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_1\right]+{\Psi }_2^{\left(3\right)}\left[{\Psi }_0^3\right].\hfill \end{array}$$

(43)

In

Table 3. Combinatorial sums for each j and computation of $B_2\left(x\right)$ coefficient of $t^{2p}$.

j	Constraint	Partitions of $\left(2-\mathit{j}\right)$	Sum
0	$i_1+i_2+i_3=2$	$3\times \left(2,0,0\right)$ & $3\times \left(1,1,0\right)$	$3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0$
1	$i_1+i_2+i_3=1$	$3\times \left(1,0,0\right)$	$3{\Psi }_0^2{\Psi }_1$
2	$i_1+i_2+i_3=0$	$\left(0,0,0\right)$	${\Psi }_0^3$

, the computation of $B_2\left(x\right)$ coefficient of $t^{2p}$ is estimated and then inserted into Equation (43).

By substituting Equation (43) into Equation (42), we ultimately obtain

$${\Psi }_3={\displaystyle \frac{\Gamma (2p+1)}{\Gamma (3p+1)}}\left\{{\Psi }_0^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0\right]+{\Psi }_1^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_1\right]+{\Psi }_2^{\left(3\right)}\left[{\Psi }_0^3\right]\right\}.$$

(44)

(iv)

Determination of ${\Psi }_4$: For $n=3$, relation (27) reads

$${\Psi }_4={\displaystyle \frac{\Gamma (3p+1)}{\Gamma (4p+1)}}{B}_3\left(x\right).$$

(45)

The coefficient $B_3\left(x\right)$ is obtained by setting $n=3$ in Equation (28):

$$\begin{array}{cc}\hfill B_3\left(x\right)& =\sum _{j=0}^3{\Psi }_j^{\left(3\right)} \left(\sum _{\begin{array}{c}{i}_1+i_2+i_3=3-j\\ i_1,i_2,i_3\ge 0\end{array}}{\Psi }_{i_1}{\Psi }_{i_2}{\Psi }_{i_3}\right)\hfill \\ \hfill & ={\Psi }_0^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_3+6{\Psi }_2{\Psi }_1{\Psi }_0+{\Psi }_1^3\right]\hfill \\ \hfill & +{\Psi }_1^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0\right]\hfill \\ \hfill & +{\Psi }_2^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_1\right]+{\Psi }_3^{\left(3\right)}\left[{\Psi }_0^3\right].\hfill \end{array}$$

(46)

In

Table 4. Combinatorial sums for each j and computation of $B_3\left(x\right)$ coefficient of $t^{3p}$.

j	Constraint	Partitions of $\left(3-\mathit{j}\right)$	Sum
0	$i_1+i_2+i_3=3$	$3\times \left(3,0,0\right)$ & $6\times \left(2,1,0\right)$ & $1\times \left(1,1,1\right)$	$3{\Psi }_0^2{\Psi }_3+6{\Psi }_2{\Psi }_1{\Psi }_0+{\Psi }_1^3$
1	$i_1+i_2+i_3=2$	$3\times \left(2,0,0\right)$ & $3\times \left(1,1,0\right)$	$3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0$
2	$i_1+i_2+i_3=1$	$3\times \left(1,0,0\right)$	$3{\Psi }_0^2{\Psi }_1$
3	$i_1+i_2+i_3=0$	$\left(0,0,0\right)$	${\Psi }_0^3$

, the computation of $B_3\left(x\right)$ coefficient of $t^{3p}$ is estimated and then inserted into Equation (46).

By substituting Equation (46) into Equation (45), we ultimately obtain

$${\Psi }_4={\displaystyle \frac{\Gamma (3p+1)}{\Gamma (4p+1)}}\left\{\begin{array}{c}{\Psi }_0^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_3+6{\Psi }_2{\Psi }_1{\Psi }_0+{\Psi }_1^3\right]\\ +{\Psi }_1^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0\right]\\ +{\Psi }_2^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_1\right]+{\Psi }_3^{\left(3\right)}\left[{\Psi }_0^3\right].\end{array}\right\}.$$

(47)

(v)

Determination of ${\Psi }_5$: For $n=4$, relation (27) reads

$${\Psi }_5={\displaystyle \frac{\Gamma (4p+1)}{\Gamma (5p+1)}}{B}_4\left(x\right).$$

(48)

The coefficient $B_4\left(x\right)$ is obtained by setting $n=4$ in Equation (28):

$$\begin{array}{cc}\hfill B_4\left(x\right)& =\sum _{j=0}^4{\Psi }_j^{\left(3\right)} \left(\sum _{\begin{array}{c}{i}_1+i_2+i_3=4-j\\ i_1,i_2,i_3\ge 0\end{array}}{\Psi }_{i_1}{\Psi }_{i_2}{\Psi }_{i_3}\right)\hfill \\ \hfill & ={\Psi }_0^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_4+6{\Psi }_3{\Psi }_1{\Psi }_0+3{\Psi }_2^2{\Psi }_0+3{\Psi }_2{\Psi }_1^2\right]\hfill \\ \hfill & +{\Psi }_1^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_3+6{\Psi }_2{\Psi }_1{\Psi }_0+{\Psi }_1^3\right]+{\Psi }_2^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0\right]\hfill \\ \hfill & +{\Psi }_3^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_1\right]+{\Psi }_4^{\left(3\right)}\left[{\Psi }_0^3\right].\hfill \end{array}$$

(49)

In

Table 5. Combinatorial sums for each j and computation of $B_4\left(x\right)$ coefficient of $t^{3p}$.

j	Constraint	Partitions of $\left(4-\mathit{j}\right)$	Sum
0	$i_1+i_2+i_3=4$	15 Partitions	$\begin{array}{c}3{\Psi }_0^2{\Psi }_4+6{\Psi }_3{\Psi }_1{\Psi }_0\\ +3{\Psi }_2^2{\Psi }_0+3{\Psi }_2{\Psi }_1^2\end{array}$
1	$i_1+i_2+i_3=3$	10 Partitions	$3{\Psi }_0^2{\Psi }_3+6{\Psi }_2{\Psi }_1{\Psi }_0+{\Psi }_1^3$
2	$i_1+i_2+i_3=2$	6 Partitions	$3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0$
3	$i_1+i_2+i_3=1$	3 Partitions	$3{\Psi }_0^2{\Psi }_1$
4	$i_1+i_2+i_3=0$	1 Partition	${\Psi }_0^3$

, the computation of $B_4\left(x\right)$ coefficient of $t^{4p}$ is estimated and then inserted into Equation (49).

By substituting Equation (46) into Equation (45), we ultimately obtain

$${\Psi }_5={\displaystyle \frac{\Gamma (4p+1)}{\Gamma (5p+1)}}\left\{\begin{array}{c}{\Psi }_0^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_4+6{\Psi }_3{\Psi }_1{\Psi }_0+3{\Psi }_2^2{\Psi }_0+3{\Psi }_2{\Psi }_1^2\right]\\ +{\Psi }_1^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_3+6{\Psi }_2{\Psi }_1{\Psi }_0+{\Psi }_1^3\right]+{\Psi }_2^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_2+3{\Psi }_1^2{\Psi }_0\right]\\ +{\Psi }_3^{\left(3\right)}\left[3{\Psi }_0^2{\Psi }_1\right]+{\Psi }_4^{\left(3\right)}\left[{\Psi }_0^3\right].\end{array}\right\}.$$

(50)

Step (7)

Another way to collect the coefficients up to fifth-order (traditional way): By removing the above steps (5) and (6) and substituting both the Caputo operator expansion (16) and the expansion of the nonlinear term into Equation (15), it yields

$$\begin{array}{cc}\hfill {\mathcal{R}}^{\left(m\right)}(x,t)& =D_t^p{\Psi }^{\left(m\right)}(x,t)-{\left({\Psi }^{\left(m\right)}(x,t)\right)}^3 {\partial }_x^3{\Psi }^{\left(m\right)}(x,t)\hfill \\ \hfill & =\sum _{k=1}^m{\Psi }_k L_{k-1}\left(p\right) t^{(k-1)p}\hfill \\ \hfill & -{\left({\Psi }_0+{\Psi }_1{t}^p+{\Psi }_2{t}^{2p}+{\Psi }_3{t}^{3p}+{\Psi }_4{t}^{4p}+{\Psi }_5{t}^{5p}+\dots \right)}^3\hfill \\ \hfill & \times {\partial }_x^3\left({\Psi }_0+{\Psi }_1{t}^p+{\Psi }_2{t}^{2p}+{\Psi }_3{t}^{3p}+{\Psi }_4{t}^{4p}+{\Psi }_5{t}^{5p}+\dots \right).\hfill \end{array}$$

(51)

Afterward, collecting like powers of $t^p$ in ${\mathcal{R}}^{\left(m\right)}(x,t)$ yields

$${\mathcal{R}}^{\left(m\right)}(x,t)=H_0\left(x\right)+H_1\left(x\right) t^p+H_2\left(x\right) t^{2p}+\dots +H_{m-1}\left(x\right) t^{(m-1)p},$$

(52)

with

$$\begin{array}{cc}\hfill H_0\left(x\right)& =L_0\left(p\right){\Psi }_1-{\Psi }_0^3{\Psi }_0^{\left(3\right)},\hfill \\ \hfill H_1\left(x\right)& =L_1\left(p\right){\Psi }_2-3{\Psi }_0^2{\Psi }_0^{\left(3\right)}{\Psi }_1-{\Psi }_0^3{\Psi }_1^{\left(3\right)},\hfill \\ \hfill H_2\left(x\right)& =L_2\left(p\right){\Psi }_3-3{\Psi }_0{\Psi }_0^{\left(3\right)}{\Psi }_1^2-3{\Psi }_0^2{\Psi }_0^{\left(3\right)}{\Psi }_2-3{\Psi }_0^2{\Psi }_1{\Psi }_1^{\left(3\right)}-{\Psi }_0^3{\Psi }_2^{\left(3\right)},\hfill \\ \hfill H_3\left(x\right)& =L_3\left(p\right){\Psi }_4-{\Psi }_1^3{\Psi }_0^{\left(3\right)}-6{\Psi }_0{\Psi }_1{\Psi }_2{\Psi }_0^{\left(3\right)}-3{\Psi }_0^2{\Psi }_0^{\left(3\right)}{\Psi }_3-3{\Psi }_0{\Psi }_1^2{\Psi }_1^{\left(3\right)}\hfill \\ \hfill & -3{\Psi }_0^2{\Psi }_2{\Psi }_1^{\left(3\right)}-3{\Psi }_0^2{\Psi }_1{\Psi }_2^{\left(3\right)}-{\Psi }_0^3{\Psi }_3^{\left(3\right)},\hfill \\ \hfill H_4\left(x\right)& =L_4\left(p\right){\Psi }_5-3{\Psi }_1^2{\Psi }_2{\Psi }_0^{\left(3\right)}-3{\Psi }_0{\Psi }_2^2{\Psi }_0^{\left(3\right)}-6{\Psi }_0{\Psi }_1{\Psi }_3{\Psi }_0^{\left(3\right)}\hfill \\ \hfill & -3{\Psi }_0^2{\Psi }_4{\Psi }_0^{\left(3\right)}-{\Psi }_1^3{\Psi }_1^{\left(3\right)}-6{\Psi }_0{\Psi }_1{\Psi }_2{\Psi }_1^{\left(3\right)}-3{\Psi }_0^2{\Psi }_3{\Psi }_1^{\left(3\right)}\hfill \\ \hfill & -3{\Psi }_0{\Psi }_1^2{\Psi }_2^{\left(3\right)}-3{\Psi }_0^2{\Psi }_2{\Psi }_2^{\left(3\right)}-3{\Psi }_0^2{\Psi }_1{\Psi }_3^{\left(3\right)}-{\Psi }_0^3{\Psi }_4^{\left(3\right)},\hfill \\ \hfill & ⋮,\hfill \end{array}$$

where the notation ${\Psi }^{\left(3\right)}\equiv {\partial }_x^3\Psi$.

Step (8)

Determination of the spatial coefficients: According to the TT, the coefficients of Equation (52) are matching the following conditions

$$H_0\left(x\right)=H_1\left(x\right)=\dots =H_{m-1}\left(x\right)=0,$$

(53)

which forms a triangular (recursive) algebraic system for the unknown functions ${\Psi }_1,\dots ,{\Psi }_m$ in terms of the known initial profile ${\Psi }_0$ and its derivatives [20,21,22,23]. The reason for collecting coefficients in this way is that it disentangles the contributions of different fractional orders. Each $H_i\left(x\right)=0$ can be viewed as a balance condition at order $t^{ip}$ between the time-fractional derivative and the nonlinear dispersive term.

Now, by solving the system (53) in ${\Psi }_1,\dots ,{\Psi }_m$, we ultimately obtain the values of ${\Psi }_1,\dots ,{\Psi }_m$ as functions of initial profile ${\Psi }_0$ and its derivatives as follows:

$$\begin{array}{cc}\hfill {\Psi }_1& =-{\displaystyle \frac{b^{\frac{3}{2}}}{{\Gamma }_1{X}^{\frac{1}{3}}}}, \hfill \\ \hfill {\Psi }_2& =-{\displaystyle \frac{b^3}{2{\Gamma }_2{X}^{\frac{4}{3}}}}, \hfill \\ \hfill {\Psi }_3& ={\displaystyle \frac{b^{\frac{9}{2}}{Q}_0}{2{\Gamma }_1^2{\Gamma }_3{X}^{\frac{7}{3}}}},\hfill \\ \hfill {\Psi }_4& ={\displaystyle \frac{b^6{Q}_1}{2^3{\Gamma }_1^3{\Gamma }_2{\Gamma }_4{X}^{\frac{10}{3}}}}, \hfill \\ \hfill {\Psi }_5& ={\displaystyle \frac{b^{\frac{15}{2}}\left(Q_2-Q_3\right)}{2^3{\Gamma }_1^4{\Gamma }_2^2{\Gamma }_3{\Gamma }_5{X}^{\frac{13}{3}}}},\hfill \end{array}$$

(54)

with

$$\begin{array}{cc}\hfill Q_0& =\left(15{\Gamma }_2-32{\Gamma }_1^2\right),\hfill \\ \hfill Q_1& =\left[443{\Gamma }_1{\Gamma }_2\left(15{\Gamma }_2-32{\Gamma }_1^2\right)+2{\Gamma }_3\left(219{\Gamma }_1^2-38{\Gamma }_2\right)\right],\hfill \\ \hfill Q_2& ={\Gamma }_4\left[1383{\Gamma }_1{\Gamma }_2^2\left(32{\Gamma }_1^2-15{\Gamma }_2\right)+4{\Gamma }_3\left(51{\Gamma }_1^4-123{\Gamma }_2{\Gamma }_1^2+7{\Gamma }_2^2\right)\right],\hfill \\ \hfill Q_3& =257{\Gamma }_1{\Gamma }_2{\Gamma }_3\times \left[443{\Gamma }_1\left(32{\Gamma }_1^2-15{\Gamma }_2\right){\Gamma }_2+2\left(38{\Gamma }_2-219{\Gamma }_1^2\right){\Gamma }_3\right],\hfill \\ \hfill {\Gamma }_k& =\Gamma (kp+1), \forall k=1,2,3,\dots .\hfill \end{array}$$

where $X=\left(a-{\displaystyle \frac{3}{2}}x\sqrt{b}\right)$.

This step is where the structure of the TT becomes particularly advantageous. Because the method works directly with the original equation and the Caputo operator, the computed coefficients ${\Psi }_m$ incorporate both the nonlinear and fractional features of the model without any auxiliary transformations. Moreover, the resulting expressions remain relatively compact, which facilitates their use in subsequent analysis.

Step (9)

Construction of the approximate solution: Once the coefficients ${\Psi }_0,\dots ,{\Psi }_5$ have been determined, the 5th-order Tantawy approximation to the solution of the model (6) is given by

$$\begin{array}{cc}\hfill {\Psi }_T^{\left(5\right)}& ={\Psi }_0+{\Psi }_1{t}^p+{\Psi }_2{t}^{2p}+\dots +{\Psi }_5{t}^{5p}\hfill \\ \hfill & =X^{{\displaystyle \frac{2}{3}}}-{\displaystyle \frac{b^{\frac{3}{2}}}{{\Gamma }_1{X}^{\frac{1}{3}}}}{t}^p-{\displaystyle \frac{b^3}{2{\Gamma }_2{X}^{\frac{4}{3}}}}{t}^{2p}+{\displaystyle \frac{b^{\frac{9}{2}}{Q}_0}{2{\Gamma }_1^2{\Gamma }_3{X}^{\frac{7}{3}}}}{t}^{3p}\hfill \\ \hfill & +{\displaystyle \frac{b^{6}Q1}{2^3{\Gamma }_1^3{\Gamma }_2{\Gamma }_4{X}^{\frac{10}{3}}}}{t}^{4p}+{\displaystyle \frac{b^{\frac{15}{2}}\left(Q_2-Q_3\right)}{2^3{\Gamma }_1^4{\Gamma }_2^2{\Gamma }_3{\Gamma }_5{X}^{\frac{13}{3}}}}{t}^{5p}+\dots .\hfill \end{array}$$

(55)

In the present work, explicit expressions are derived up to $m=5$, yielding a fifth-order approximation that remains compact enough to be expressed in closed form, unlike the corresponding higher-order series obtained by other methods. The approximation (55) is entirely congruent with the solution derived by the HPM as given in Ref. [55]. We emphasize that the HPM results do not always need to be consistent with the TT; rather, that depends on the form and structure of the fractional differential equation to be studied.

Step (10)

Consistency checks and further remarks: Using the NIM algorithm [57], the following approximate solution up to the second-order is obtained:

$$\begin{array}{cc}\hfill {\Psi }_N^{\left(2\right)}& ={\Psi }_0+{\Psi }_1+{\Psi }_2+\dots \hfill \\ \hfill & =X^{{\displaystyle \frac{2}{3}}}-{\displaystyle \frac{b^{\frac{3}{2}}}{{\Gamma }_1{X}^{\frac{1}{3}}}}{t}^p-{\displaystyle \frac{b^3}{2{\Gamma }_2{X}^{\frac{4}{3}}}}{t}^{2p}+{\displaystyle \frac{15b^{\frac{9}{2}}{\Gamma }_2}{2{\Gamma }_1^2{\Gamma }_3{X}^{\frac{7}{3}}}}{t}^{3p}\hfill \\ \hfill & -{\displaystyle \frac{19b^6{\Gamma }_3}{2{\Gamma }_1^3{\Gamma }_4{X}^{\frac{10}{3}}}}{t}^{4p}+{\displaystyle \frac{7b^{\frac{15}{2}}{\Gamma }_4}{2{\Gamma }_1^4{\Gamma }_5{X}^{\frac{13}{3}}}}{t}^{5p}.\hfill \end{array}$$

(56)

A third-order expression using NIM can also be generated, but its cumbersome length renders it unsuitable for inclusion in the main text. This stands in sharp contrast to the TT, through which a fifth-order solution is derived in a notably compact form that fits naturally within the manuscript. In addition, the same strategy can be employed to obtain even higher-order approximations, which can be incorporated into the analysis without difficulty.

Now, we can perform two tests to verify the high accuracy and efficiency of the generated approximations using the TT. First, in the case of limit $p\to 1$ (the integer case), the derived approximations (55) reduce to the exact solution (8) of the classical HD model (1). This confirms that the fractional solution generated by TT is consistent with the established integer-order theorem. Second, the NIM approximation (56) will be used in to compare absolute error and global maximum residual error (MRE) against the approximations (55).

Step (11)

The steps for analyzing any fractional differential equation using the TT are summarized in the flowchart as illustrated in Figure 1.

3. Results and Discussion

In this section, we assess the accuracy, convergence, and stability behavior of the generated approximations (55) constructed for the FHD model (6). The analysis is carried out for representative fractional orders $p\in (0,1]$ and over the computational domain $\left(x,t\right)\in [0,2]\times [0,1]$, with a particular emphasis on the benchmark case $p=1$, where the exact (integer-order) solution (8) is available [42]. We focus on the TT-truncations (14) (notably the third-, fourth- and fifth-order forms) and compare them with the NIM third-order approximation (56) [57]. The central diagnostics used throughout are the pointwise absolute error and the residual of the governing equation, which together provide a robust measure of both accuracy and stability for fractional evolution equations.

• Accuracy diagnostics: For the integer case $\left(p=1\right)$, the absolute error for the generated approximations (55) using the TT (third-order approximation ${\Psi }_T^{\left(3\right)}$, fourth-order approximation ${\Psi }_T^{\left(4\right)}$, & fifth-order approximation ${\Psi }_T^{\left(5\right)}$) and the approximation (56) using the the NIM (third-order approximation ${\Psi }_N^{\left(3\right)}$) as compared to the exact reference solution (8), are defined by

  $$\begin{array}{cc}\hfill E_{{\Psi }_T^{\left(3\right)}}& =\left|{\Psi }_T^{\left(3\right)}-{\Psi }_{\mathrm{exact}}\right|,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill E_{{\Psi }_T^{\left(4\right)}}& =\left|{\Psi }_T^{\left(4\right)}-{\Psi }_{\mathrm{exact}}\right|,\hfill \\ \hfill E_{{\Psi }_T^{\left(5\right)}}& =\left|{\Psi }_T^{\left(5\right)}-{\Psi }_{\mathrm{exact}}\right|,\hfill \end{array}$$

  (57)

  $$\begin{array}{cc}\hfill E_{{\Psi }_N^{\left(3\right)}}& =\left|{\Psi }_N^{\left(3\right)}-{\Psi }_{\mathrm{exact}}\right|.\hfill \end{array}$$

  (58)
  Since in fractional models the quality of the generated approximation is also tied to how well it satisfies the differential equation itself, we further introduce the following (residual) defect:

  $${\mathcal{E}}_{\mathrm{res}}(x,t)=\left|D_t^p{\Psi }_{\mathrm{app}}-{\left({\Psi }_{\mathrm{app}}\right)}^3 {\partial }_x^3{\Psi }_{\mathrm{app}}\right|,$$

  and its maximum over the computational domain $\Omega =[0,2]\times [0,0.1]$, reads

  $$L_{\mathrm{app}}=\underset{(x,t)\in \Omega }{max} {\mathcal{E}}_{\mathrm{res}}(x,t),$$

  (59)

  where ${\Psi }_{\mathrm{app}}$ indicates any generated approximation (${\Psi }_T^{\left(3\right)}$, ${\Psi }_T^{\left(4\right)}$, ${\Psi }_T^{\left(5\right)}$, ${\Psi }_N^{\left(3\right)}$, etc.) for the FHD model (6) and ${\Psi }_{\mathrm{exact}}$ represents the exact reference solution for the integer case given in Equation (8). Here, the quantity (59) indicates the global MRE, which is widely used as a stability/consistency indicator in fractional computations. Accordingly, for the generated approximations (55) and (56), the global MRE for the approximation ${\Psi }_T^{\left(3\right)}$, ${\Psi }_T^{\left(4\right)}$, ${\Psi }_T^{\left(5\right)}$, and ${\Psi }_N^{\left(3\right)}$ can be written as follows:

  $$\begin{array}{cc}\hfill L_{T\left(3\right)}& =\underset{(x,t)\in \Omega }{max} {\mathcal{E}}_{\mathrm{res}}(x,t)=\underset{\Omega }{max}\left|D_t^p{\Psi }_T^{\left(3\right)}-{\left({\Psi }_T^{\left(3\right)}\right)}^3{\partial }_x^3{\Psi }_T^{\left(3\right)}\right|,\hfill \end{array}$$

  $$\begin{array}{cc}\hfill L_{T\left(4\right)}& =\underset{(x,t)\in \Omega }{max} {\mathcal{E}}_{\mathrm{res}}(x,t)=\underset{\Omega }{max}\left|D_t^p{\Psi }_T^{\left(4\right)}-{\left({\Psi }_T^{\left(4\right)}\right)}^3{\partial }_x^3{\Psi }_T^{\left(4\right)}\right|,\hfill \\ \hfill L_{T\left(5\right)}& =\underset{(x,t)\in \Omega }{max} {\mathcal{E}}_{\mathrm{res}}(x,t)=\underset{\Omega }{max}\left|D_t^p{\Psi }_T^{\left(5\right)}-{\left({\Psi }_T^{\left(5\right)}\right)}^3{\partial }_x^3{\Psi }_T^{\left(5\right)}\right|,\hfill \end{array}$$

  (60)

  $$\begin{array}{cc}\hfill L_{N\left(3\right)}& =\underset{(x,t)\in \Omega }{max} {\mathcal{E}}_{\mathrm{res}}(x,t)=\underset{\Omega }{max}\left|D_t^p{\Psi }_N^{\left(3\right)}-{\left({\Psi }_N^{\left(3\right)}\right)}^3{\partial }_x^3{\Psi }_N^{\left(3\right)}\right|.\hfill \end{array}$$

  (61)
• Effect of the fractional order: Figure 2 illustrates the generated approximation (55) (truncated at third order ${\Psi }_T^{\left(3\right)}$) for different values of the fractional order p, namely $p=0.25$, $p=0.5$, and $p=1$. A clear and physically consistent trend is observed: decreasing p slows the time evolution and produces a more pronounced “memory-dominated” response, as expected in Caputo-type fractional dynamics [13,14,15]. In contrast, as $p\to 1$, the fractional dynamics continuously approaches the classical HD model behavior, and the approximation (55) surface becomes nearly indistinguishable from the integer-order solution at $p=1$ (see also Figure 3), which provides an important internal consistency check [42,55]. This smooth recovery of the classical limit is essential for fractional generalizations intended for physical modeling [13,14,15].

• Matching with exact solution: To quantify the agreement of the generated approximations (55) with the exact solution (8), Figure 3 compares the generated approximation (55) (third-order truncation ${\Psi }_T^{\left(3\right)}$) with the exact profile at $p=1$. It is observed that the two curves are visually indistinguishable at the scale of the figure over the considered domain, indicating that even the third-order TT-truncation already captures the dominant nonlinear dispersive dynamics with high fidelity. This visual agreement is reinforced by the numerical error levels reported in

  Table 6. A comparison between the absolute error of the third-order approximation (56) using the NIM and the third-approximation (55) using the TT at $t=0.1$.

  x	${\mathit{\Psi }}_{\mathit{exact}}$	${\mathit{\Psi }}_{\mathit{N}}^{\left(3\right)}$	${\mathit{\Psi }}_{\mathit{T}}^{\left(3\right)}$	${\mathit{E}}_{{\mathit{\Psi }}_{\mathit{N}}^{\left(3\right)}}$	${\mathit{E}}_{{\mathit{\Psi }}_{\mathit{T}}^{\left(3\right)}}$
  0	2.45645	2.45655	2.45645	0.102821 × ${10}^{-3}$	0.147234 × ${10}^{-6}$
  0.1	2.39222	2.39233	2.39222	0.11232 × ${10}^{-3}$	0.167408 × ${10}^{-6}$
  0.2	2.32712	2.32724	2.32712	0.123125 × ${10}^{-3}$	0.191328 × ${10}^{-6}$
  0.3	2.2611	2.26123	2.2611	0.135478 × ${10}^{-3}$	0.219888 × ${10}^{-6}$
  0.4	2.1941	2.19425	2.1941	0.149681 × ${10}^{-3}$	0.254249 × ${10}^{-6}$
  0.5	2.12605	2.12622	2.12606	0.16611 × ${10}^{-3}$	0.295929 × ${10}^{-6}$
  0.6	2.05691	2.05709	2.05691	0.185241 × ${10}^{-3}$	0.346948 × ${10}^{-6}$
  0.7	1.98658	1.98678	1.98658	0.207683 × ${10}^{-3}$	0.41002 × ${10}^{-6}$
  0.8	1.91498	1.91521	1.91498	0.234219 × ${10}^{-3}$	0.488848 × ${10}^{-6}$
  0.9	1.84202	1.84228	1.84202	0.265877 × ${10}^{-3}$	0.58857 × ${10}^{-6}$
  1	1.76758	1.76788	1.76758	0.304022 × ${10}^{-3}$	0.716443 × ${10}^{-6}$

  , where the pointwise absolute errors remain extremely small across the sampled spatial points for $t=0.1$.
• Estimated error and comparison: The comparison between the absolute error of the third-order approximation (56) ${\Psi }_N^{\left(3\right)}$ using the NIM and the third-order approximation (55) ${\Psi }_T^{\left(3\right)}$ using the TT is numerically computed at $t=0.1$ as illustrated in Table 5. Additionally, the absolute error of the third-order approximations (56) ${\Psi }_N^{\left(3\right)}$ and (55) ${\Psi }_T^{\left(3\right)}$ is graphically compared with each other as evident in Figure 4a,b, respectively. One can notice that the accuracy of the third-order approximation ${\Psi }_T^{\left(3\right)}$ (55) using the TT surpasses the third-order approximation ${\Psi }_N^{\left(3\right)}$ (56) using the NIM at the same order. Moreover, the absolute error of the third-, fourth-, and fifth-order approximations (55) using the TT at $t=0.1$ is numerically estimated as illustrated in

  Table 7. A comparison between the absolute error $E_{{\Psi }_T^{\left(m\right)}}$ of the third-, fourth-, and fifth-order approximations (55) using the TT at $t=0.1$.

  x	${\mathit{\Psi }}_{\mathit{exact}}$	${\mathit{\Psi }}_{\mathit{T}}^{\left(3\right)}$	${\mathit{\Psi }}_{\mathit{T}}^{\left(4\right)}$	${\mathit{\Psi }}_{\mathit{T}}^{\left(5\right)}$	${\mathit{E}}_{{\mathit{\Psi }}_{\mathit{T}}^{\left(3\right)}}$	${\mathit{E}}_{{\mathit{\Psi }}_{\mathit{T}}^{\left(4\right)}}$	${\mathit{E}}_{{\mathit{\Psi }}_{\mathit{T}}^{\left(5\right)}}$
  0	2.45645	2.45645	2.45645	2.45645	0.147234 × ${10}^{-6}$	0.036887 × ${10}^{-7}$	0.100054 × ${10}^{-9}$
  0.1	2.39222	2.39222	2.39222	2.39222	0.167408 × ${10}^{-6}$	0.0435789 × ${10}^{-7}$	0.122819 × ${10}^{-9}$
  0.2	2.32712	2.32712	2.32712	2.32712	0.191328 × ${10}^{-6}$	0.0518295 × ${10}^{-7}$	0.152003 × ${10}^{-9}$
  0.3	2.2611	2.2611	2.2611	2.2611	0.219888 × ${10}^{-6}$	0.0620893 × ${10}^{-7}$	0.189801 × ${10}^{-9}$
  0.4	2.1941	2.1941	2.1941	2.1941	0.254249 × ${10}^{-6}$	0.074967 × ${10}^{-7}$	0.239295 × ${10}^{-9}$
  0.5	2.12605	2.12606	2.12605	2.12605	0.295929 × ${10}^{-6}$	0.091295 × ${10}^{-7}$	0.304891 × ${10}^{-9}$
  0.6	2.05691	2.05691	2.05691	2.05691	0.346948 × ${10}^{-6}$	0.112228 × ${10}^{-7}$	0.392972 × ${10}^{-9}$
  0.7	1.98658	1.98658	1.98658	1.98658	0.41002 × ${10}^{-6}$	0.139394 × ${10}^{-7}$	0.512966 × ${10}^{-9}$
  0.8	1.91498	1.91498	1.91498	1.91498	0.488848 × ${10}^{-6}$	0.175125 × ${10}^{-7}$	0.679055 × ${10}^{-9}$
  0.9	1.84202	1.84202	1.84202	1.84202	0.58857 × ${10}^{-6}$	0.222824 × ${10}^{-7}$	0.913035 × ${10}^{-9}$
  1	1.76758	1.76758	1.76758	1.76758	0.716443 × ${10}^{-6}$	0.287566 × ${10}^{-7}$	1.2492 × ${10}^{-9}$

  . Furthermore, the absolute error $E_{{\Psi }_T^{\left(m\right)}}$ of the third-, fourth-, and fifth-order approximations (55) using the TT is graphically compared with each other as elucidated in Figure 5. It is observed that the absolute error $E_{{\Psi }_T^{\left(m\right)}}$ decreases with increasing the order of approximation, i.e., $E_{{\Psi }_T^{\left(5\right)}}<E_{{\Psi }_T^{\left(4\right)}}<E_{{\Psi }_T^{\left(3\right)}}$, which confirms that the accuracy of the generated approximation increases with increasing the order of approximation. This, in turn, enhances the efficiency of the used approach and confirms its convergence and stability throughout the entire study domain. This behavior is consistent with the algebraic structure of the TT coefficients, which remain manageable at higher orders, unlike many iterative expansions where terms grow quickly in complexity and may degrade numerical robustness.
  Furthermore, the global MRE according to relation (61) for the third-order approximation (56) using the NIM and third-, fourth-, and fifth-order approximations (55) using the TT, through the domain $\Omega \equiv \left(x,t\right)\equiv \left[0,2\right]\times \left[0,0.1\right]$ is estimated as follows:

  $$\begin{array}{cc}\hfill L_{T\left(3\right)}& =0.00265179, \mathrm{at} \left(x,t\right)=\left(2,0.1\right),\hfill \\ \hfill L_{T\left(4\right)}& =0.000546935, \mathrm{at} \left(x,t\right)=\left(2,0.1\right),\hfill \\ \hfill L_{T\left(5\right)}& =0.000104494, \mathrm{at} \left(x,t\right)=\left(2,0.1\right),\hfill \\ \hfill L_{N\left(3\right)}& =10.0346, \mathrm{at} \left(x,t\right)=\left(1.98988,0.0974508\right),\hfill \end{array}$$
  Thus, even at the same nominal order, the NIM-truncation exhibits a much larger residual than the TT-truncation, while the fourth- and fifth-order TT correction drive the residual down to the $\mathcal{O}\left({10}^{-4}\right)$ level. The monotone decrease of $L_{T\left(m\right)}$ is the signature of a stable, rapidly convergent construction of the generated approximations [39,40,41]. In practical terms, these residual levels imply that the TT approximation not only matches the benchmark solution at $p=1$ but also remains consistent with the governing nonlinear operator, which is crucial when the same approximation is used for parameter sweeps or as an initial guess in hybrid symbolic–numerical procedures [60].
• Physical interpretation and computational implications: Physically, the observed dependence on the fractional order p reflects the role of history effects in dispersive nonlinear media, which for smaller p delays the temporal development of the wave profile and effectively spreads the influence of past states over the present dynamics, in line with fractional relaxation mechanisms [61,62]. From a computational standpoint, the present results highlight a key advantage of the TT: high-order corrections can be derived in closed form while retaining compact expressions (cf. (55)), which makes the method attractive for fast evaluation, symbolic manipulation, and residual-based validation [39,40,41]. In contrast, the NIM series becomes less reliable in the present problem as indicated by its large residual at the third order, which can be interpreted as slower convergence and weaker stability for this nonlinear dispersive fractional operator [32,57]. It should be emphasized that the TT is not necessarily superior to the NIM in every case, as each problem is independent. While it may have been superior to the NIM in the current problem and some other studied problems [39], the accuracy might differ when analyzing other problems. However, in general, the method is easily applicable to all types of fractional differential equations, posing no particular challenge to researchers. These features make the TT a strong candidate for the analysis of other fractional nonlinear evolution equations arising in plasma physics and nonlinear wave theory, where both accuracy and stability are mandatory.

4. Conclusions

In this work, we have examined the time-fractional Harry Dym (FHD) equation in the Caputo sense as a model problem for nonlinear dispersive waves in media with long-range temporal memory. Starting from the classical/integer HD model and its known exact traveling-wave solution, we introduced a fractional generalization that preserves the physical meaning of the initial conditions and makes explicit the role of the fractional order in modulating the dynamics. On this basis, the Tantawy technique (TT) was formulated as a direct fractional-power series method, free of perturbation assumptions, decomposition polynomials, linearization, or integral transforms, and was implemented to generate compact approximations of arbitrary order approximation. In this study, we generated approximations up to a fifth-order for the FHD model that remain amenable to further extension. By comparing these approximations with the exact integer-order solution and with results obtained via the new iterative method (NIM), we demonstrated that the TT yields smaller absolute and maximum residual errors over the computational domain and exhibits favorable stability properties, capturing both the global evolution and the local structure of the wave profile as the order of the fractional derivative varies. At matching orders, it was found that Tantawy’s approximations hug the exact integer-order solution far more tightly than the NIM, with absolute errors dropping from ${10}^{-6}$ (third order) to ${10}^{-9}$ (fifth order). Moreover, increasing the approximation order leads to a monotonic decrease in the residual, indicating rapid convergence and structural robustness of the TT. Due to its balance between analytical transparency and numerical reliability, the Tantawy technique can be employed in practical applications where FHD-type models arise, such as nonlinear wave propagation in complex fluids, anomalous transport in plasmas, and other dispersive systems with pronounced memory effects. On this foundation, future work may explore systematic extensions to coupled systems, variable-order fractional operators, and stochastic perturbations, as well as the integration of the TT with numerical time-stepping schemes to treat problems with more intricate boundary conditions and external forcing.

Future work: Several natural extensions arise from the present study:

(i)

In this study, the time fractional one-dimensional HD model (6) has been investigated. However, with incoming works, it would be worthwhile to extend this remarkably precise technique to analyze the following $\left(2+1\right)$-dimensional HD model [63]:

$$D_t^p\Psi +{\Psi }^3{\partial }_x^3\Psi +{\Psi }^3{\partial }_y^3\Psi +3{\Psi }^3{\partial }_{xyy}\Psi +3{\Psi }^3{\partial }_{xxy}\Psi =0.$$

Additionally, the study can be extended to include multi-dimensional fractional structures [64].

(ii)

A second line of work worth pursuing involves applying the TT to fractional evolutionary equations with richer structures: think higher-order nonlinearities, stronger dispersion terms, or added higher-order dissipation that crop up in plasma waves, condensed matter flows, or even complex fluids. We have already seen it work well with fractional Burgers equations, easily handling shock fronts and dissipation; it also deals with multi-dimensional diffusion problems, following unusual transport patterns; and it effectively manages fourth-order fractional Cahn–Hilliard models, clearly identifying phase boundaries and spinodal decomposition. At the same time, using the technique on fractional third-order dispersion KdV-type equations has shown that it can effectively model solitary and cnoidal wave patterns in nonthermal plasmas. Based on these results, it is natural to envisage systematic studies of coupled fractional systems, such as higher-order dissipative fractional Burgers-type equations, higher-order dispersion and nonlinearity KdV-type equations, fractional Zakharov–Kuznetsov models, and fractional nonlinear Schrödinger-type equations, where multiple mechanisms of nonlinearity, dispersion, dissipation, diffusion, and memory act simultaneously [65,66].

(iii)

By developing hybrid TT-spectral or TT-finite-difference schemes for large-scale simulations, compact analytical approximations can serve as highly accurate initial predictors.