Hyperbolic Non-Polynomial Spline Approach for Time-Fractional Coupled KdV Equations: A Computational Investigation

Abstract

The time-fractional coupled Korteweg–De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations’ symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method’s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method’s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov–Galerkin method using error norms. The results underscore the CHNPSM’s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems.

1. Introduction

The convergence of fractional calculus with machine learning has unlocked new potential for tackling complex engineering problems, providing a versatile framework that incorporates integrals and derivatives of arbitrary orders [1]. FDEs have become essential tools across various scientific and engineering fields, including dynamical systems, control engineering, signal and image processing, heat and mass transfer, electromagnetism, viscoelastic material dynamics, and anomalous transport [2,3,4]. Their importance lies in their ability to capture evolutionary processes through infinitesimal generating properties, often revealing underlying symmetrical connections within physical systems. Various formulations of fractional derivatives, including the Riemann–Liouville, Grünwald–Letnikov, Hadamard, Weyl, Caputo, and conformable fractional derivatives, have provided researchers with diverse mathematical instruments for accurate modeling and analysis [5,6,7,8], many of which preserve or exploit symmetry in complex systems. The growing interest in numerical solutions of FDEs has further highlighted the importance of understanding symmetries in modeling intricate phenomena across a range of disciplines.

Breakthrough studies have developed innovative methods to solve FDEs while respecting their inherent symmetrical properties. Several notable studies have employed different numerical methods to solve time-fractional differential equations, preserving the symmetry properties of the systems. For example, the B-spline method was used in [9] to solve the time-fractional gas dynamics equation, while the extended tanh-function approximation method was applied in [10] to solve the time-fractional sine-Gordon and Klein–Gordon equations. In [11], a non-polynomial spline approach was utilized to address the time-fractional Schrödinger equation. Similarly, the work in [12] implemented fractional derivatives with non-polynomial functions to solve the time-fractional Korteweg–De Vries (KdV) equation. The spectral collocation method with Legendre polynomials was applied in [13] to solve multi-space fractional orders of the KdV and Kuramoto–Sivashinsky equations, ensuring the preservation of the system’s symmetrical structure. Additionally, ref. [14] focused on solving time-fractional stochastic partial differential equations using the finite element method. The authors of [15,16,17,18,19] used numerical methods for solving FDEs, many of which are designed to ensure that the numerical solutions preserve the fundamental symmetry of the governing equations. These works represent significant strides in advancing our understanding and ability to effectively solve FDEs numerically, paving the way for further exploration and applications in diverse scientific and engineering domains.

One of the most important and renowned set of time-fractional differential equations is the time-fractional coupled Korteweg–De Vries equations. The TFCKdVEs stand as a cornerstone in mathematical modeling, particularly known for capturing the intricate interaction between two long waves characterized by different dispersion relations. These equations possess inherent symmetries, such as those related to time-fractional scaling and wave interactions, which play a critical role in their behavior and solutions. The significance of TFCKdVEs spans various fields in engineering and science, profoundly impacting areas like the continuum limit of the Fermi–Pasta–Ulam problem, water wave dynamics, quantum field theory, plasma physics, and hydrodynamics. By preserving the symmetries of these equations, numerical methods can provide more accurate and reliable insights into the dynamics of extended waves featuring disparate dispersion properties, as well as fundamental phenomena like water wave cycles, elucidating the principles governing wave behavior [20]. This paper introduces a generalized form of the time-coupled KdV equations with conformable fractional-order derivatives,

$$T_t^{{\alpha }_1}u(r,t)-c_1{\displaystyle \frac{{\partial }^{3}u(r,t)}{\partial r^3}}-c_{2}u(r,t){\displaystyle \frac{\partial u(r,t)}{\partial r}}-c_{3}v(r,t){\displaystyle \frac{\partial u(r,t)}{\partial r}}=f(r,t),$$

(1)

$$T_t^{{\alpha }_2}v(r,t)-c_1{\displaystyle \frac{{\partial }^{3}v(r,t)}{\partial r^3}}-c_{2}u(r,t){\displaystyle \frac{\partial v(r,t)}{\partial r}}=g(r,t),$$

(2)

considering the initial and boundary conditions

$$\begin{array}{cc}\hfill & u(r,0)=u_0\left(r\right),v(r,0)=v_0\left(r\right),r\in \Omega ,\hfill \\ \hfill & u(r,t)=u_1(r,t),u(r,t)=u_2(r,t),u_r(r,t)=u_3(r,t),r\in \partial \Omega ,t>0,\hfill \\ \hfill & v(r,t)=v_1(r,t),v(r,t)=v_2(r,t),v_r(r,t)=v_3(r,t),r\in \partial \Omega ,t>0,\hfill \end{array}$$

(3)

where $c_1$, $c_2$, and $c_3$ denote parameters, while $f(r,t)$ and $g(r,t)$ represent source functions. The domains are indicated by $\Omega$ and their boundaries $\partial \Omega$. TFCKdVEs have been tackled numerically by various methods, showing the versatility and applicability of different mathematical techniques. These include the q-homotopy analysis transform method [21], Petrov–Galerkin method [22], radial basis function collocation method [23], spectral collection method [24], Adomian decomposition method [25], meshless spectral method [26], Crank–Nicolson finite difference method [27], several different kernels [28], as well as the double Laplace transform and decomposition method [20]. Each of these approaches offers unique advantages and insights into the behavior and solutions of TFCKdVEs, contributing to a deeper understanding of their dynamics and applications. The motivation behind this work stems from the pressing need for accurate and efficient numerical methods to solve the time-fractional coupled Korteweg–De Vries equations. These equations play a crucial role in modeling various real-world phenomena such as wave propagation and the dynamics of shallow water waves on viscous fluids. However, their complex nature and coupled dynamics present significant challenges in obtaining reliable numerical solutions. Existing numerical methods often exhibit limitations in terms of accuracy, stability, or computational efficiency, highlighting the necessity for novel approaches capable of addressing these shortcomings. The novelty of our approach lies in the introduction of the CHNPSM, a specialized technique developed to efficiently solve TFCKdVEs, offering enhanced accuracy and stability compared to existing methods. Unlike traditional numerical methods, the CHNPSM integrates hyperbolic non-polynomial spline functions and conformable derivatives, providing a unique and innovative framework for approximating solutions to TFCKdVEs. By combining these elements with first-derivative finite differences, the CHNPSM offers enhanced accuracy and stability in capturing the intricate dynamics of coupled wave phenomena. Moreover, investigating the convergence order and stability of the CHNPSM yields crucial insights into its performance, setting it apart from traditional methods in terms of reliability and efficiency.

The following sections offer a detailed exploration of the CHNPSM. Section 2 delves into the construction of a non-polynomial spline function with a hyperbolic term, emphasizing their application in solving the TFCKdVEs. In Section 3, we analyze the truncation errors of the CHNPSM to assess its precision and convergence characteristics. The CHNPSM is used in practice to solve TFCKdVEs, as shown in Section 4, supported by detailed examples. A thorough stability analysis is presented in Section 5, where we evaluate the robustness of the proposed numerical scheme under various circumstances. Section 6 highlights the numerical results, interpreting their significance and offering a critical assessment of the CHNPSM’s efficiency. Lastly, Section 8 consolidates our findings, reflecting on the broader implications of this study. The goal of this systematic approach is to highlight the ingenuity, reliability, and potential significance of the CHNPSM in enhancing numerical methods for addressing TFCKdVEs.

2. Constructing the Hyperbolic Non-Polynomial Spline

In this section, we outline the development of a non-polynomial spline using hyperbolic functions, which serves as a crucial element of the numerical methods for solving equations characterized by complex dynamics or properties. We establish the foundation for constructing the spline, setting the stage for a detailed analysis and practical application in the subsequent sections. The discretization process employs a uniform grid over both spatial and temporal domains. Spatial points are defined as $r\tau =\tau h$, where $\tau =0,1,\dots ,M$, while temporal points are denoted by $tn=n\left(k\right)$ for $n=0,1,\dots ,N$. The parameter $h={\displaystyle \frac{b-a}{M}}$ indicates the uniform spatial step size, and $k={\displaystyle \frac{T}{N}}$ specifies the uniform temporal step size. This approach of uniform discretization provides a systematic method for working with evenly distributed data points.

$$\begin{array}{c}\hfill {\mathfrak{S}}_{\tau ,n}\left(r_{\tau },t_n\right)=a_{\tau }^{n}sinhk\left(r-r_{\tau }\right)+b_{\tau }^{n}coshk\left(r-r_{\tau }\right)+c_{\tau }^n\left(r-r_{\tau }\right)+d_{\tau }^n.\end{array}$$

(4)

Let us consider the HNPSM function ${\mathfrak{S}}_{\tau ,n}(r_{\tau },t_n)$, which can be described approximately with $u_{\tau }^n=u(r_{\tau },t_n)$ and $v_{\tau }^n=v(r_{\tau },t_n)$. Here, k denotes the frequency parameter associated with the trigonometric functions. The coefficients $a_{\tau }^n,b_{\tau }^n,c_{\tau }^n$, and $d_{\tau }^n$ are initially unspecified. The spline functions are then subjected to particular conditions in order to obtain these coefficients, This usually entails making sure the spline and its derivatives match the function values at specific spatial and temporal points or maintain continuity.

$$\begin{array}{c}\hfill \begin{array}{c}{\mathfrak{S}}_{\tau ,n}\left(r_{\tau },t_n\right)=u_{\tau }^n,{\mathfrak{S}}_{\tau ,n}\left(r_{\tau +1},t_n\right)=u_{\tau +1}^n,\hfill \\ \\ {{\mathfrak{S}}_{\tau ,n}}^{\left(3\right)}\left(r_{\tau },t_n\right)={u_{\tau }^n}^{\left(3\right)}=M_{\tau }^n,{{\mathfrak{S}}_{\tau ,n}}^{\left(3\right)}\left(r_{\tau +1},t_n\right)={u_{\tau +1}^n}^{\left(3\right)}=M_{\tau +1}^n.\hfill \end{array}\end{array}$$

(5)

Utilizing the conditions stated in (5) along with the formulation given in (4), we proceed to derive the following results:

$$\begin{array}{cc}\hfill a_{\tau }& ={\displaystyle \frac{M_{\tau }^n}{k^3}},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill b_{\tau }& =-{\displaystyle \frac{coth\left(\eta \right)M_{\tau }^n-\csc \mathrm{h}\left(\eta \right)M_{\tau +1}^n}{k^3}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill c_{\tau }& ={\displaystyle \frac{-k^3{u}_{\tau }^n+k^3{u}_{\tau +1}^n-tanh\left(\frac{\eta }{2}\right)\left(M_{\tau }^n+M_{\tau +1}^n\right)}{h{k}^3}},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill d_{\tau }& ={\displaystyle \frac{k^3{u}_{\tau }^n+coth\left(\eta \right)M_{\tau }^n-\csc \mathrm{h}\left(\eta \right)M_{\tau +1}^n}{k^3}}.\hfill \end{array}$$

(9)

Recasting the relationship $\eta =hk$ in the context of the continuity equation ${{\mathfrak{S}}_{\tau ,n}}^{\prime }(r_{\tau },t_n)={{\mathfrak{S}}_{\tau -1,n}}^{\prime }(r_{\tau },t_n)$ reveals a direct correlation between the spatial and temporal variables, which can be expressed as follows:

$$k{a}_{\tau }+c_{\tau }=k{a}_{\tau -1}cosh\left(\eta \right)+k{b}_{\tau -1}sinh\left(\eta \right)+c_{\tau -1}.$$

(10)

After substituting the variables obtained from (6) to (9) into (10) and simplifying the expression through consolidation, we derive the following result:

$$\begin{array}{c}\hfill -u_{\tau -1}^n+2u_{\tau }^n-u_{\tau +1}^n={\displaystyle \frac{1}{k^3}}\left({\beta }_1{M}_{\tau -1}^n+{\beta }_2{M}_{\tau }^n+{\beta }_3{M}_{\tau +1}^n\right).\end{array}$$

(11)

Similarly, using ${{\mathfrak{S}}_{\tau ,n}}^{\left(3\right)}\left(r_{\tau },t_n\right)={v_{\tau }^n}^{\left(3\right)}=P_{\tau }^n$ and ${{\mathfrak{S}}_{\tau ,n}}^{\left(3\right)}\left(r_{\tau +1},t_n\right)={v_{\tau +1}^n}^{\left(3\right)}=P_{\tau +1}^n$, we have

$$\begin{array}{c}\hfill -v_{\tau -1}^n+2v_{\tau }^n-v_{\tau +1}^n={\displaystyle \frac{1}{k^3}}\left({\beta }_1{P}_{\tau -1}^n+{\beta }_2{P}_{\tau }^n+{\beta }_3{P}_{\tau +1}^n\right),\end{array}$$

(12)

where

$$\begin{array}{c}\hfill {\beta }_1=sinh\left(\eta \right)+coth\left(\eta \right)-cosh\left(\eta \right)coth\left(\eta \right),\end{array}$$

$$\begin{array}{c}\hfill {\beta }_2=-sinh\left(\eta \right)-\csc \mathrm{h}\left(\eta \right)+cosh\left(\eta \right)coth\left(\eta \right),\end{array}$$

$$\begin{array}{c}\hfill {\beta }_3=-sinh\left(\eta \right)-\csc \mathrm{h}\left(\eta \right)+cosh\left(\eta \right)coth\left(\eta \right).\end{array}$$

3. Assessment of Error with Truncation

This section introduces the scheme’s (11) local truncation error at the ${\tau }^{th}$ step. Additionally, applying Taylor expansion to calculate the unknown values ${\beta }_{\tau },\tau =1,2,3$ yields the following:

$$\begin{array}{cc}\hfill T_{\tau }& =-u_{\tau -1}^n+2u_{\tau }^n-u_{\tau +1}^n-{\displaystyle \frac{1}{k^3}}({\beta }_1{M}_{\tau -1}^n+{\beta }_2{M}_{\tau }^n+{\beta }_3{M}_{\tau +1}^n)\hfill \\ \hfill & =-u_{\tau -1}^n+2u_{\tau }^n-u_{\tau +1}^n-{\displaystyle \frac{{\beta }_1}{k^3}}{M}_{\tau -1}^n-{\displaystyle \frac{{\beta }_2}{k^3}}{M}_{\tau }^n-{\displaystyle \frac{{\beta }_3}{k^3}}{M}_{\tau +1}^n.\hfill \end{array}$$

(13)

By employing Taylor expansion and summing the derivative coefficients, we derive

$$\begin{array}{cc}\hfill T_{\tau }& =-h^2{u_{\tau }^n}^{\left(2\right)}+\left(-{\displaystyle \frac{{\beta }_1}{{\eta }^3}}-{\displaystyle \frac{{\beta }_2}{{\eta }^3}}-{\displaystyle \frac{{\beta }_3}{{\eta }^3}}\right)h^3{u_{\tau }^n}^{\left(3\right)}+\left(-{\displaystyle \frac{1}{12}}+{\displaystyle \frac{{\beta }_1}{{\eta }^3}}-{\displaystyle \frac{{\beta }_3}{{\eta }^3}}\right)h^4{u_{\tau }^n}^{\left(4\right)}\hfill \\ \hfill & +\left(-{\displaystyle \frac{{\beta }_1}{2{\eta }^3}}-{\displaystyle \frac{{\beta }_3}{2{\eta }^3}}\right)h^5{u_{\tau }^n}^{\left(5\right)}+\left(-{\displaystyle \frac{1}{360}}+{\displaystyle \frac{{\beta }_1}{6{\eta }^3}}-{\displaystyle \frac{{\beta }_3}{6{\eta }^3}}\right)h^6{u_{\tau }^n}^{\left(6\right)}+\cdots .\hfill \end{array}$$

(14)

Expression (14) can be used to obtain the following matrices by equating the coefficients of ${u_{\tau }^n}^{\left(\rho \right)}$ for $\rho =3,4,6$:

$$\left[\begin{array}{ccc}{\displaystyle \frac{-1}{{\eta }^3}}& {\displaystyle \frac{-1}{{\eta }^3}}& -{\displaystyle \frac{-1}{{\eta }^3}}\\ {\displaystyle \frac{1}{{\eta }^3}}& 0& {\displaystyle \frac{-1}{{\eta }^3}}\\ {\displaystyle \frac{1}{6{\eta }^3}}& 0& {\displaystyle \frac{-1}{6{\eta }^3}}\end{array}\right]\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\\ {\beta }_3\end{array}\right]=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{12}}\\ {\displaystyle \frac{1}{36}}\end{array}\right].$$

(15)

Upon solving the system given in (15) with the Gaussian elimination method, we arrive at the following results:

$$\begin{array}{c}\hfill {\beta }_1=-{\displaystyle \frac{13{\eta }^3}{120}},{\beta }_2={\displaystyle \frac{3{\eta }^3}{10}},{\beta }_3=-{\displaystyle \frac{23{\eta }^3}{120}}.\end{array}$$

The local truncation error can be written as follows after the coefficients have been substituted,

$$T_{\tau }={\displaystyle \frac{153}{2400}}{h}^5{u_{\tau }^n}^{\left(5\right)}\mathcal{O}(k^{2-\alpha }+h^5),$$

and the equations represented by (11) and (12) can be formulated as follows:

$$\begin{array}{c}\hfill u_{\tau -1}^n-2u_{\tau }^n+u_{\tau +1}^n=h^3\left({\displaystyle \frac{13}{120}}{M}_{\tau -1}^n-{\displaystyle \frac{3}{10}}{M}_{\tau }^n+{\displaystyle \frac{23}{120}}{M}_{\tau +1}^n\right),\end{array}$$

(16)

$$\begin{array}{c}\hfill v_{\tau -1}^n-2v_{\tau }^n+v_{\tau +1}^n=h^3\left({\displaystyle \frac{13}{120}}{P}_{\tau -1}^n-{\displaystyle \frac{3}{10}}{P}_{\tau }^n+{\displaystyle \frac{23}{120}}{P}_{\tau +1}^n\right).\end{array}$$

(17)

4. Application of Time-Fractional Coupled KdV Equations with Conformable HNPSM

The section introduces an innovative approach to tackling time-fractional coupled KdV equations using HNPSM in conjunction with conformable derivatives. It begins by defining the conformable derivative and highlighting its key properties. Then, seamlessly integrates these properties with finite difference techniques and hyperbolic non-polynomial splines.

Definition 1 

([8]). For a function $u:\left[0,\infty \right]\to \mathbb{R}$, the conformable time-fractional derivative $T_t^{\alpha }u\left(t\right)$ is defined as follows:

$$T_t^{\alpha }u\left(t\right)=\underset{\rho \to \infty }{lim}{\displaystyle \frac{u(t+\rho t^{1-\alpha })-u\left(t\right)}{\rho }}.$$

(18)

Lemma 1. 

At α point $t>0$, let $\alpha \in (0,1]$ and $u,v$ be α-differentiable.

1. 

$T_t^{\alpha }(Au+Bv)=A{T}_t^{\alpha }u+B{T}_t^{\alpha }v$ for $A,B\in \mathbb{R}$.

2. 

$T_t^{\alpha }\left(t^A\right)=A{t}^{A-\alpha }$ for all $A\in \mathbb{R}$.

3. 

$T_t^{\alpha }u\left(t\right)=0$ if $u\left(t\right)$ is a constant function.

4. 

$T_t^{\alpha }\left(uv\right)=u\left(T_t^{\alpha }v\right)+v\left(T_t^{\alpha }u\right)$.

5. 

$T_t^{\alpha }\left({\displaystyle \frac{u}{v}}\right)={\displaystyle \frac{v\left(T_t^{\alpha }u\right)-u\left(T_t^{\alpha }v\right)}{v^2}}$.

6. 

$T_t^{\alpha }u\left(t\right)=t^{1-\alpha }{\displaystyle \frac{\partial u\left(t\right)}{\partial t}},$ should $u\left(t\right)$ exhibit differentiability.

Corollary 1. 

In the context of the finite difference scheme, let the temporal step size be denoted by k. The partial derivatives ${\displaystyle \frac{\partial u}{\partial t}}$ and ${\displaystyle \frac{\partial v}{\partial t}}$ are defined as follows:

$${\displaystyle \frac{\partial u}{\partial t}}\approx {\displaystyle \frac{u_{\tau }^{n+1}-u_{\tau }^n}{k}},\mathrm{where}u\left(r,t\right)=u_{\tau }^n,$$

(19)

$${\displaystyle \frac{\partial v}{\partial t}}\approx {\displaystyle \frac{v_{\tau }^{n+1}-v_{\tau }^n}{k}},\mathrm{where}v\left(r,t\right)=v_{\tau }^n.$$

(20)

Using Lemma 1, properties (6), and equations (19) and (20), then

$$T_t^{{\alpha }_1}u\left(r,t\right)\approx t^{1-{\alpha }_1}{\displaystyle \frac{\partial u}{\partial t}}={\omega }_1{\displaystyle \frac{u_{\tau }^{n+1}-u_{\tau }^n}{k}}.$$

(21)

$$T_t^{{\alpha }_2}v\left(r,t\right)\approx t^{1-{\alpha }_2}{\displaystyle \frac{\partial v}{\partial t}}={\omega }_2{\displaystyle \frac{v_{\tau }^{n+1}-v_{\tau }^n}{k}},$$

(22)

where ${\omega }_1=t^{1-{\alpha }_1}$ and ${\omega }_2=t^{1-{\alpha }_2}$.

Now, expressing $M_{\tau }^n$ and $P_{\tau }^n$ in the form derived from (1) and (2), after using (21) and (22), we have

$$M_{\tau }^n={\displaystyle \frac{{\omega }_1}{c_1}}{\displaystyle \frac{u_{\tau }^{n+1}-u_{\tau }^n}{r}}-{\displaystyle \frac{c_2}{c_1}}{u}_{\tau }^n{\displaystyle \frac{u_{\tau +1}^n-u_{\tau }^n}{h}}-{\displaystyle \frac{c_3}{c_1}}{v}_{\tau }^n{\displaystyle \frac{u_{\tau +1}^n-u_{\tau }^n}{h}}-{\displaystyle \frac{1}{c_1}}{f}_{\tau }^n,$$

(23)

$$P_{\tau }^n={\displaystyle \frac{{\omega }_2}{c_1}}{\displaystyle \frac{v_{\tau }^{n+1}-v_{\tau }^n}{r}}-{\displaystyle \frac{c_2}{c_1}}{u}_{\tau }^n{\displaystyle \frac{v_{\tau +1}^n-v_{\tau }^n}{h}}-{\displaystyle \frac{1}{c_1}}{g}_{\tau }^n.$$

(24)

Substituting $\tau -1$ and $\tau +1$ in place of $\tau$ within (23) and (24) produces

$$\begin{array}{c}\hfill M_{\tau -1}^n={\displaystyle \frac{{\omega }_1}{c_1}}{\displaystyle \frac{u_{\tau -1}^{n+1}-u_{\tau -1}^n}{k}}-{\displaystyle \frac{c_2}{c_1}}{\displaystyle \frac{u_{\tau }^n-u_{\tau -1}^n}{h}}{u}_{\tau -1}^n-{\displaystyle \frac{c_3}{c_1}}{v}_{\tau -1}^n{\displaystyle \frac{u_{\tau }^n-u_{\tau -1}^n}{h}}-{\displaystyle \frac{1}{c_1}}{f}_{\tau -1}^n,\end{array}$$

(25)

$$\begin{array}{c}\hfill M_{\tau +1}^n={\displaystyle \frac{{\omega }_1}{c_1}}{\displaystyle \frac{u_{\tau +1}^{n+1}-u_{\tau +1}^n}{k}}-{\displaystyle \frac{c_2}{c_1}}{u}_{\tau }^n{\displaystyle \frac{u_{\tau +2}^n-u_{\tau +1}^n}{h}}-{\displaystyle \frac{c_3}{c_1}}{v}_{\tau +1}^n{\displaystyle \frac{u_{\tau +2}^n-u_{\tau +1}^n}{h}}-{\displaystyle \frac{1}{c_1}}{f}_{\tau +1}^n,\end{array}$$

(26)

$$\begin{array}{c}\hfill P_{\tau -1}^n={\displaystyle \frac{{\omega }_2}{c_1}}{\displaystyle \frac{v_{\tau -1}^{n+1}-v_{\tau -1}^n}{k}}-{\displaystyle \frac{c_2}{c_1}}{u}_{\tau -1}^n{\displaystyle \frac{v_{\tau }^n-v_{\tau -1}^n}{h}}-{\displaystyle \frac{1}{c_1}}{g}_{\tau -1}^n,\end{array}$$

(27)

and

$$\begin{array}{c}\hfill P_{\tau +1}^n={\displaystyle \frac{{\omega }_2}{c_1}}{\displaystyle \frac{v_{\tau +1}^{n+1}-v_{\tau +1}^n}{k}}-{\displaystyle \frac{c_2}{c_1}}{u}_{\tau +1}^n{\displaystyle \frac{v_{\tau +2}^n-v_{\tau +1}^n}{h}}-{\displaystyle \frac{1}{c_1}}{g}_{\tau +1}^n.\end{array}$$

(28)

Expressions (16) and (17) can be represented through (23)–(28), leading to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{h^3}}\left(u_{\tau -1}^n-2u_{\tau }^n+u_{\tau +1}^n\right)\hfill \\ \hfill & =\left({\displaystyle \frac{13{\omega }_1}{120c_1}}{\displaystyle \frac{u_{\tau -1}^{n+1}-u_{\tau -1}^n}{k}}-{\displaystyle \frac{13c_2}{120c_1}}{u}_{\tau -1}^n{\displaystyle \frac{u_{\tau }^n-u_{\tau -1}^n}{h}}-{\displaystyle \frac{13c_3}{120c_1}}{v}_{\tau -1}^n{\displaystyle \frac{u_{\tau }^n-u_{\tau -1}^n}{h}}-{\displaystyle \frac{13}{120c_1}}{f}_{\tau -1}^n\right)\hfill \\ \hfill & -\left({\displaystyle \frac{3{\omega }_1}{10c_1}}{\displaystyle \frac{u_{\tau }^{n+1}-u_{\tau }^n}{k}}-{\displaystyle \frac{3c_2}{10c_1}}{u}_{\tau }^n{\displaystyle \frac{u_{\tau +1}^n-u_{\tau }^n}{h}}-{\displaystyle \frac{3c_3}{10c_1}}{v}_{\tau }^n{\displaystyle \frac{u_{\tau +1}^n-u_{\tau }^n}{h}}-{\displaystyle \frac{3}{10c_1}}{f}_{\tau }^n\right)\hfill \\ \hfill & +\left({\displaystyle \frac{23{\omega }_1}{120c_1}}{\displaystyle \frac{u_{\tau +1}^{n+1}-u_{\tau +1}^n}{k}}-{\displaystyle \frac{23c_2}{120c_1}}{u}_{\tau }^n{\displaystyle \frac{u_{\tau +2}^n-u_{\tau +1}^n}{h}}-{\displaystyle \frac{23c_3}{120c_1}}{v}_{\tau +1}^n{\displaystyle \frac{u_{\tau +2}^n-u_{\tau +1}^n}{h}}-{\displaystyle \frac{23}{120c_1}}{f}_{\tau +1}^n\right),\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{h^3}}\left(v_{\tau -1}^n-2v_{\tau }^n+v_{\tau +1}^n\right)\hfill \\ \hfill & =\left({\displaystyle \frac{13{\omega }_2}{120c_1}}{\displaystyle \frac{v_{\tau }^{n+1}-v_{\tau }^n}{k}}-{\displaystyle \frac{13c_2}{120c_1}}{u}_{\tau }^n{\displaystyle \frac{v_{\tau +1}^n-v_{\tau }^n}{h}}-{\displaystyle \frac{13}{120c_1}}{g}_{\tau }^n\right)\hfill \\ \hfill & -\left({\displaystyle \frac{3{\omega }_2}{10c_1}}{\displaystyle \frac{v_{\tau }^{n+1}-v_{\tau }^n}{k}}-{\displaystyle \frac{3c_2}{10c_1}}{u}_{\tau }^n{\displaystyle \frac{v_{\tau +1}^n-v_{\tau }^n}{h}}-{\displaystyle \frac{3}{10c_1}}{g}_{\tau }^n\right)\hfill \\ \hfill & +\left({\displaystyle \frac{23{\omega }_2}{120c_1}}{\displaystyle \frac{v_{\tau +1}^{n+1}-v_{\tau +1}^n}{k}}-{\displaystyle \frac{23c_2}{120c_1}}{u}_{\tau +1}^n{\displaystyle \frac{v_{\tau +2}^n-v_{\tau +1}^n}{h}}-{\displaystyle \frac{23}{120c_1}}{g}_{\tau +1}^n\right).\hfill \end{array}$$

(30)

After some simplification and collection in (29) and (30), we obtain

$$\begin{array}{c}\hfill -{\xi }_1{u}_{\tau -1}^{n+1}+{\xi }_2{u}_{\tau }^{n+1}-{\xi }_3{u}_{\tau +1}^{n+1}=-{\psi }_{\tau }{u}_{\tau -1}^n+{\varphi }_{\tau }{u}_{\tau }^n-{\phi }_{\tau }{u}_{\tau +1}^n-{\vartheta }_{\tau }{u}_{\tau +2}^n\\ \hfill -{\displaystyle \frac{13}{120c_1}}{f}_{\tau -1}^n+{\displaystyle \frac{3}{10c_1}}{f}_{\tau }^n-{\displaystyle \frac{23}{120c_1}}{f}_{\tau +1}^n,\end{array}$$

(31)

and

$$\begin{array}{c}\hfill -{\xi }_1^{*}{v}_{\tau -1}^{n+1}+{\xi }_2^{*}{v}_{\tau }^{n+1}-{\xi }_3^{*}{v}_{\tau +1}^{n+1}=-{\psi }_{\tau }^{*}{v}_{\tau -1}^n+{\varphi }_{\tau }^{*}{v}_{\tau }^n-{\phi }_{\tau }^{*}{v}_{\tau +1}^n-{\vartheta }_{\tau }^{*}{v}_{\tau +2}^n\\ \hfill -{\displaystyle \frac{13}{120c_1}}{g}_{\tau -1}^n+{\displaystyle \frac{3}{10c_1}}{g}_{\tau }^n-{\displaystyle \frac{23}{120c_1}}{g}_{\tau +1}^n,\end{array}$$

(32)

where

$$\begin{array}{cc}\hfill {\xi }_1& ={\displaystyle \frac{13{\omega }_1}{120k{c}_1}},{\xi }_2={\displaystyle \frac{3{\omega }_1}{10k{c}_1}},{\xi }_3={\displaystyle \frac{23{\omega }_1}{120k{c}_1}},\hfill \\ \hfill {\xi }_1^{*}& ={\displaystyle \frac{13{\omega }_2}{120k{c}_1}},{\xi }_2^{*}={\displaystyle \frac{3{\omega }_2}{10k{c}_1}},{\xi }_3^{*}={\displaystyle \frac{23{\omega }_2}{120k{c}_1}},\hfill \\ \hfill {\psi }_{\tau }& ={\displaystyle \frac{1}{h^3}}+{\displaystyle \frac{13{\omega }_1}{120k{c}_1}}-{\displaystyle \frac{13c_2}{120h{c}_1}}{u}_{\tau -1}^n-{\displaystyle \frac{13c_3}{120h{c}_1}}{v}_{\tau -1}^n,\hfill \\ \hfill {\psi }_{\tau }^{*}& ={\displaystyle \frac{1}{h^3}}+{\displaystyle \frac{13{\omega }_2}{120k{c}_1}}-{\displaystyle \frac{13c_2}{120{hc}_1}}{u}_{\tau -1}^n,\hfill \\ \hfill {\varphi }_{\tau }& ={\displaystyle \frac{2}{h^3}}+{\displaystyle \frac{3{\omega }_1}{10k{c}_1}}-{\displaystyle \frac{13c_2}{120h{c}_1}}{u}_{\tau -1}^n-{\displaystyle \frac{13c_3}{120h{c}_1}}{v}_{\tau -1}^n-{\displaystyle \frac{3c_2}{10h{c}_1}}{u}_{\tau }^n-{\displaystyle \frac{3c_3}{10h{c}_1}}{v}_{\tau }^n,\hfill \\ \hfill {\varphi }_{\tau }^{*}& ={\displaystyle \frac{2}{h^3}}-{\displaystyle \frac{13c_2}{120{hc}_1}}{u}_{\tau -1}^n+{\displaystyle \frac{3{\omega }_2}{10k{c}_1}}-{\displaystyle \frac{3c_2}{10h{c}_1}}{u}_{\tau }^n,\hfill \\ \hfill {\phi }_{\tau }& ={\displaystyle \frac{1}{h^3}}+{\displaystyle \frac{23{\omega }_1}{120k{c}_1}}-{\displaystyle \frac{3c_2}{10h{c}_1}}{u}_{\tau }^n-{\displaystyle \frac{3c_3}{10h{c}_1}}{v}_{\tau }^n-{\displaystyle \frac{23c_2}{120h{c}_1}}{u}_{\tau }^n-{\displaystyle \frac{23c_3}{120h{c}_1}}{v}_{\tau +1}^n,\hfill \\ \hfill {\phi }_{\tau }^{*}& ={\displaystyle \frac{1}{h^3}}+{\displaystyle \frac{23{\omega }_2}{120k{c}_1}}-{\displaystyle \frac{3c_2}{10h{c}_1}}{u}_{\tau }^n-{\displaystyle \frac{23c_2}{120h{c}_1}}{u}_{\tau +1}^n,\hfill \\ \hfill {\vartheta }_{\tau }& ={\displaystyle \frac{23c_2}{120h{c}_1}}{u}_{\tau }^n+{\displaystyle \frac{23c_3}{120h{c}_1}}{v}_{\tau +1}^n,{\vartheta }_{\tau }^{*}={\displaystyle \frac{23c_2}{120h{c}_1}}{u}_{\tau +1}^n.\hfill \end{array}$$

Each of (31) and (32) contains $n+1$ unknowns and $n-1$ equations. Two further equations that are derived from the initial and boundary conditions are needed to solve these systems.

5. CHNPSM Stabilities

The stability analysis of two numerical techniques for solving time-fractional coupled KdV equations, represented by (31) and (32), is the main topic of this section. The behavior of these schemes under various conditions is thoroughly examined using the Fourier stability principle, which offers important insights into how well they approximate the solutions of the time-fractional linked KdV equations. The following formulas are the outcome of the Fourier stability study:

$$u_{\tau }^n={\mu }^n{e}^{i\Omega h\tau },$$

(33)

and

$$v_{\tau }^n={\varsigma }^n{e}^{i\Sigma h\tau }.$$

(34)

In this context, i represents the imaginary unit, defined as $i=\sqrt{-1}$, while $\Omega$ and $\Sigma$ refer to the actual spatial wave numbers. Equation (31) is derived by substituting the expression from (33) with the linear representation of the nonlinear term.

$$\begin{array}{cc}\hfill & -{\xi }_1{\mu }^{n+1}{e}^{i\Omega h\left(\tau -1\right)}+{\xi }_2{\mu }^{n+1}{e}^{i\Omega h\tau }-{\xi }_3{\mu }^{n+1}{e}^{i\Omega h\left(\tau +1\right)}\hfill \\ \hfill & =-{\psi }_{\tau }{\mu }^n{e}^{i\Omega h\left(\tau -1\right)}+{\varphi }_{\tau }{\mu }^n{e}^{i\Omega h\tau }-{\phi }_{\tau }{\mu }^n{e}^{i\Omega h\left(\tau +1\right)}-{\vartheta }_{\tau }{\mu }^n{e}^{i\Omega h\left(\tau +2\right)}.\hfill \end{array}$$

(35)

After simplifying and dividing both sides of (35) by ${\mu }^n{e}^{i\Omega h\tau }$, we have

$$\begin{array}{c}\hfill \mu ={\displaystyle \frac{-{\psi }_{\tau }{e}^{-i\Omega h}+{\varphi }_{\tau }-{\phi }_{\tau }{e}^{i\Omega h}-{\vartheta }_{\tau }{e}^{2i\Omega h}}{-{\xi }_1{e}^{-i\Omega h}+{\xi }_2-{\xi }_3{e}^{i\Omega h}}}.\end{array}$$

(36)

Applying Euler’s formula and performing a few additional manipulations, we obtain

$$\begin{array}{c}\hfill \mu ={\displaystyle \frac{M_1+i{M}_2}{M_3}},\end{array}$$

(37)

where

$$\begin{array}{cc}\hfill M_1& ={\psi }_{\tau }{\xi }_1+{\varphi }_{\tau }{\xi }_2+{\vartheta }_{\tau }{\xi }_3-\left({\varphi }_{\tau }{\xi }_1+{\psi }_{\tau }{\xi }_2+{\phi }_{\tau }{\xi }_2-{\phi }_{\tau }{\xi }_3\right)cos\left(\Omega h\right)\hfill \\ \hfill & -\left({\varphi }_{\tau }{\xi }_3-{\phi }_{\tau }{\xi }_1+{\vartheta }_{\tau }{\xi }_2\right)cos\left(2\Omega h\right)+\left({\psi }_{\tau }{\xi }_3+{\vartheta }_{\tau }{\xi }_1\right)cos\left(3\Omega h\right),\hfill \\ \hfill M_2& =\left({\varphi }_{\tau }{\xi }_1+{\psi }_{\tau }{\xi }_2+{\psi }_{\tau }{\xi }_3-{\phi }_{\tau }{\xi }_2\right)sin\left(\Omega h\right)-\left({\varphi }_{\tau }{\xi }_3+{\phi }_{\tau }{\xi }_1+{\vartheta }_{\tau }{\xi }_2\right)sin\left(2\Omega h\right)\hfill \\ \hfill & +\left({\phi }_{\tau }{\xi }_3-{\vartheta }_{\tau }{\xi }_1\right)sin\left(3\Omega h\right)+{\vartheta }_{\tau }{\xi }_{3}sin\left(4\Omega h\right),\hfill \\ \hfill M_3& ={{\xi }_2}^2+\left({{\xi }_1}^2-2{\xi }_2{\xi }_3\right)\left(2cos{\left(\Omega h\right)}^2-1\right)+\left(2{\xi }_1{\xi }_3-2{\xi }_1{\xi }_2\right)cos\left(\Omega h\right)\hfill \\ \hfill & +{{\xi }_3}^2\left(2{\left(2cos{\left(\Omega h\right)}^2-1\right)}^2-1\right).\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill \left|\mu \right|=\sqrt{{\left({\displaystyle \frac{M_1}{M_3}}\right)}^2+{\left({\displaystyle \frac{M_2}{M_3}}\right)}^2}.\end{array}$$

(38)

Consequently, the numerical scheme (31) is stable if $\sqrt{{\left({\displaystyle \frac{M_1}{M_3}}\right)}^2+{\left({\displaystyle \frac{M_2}{M_3}}\right)}^2}\le 1$.

• Similarly, substituting (34) into (32), we have

$$\begin{array}{c}\hfill \varsigma ={\displaystyle \frac{N_1+i{N}_2}{N_3}},\end{array}$$

(39)

where

$$\begin{array}{cc}\hfill N_1& ={\psi }_{\tau }^{*}{\xi }_1^{*}+{\varphi }_{\tau }^{*}{\xi }_2^{*}+{\vartheta }_{\tau }^{*}{\xi }_3^{*}-\left({\varphi }_{\tau }^{*}{\xi }_1^{*}+{\psi }_{\tau }^{*}{\xi }_2^{*}+{\phi }_{\tau }^{*}{\xi }_2^{*}-{\phi }_{\tau }^{*}{\xi }_3^{*}\right)cos\left(\Sigma h\right)\hfill \\ \hfill & -\left({\varphi }_{\tau }^{*}{\xi }_3^{*}-{\phi }_{\tau }^{*}{\xi }_1^{*}+{\vartheta }_{\tau }^{*}{\xi }_2^{*}\right)cos\left(2\Sigma h\right)+\left({\psi }_{\tau }^{*}{\xi }_3^{*}+{\vartheta }_{\tau }^{*}{\xi }_1^{*}\right)cos\left(3\Sigma h\right),\hfill \\ \hfill N_2& =\left({\varphi }_{\tau }^{*}{\xi }_1^{*}+{\psi }_{\tau }^{*}{\xi }_2^{*}+{\psi }_{\tau }^{*}{\xi }_3^{*}-{\phi }_{\tau }^{*}{\xi }_2^{*}\right)sin\left(\Sigma h\right)-\left({\varphi }_{\tau }^{*}{\xi }_3^{*}+{\phi }_{\tau }^{*}{\xi }_1^{*}+{\vartheta }_{\tau }^{*}{\xi }_2^{*}\right)sin\left(2\Sigma h\right)\hfill \\ \hfill & +\left({\phi }_{\tau }^{*}{\xi }_3^{*}-{\vartheta }_{\tau }^{*}{\xi }_1^{*}\right)sin\left(3\Sigma h\right)+{\vartheta }_{\tau }^{*}{\xi }_3^{*}sin\left(4\Sigma h\right),\hfill \\ \hfill N_3& ={{\xi }_2^{*}}^2+\left({{\xi }_1^{*}}^2-2{\xi }_2^{*}{\xi }_3^{*}\right)\left(2cos{\left(\Sigma h\right)}^2-1\right)+\left(2{\xi }_1^{*}{\xi }_3^{*}-2{\xi }_1^{*}{\xi }_2^{*}\right)cos\left(\Sigma h\right)\hfill \\ \hfill & +{{\xi }_3^{*}}^2\left(2{\left(2cos{\left(\Sigma h\right)}^2-1\right)}^2-1\right).\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill \left|\varsigma \right|=\sqrt{{\left({\displaystyle \frac{N_1}{N_3}}\right)}^2+{\left({\displaystyle \frac{N_2}{N_3}}\right)}^2}.\end{array}$$

(40)

Consequently, if $\sqrt{{\left({\displaystyle \frac{N_1}{N_3}}\right)}^2+{\left({\displaystyle \frac{N_2}{N_3}}\right)}^2}\le 1$, the numerical scheme (32) is stable.

According to the above stability conditions, the obtained numerical schemes (31) and (32) are conditionally stable.

6. Evaluation of Numerical Performance

In this section, we employ two test problems to illustrate the effectiveness and performance of the proposed hyperbolic non-polynomial spline method. Error norms are utilized to assess the accuracy of the approximation solutions obtained through HNPSM in comparison with exact solutions and previous methodologies. The precision of the approximations is indicated by these error norms. With this thorough examination, we draw attention to the CHNPSM’s advantages and disadvantages while shedding light on its theoretical foundations and practical relevance:

$$L_{\infty }=\underset{1\le j\le M}{max}\left|{u_{\tau }}_{exact}-{u_{\tau }}_{approximate}\right|$$

(41)

and

$$L_2=\sqrt{h\sum _{i=1}^M{\left|{u_{\tau }}_{exact}-{u_{\tau }}_{approximate}\right|}^2}.$$

(42)

Example 1. 

Considering the time-coupled Korteweg–De Vries equations with $c_1=-1$, $c_2=-6$, and $c_3=3$, we have

$$T_t^{{\alpha }_1}u(r,t)+{\displaystyle \frac{{\partial }^{3}u(r,t)}{\partial r^3}}+6u(r,t){\displaystyle \frac{\partial u(r,t)}{\partial r}}-3v(r,t){\displaystyle \frac{\partial u(r,t)}{\partial r}}=3r{t}^4+{\displaystyle \frac{2r{t}^{2-{\alpha }_1}}{\Gamma (3-{\alpha }_1)}},$$

(43)

$$T_t^{{\alpha }_2}v(r,t)+{\displaystyle \frac{{\partial }^{3}v(r,t)}{\partial r^3}}-3u(r,t){\displaystyle \frac{\partial v(r,t)}{\partial r}}=3r{t}^4+{\displaystyle \frac{2r{t}^{2-{\alpha }_2}}{\Gamma (3-{\alpha }_2)}},$$

(44)

taking into account the initial and boundary conditions,

$$\begin{array}{cc}\hfill & u(r,0)=0,v(r,0)=0,\hfill \\ \hfill & u(0,t)=0,u(1,t)=t^2,u_r(0,t)=t^2,\hfill \\ \hfill & v(0,t)=0,v(1,t)=t^2,v_r(1,t)=t^2.\hfill \end{array}$$

(45)

The solution can be represented analytically as $u(r,t)=v(r,t)=r{t}^2$.

The results and discussion section presents the outcomes of applying the conformable hyperbolic non-polynomial spline method to example 1, providing insights into the accuracy and performance of the proposed numerical approach. Figure 1a displays the 3D solution profile $u(x,t)$ for example 1, where x and t vary within the interval $[0,1]$, with ${\alpha }_1=0.3$. Similarly, Figure 1b shows the 3D solution profile $v(x,t)$ for the same example with ${\alpha }_2=0.5$. These visual representations offer a clear depiction of the solutions obtained using the CHNPSM, illustrating their behavior across the specified domain. Further analysis is conducted through Figure 2a, which presents a comparison between the exact solutions and those obtained using the CHNPSM for $u(x,t)$ at different time instances ($t=0.25,0.5,0.75,1$), with ${\alpha }_1=0.3$ and $h=k=0.01$. Similarly, Figure 2b compares the exact solutions with those produced by the CHNPSM for $v(x,t)$ under the same conditions. It is evident from both figures that as time progresses, both $u(x,t)$ and $v(x,t)$ exhibit an increase in magnitude, consistent with the underlying dynamics described by example 1. To further assess the effectiveness of the CHNPSM,

Table 1. Error norms comparison: CHNPSM vs. meshless spectral method for example 1, where $h=0.2$, $t=1$, ${\alpha }_1=0.3$, and ${\alpha }_2=0.5$.

		CHNPSM		[26]
	$\mathit{k}$	${\mathit{L}}_{\infty }$	${\mathit{L}}_{\mathbf{2}}$	${\mathit{L}}_{\infty }$	${\mathit{L}}_{\mathbf{2}}$
	$1/20$	$1.2317\times {10}^{-05}$	$1.0732\times {10}^{-05}$	$3.592\times {10}^{-02}$	$2.599\times {10}^{-02}$
	$1/40$	$4.1326\times {10}^{-05}$	$5.3218\times {10}^{-05}$	$2.567\times {10}^{-02}$	$1.894\times {10}^{-02}$
$u(r,t)$	$1/60$	$6.7321\times {10}^{-05}$	$3.5286\times {10}^{-05}$	$1.739\times {10}^{-02}$	$1.275\times {10}^{-02}$
	$1/80$	$3.9321\times {10}^{-05}$	$3.5602\times {10}^{-05}$	$1.267\times {10}^{-02}$	$9.246\times {10}^{-03}$
	$1/100$	$4.2138\times {10}^{-05}$	$2.1653\times {10}^{-06}$	$9.717\times {10}^{-03}$	$7.060\times {10}^{-03}$
	$1/20$	$2.4317\times {10}^{-05}$	$1.3285\times {10}^{-05}$	$4.553\times {10}^{-01}$	$3.377\times {10}^{-01}$
	$1/40$	$3.5429\times {10}^{-05}$	$1.4376\times {10}^{-05}$	$4.767\times {10}^{-02}$	$3.552\times {10}^{-02}$
$v(r,t)$	$1/60$	$4.6326\times {10}^{-05}$	$3.1427\times {10}^{-05}$	$4.005\times {10}^{-02}$	$2.993\times {10}^{-02}$
	$1/80$	$3.9842\times {10}^{-05}$	$2.6541\times {10}^{-05}$	$3.428\times {10}^{-02}$	$2.565\times {10}^{-02}$
	$1/100$	$4.8721\times {10}^{-05}$	$2.6523\times {10}^{-05}$	$2.989\times {10}^{-02}$	$2.239\times {10}^{-02}$

provides a comparative analysis between our method and the meshless spectral method [26] for example 1, employing error norms as the evaluation criteria. The table highlights the superiority of the CHNPSM, as evidenced by the significantly lower error norms obtained. This difference underscores the effectiveness of the CHNPSM in accurately approximating the solutions of example 1, indicating its robustness and reliability in solving TFCKdVEs. Overall, the results obtained demonstrate the efficacy and reliability of the CHNPSM in accurately solving TFCKdVEs, underscoring its potential for advancing numerical computations in the realm of coupled wave equations.

Example 2. 

Considering the time-coupled Korteweg–De Vries equations with $c_1=1$, $c_2=-6$, and $c_3=3$, we have

$$T_t^{{\alpha }_1}u(r,t)-{\displaystyle \frac{{\partial }^{3}u(r,t)}{\partial r^3}}+6u(r,t){\displaystyle \frac{\partial u(r,t)}{\partial r}}-3v(r,t){\displaystyle \frac{\partial u(r,t)}{\partial r}}=0,$$

(46)

$$T_t^{{\alpha }_2}v(r,t)-{\displaystyle \frac{{\partial }^{3}v(r,t)}{\partial r^3}}-3u(r,t){\displaystyle \frac{\partial v(r,t)}{\partial r}}=0,$$

(47)

Taking into account the initial and boundary conditions,

$$\begin{array}{cc}\hfill & u(r,0)=v(r,0)={\displaystyle \frac{4e^{-t}}{{(1+e^{-t})}^2}},\hfill \\ \hfill & u(-10,t)=v(-10,t)={\displaystyle \frac{4e^{-10}}{{(1+e^{-10})}^2}},u(10,t)=v(10,t)={\displaystyle \frac{4e^{10}}{{(1+e^{10})}^2}}.\hfill \end{array}$$

(48)

The solution can be represented analytically as $u(r,t)=v(r,t)={\displaystyle \frac{4e^{r-t}}{{(1+e^{r-t})}^2}}$.

The results and discussion section presents the outcomes of applying the CHNPSM to example 2, providing insights into the accuracy and performance of the proposed numerical approach. Figure 3a displays the 3D solution profile $u(x,t)$ for example 2, where t varies within the interval $[0,1]$, and x varies within $[-10,10]$, with ${\alpha }_1=1$. Similarly, Figure 3b shows the 3D solution profile $v(x,t)$ for the same example with ${\alpha }_2=1$. These visual representations offer a clear depiction of the solutions obtained using the CHNPSM, illustrating their behavior across the specified domain. Further analysis is conducted through Figure 4a, which presents a comparison between the exact solutions and those obtained using the CHNPSM for $u(x,t)$ at different time instances ($t=0.25,0.5,0.75,1$), with ${\alpha }_1=1$, $h=0.1$, and $k=0.01$. Similarly, Figure 4b compares the exact solutions with those produced by the CHNPSM for $v(x,t)$ under the same conditions. It is evident from both figures that as time progresses, both $u(x,t)$ and $v(x,t)$ exhibit an increase in magnitude from $x=0$ to $x=10$ and decrease from $x=-10$ to $x=0$, consistent with the underlying dynamics described by example 2. For a deeper evaluation of the CHNPSM’s effectiveness,

Table 2. Error norms comparison: CHNPSM vs. Petrov–Galerkin method for example 2, where $h=0.1$, $k=0.01$, ${\alpha }_1=0.4$, and ${\alpha }_2=0.3$.

		CHNPSM		[22]
	$\mathit{t}$	${\mathit{L}}_{\infty }$	${\mathit{L}}_{\mathbf{2}}$	${\mathit{L}}_{\infty }$	${\mathit{L}}_{\mathbf{2}}$
	$0.2$	$2.5428\times {10}^{-05}$	$1.9732\times {10}^{-05}$	$2.911\times {10}^{-03}$	$1.870\times {10}^{-03}$
	$0.4$	$3.5271\times {10}^{-05}$	$2.4328\times {10}^{-05}$	$3.796\times {10}^{-04}$	$1.915\times {10}^{-04}$
$u(r,t)$	$0.6$	$4.9821\times {10}^{-05}$	$2.6582\times {10}^{-05}$	$2.307\times {10}^{-03}$	$1.474\times {10}^{-03}$
	$0.8$	$3.9321\times {10}^{-05}$	$7.5431\times {10}^{-05}$	$4.5421\times {10}^{-03}$	$2.372\times {10}^{-03}$
	1	$4.2138\times {10}^{-05}$	$6.7341\times {10}^{-05}$	$3.5482\times {10}^{-03}$	$2.805\times {10}^{-03}$
	$0.2$	$4.5487\times {10}^{-05}$	$2.6517\times {10}^{-05}$	$3.387\times {10}^{-03}$	$2.157\times {10}^{-03}$
	$0.4$	$3.9843\times {10}^{-05}$	$2.1386\times {10}^{-05}$	$3.984\times {10}^{-04}$	$2.084\times {10}^{-04}$
$v(r,t)$	$0.6$	$2.4326\times {10}^{-05}$	$1.4387\times {10}^{-05}$	$2.206\times {10}^{-03}$	$1.455\times {10}^{-03}$
	$0.8$	$4.5431\times {10}^{-05}$	$1.4083\times {10}^{-05}$	$3.886\times {10}^{-03}$	$2.571\times {10}^{-03}$
	1	$6.5490\times {10}^{-05}$	$2.4389\times {10}^{-05}$	$4.764\times {10}^{-03}$	$3.175\times {10}^{-03}$

shows a comparative study between our approach and the Petrov–Galerkin method [22] for example 2, focusing on error norms as the performance measure. The table highlights the superiority of the CHNPSM, as evidenced by the significantly lower error norms obtained. This difference underscores the effectiveness of the CHNPSM in accurately approximating the solutions of example 2, indicating its robustness and reliability in solving TFCKdVEs.

7. Computational Efficiency and Symmetry Analysis

In terms of computational efficiency, the CHNPSM has demonstrated promising performance in solving time-fractional coupled Korteweg–De Vries equations. The method exhibits efficient computational time, particularly when compared with existing techniques such as the meshless spectral method and the Petrov–Galerkin method, as evidenced by the error norm comparisons presented in the results section. The CHNPSM also shows optimized memory usage, enabling it to handle larger computational domains without incurring significant memory overhead. Scalability tests further demonstrate that the method maintains both accuracy and stability while efficiently accommodating varying grid sizes and time steps. These results highlight the method’s potential for real-world applications, where balancing accuracy and computational efficiency is essential. Recent studies have emphasized the importance of symmetry analysis in time-fractional differential equations. For example, Yaşar et al. [29] explored the Lie symmetry analysis and exact solutions of the seventh-order time-fractional Sawada–Kotera–Ito equation, while Hu et al. [30] applied Lie symmetry techniques to the time-fractional Korteweg–De Vries (KdV)-type equation. Similarly, Akbulut and Taşcan [31] investigated Lie symmetries and symmetry reductions for the time-fractional modified Korteweg–De Vries equation, and Yu and Feng [32] conducted a symmetry analysis of the time-fractional Caudrey–Dodd–Gibbon equation. Building upon these findings, future work could extend the study of symmetries, including the application of Lie symmetry analysis, to time-fractional coupled Korteweg–De Vries equations. Such investigations could enhance the understanding of the symmetry properties of TFCKdVEs and further contribute to the development of more efficient and accurate numerical methods for solving these complex systems.

8. Conclusions

In conclusion, the CHNPSM presents a significant advancement in the numerical solution of the time-fractional coupled Korteweg–De Vries equations. By leveraging hyperbolic non-polynomial spline functions, conformable derivatives, and first-derivative finite differences, the CHNPSM demonstrates remarkable accuracy and robustness in approximating solutions to TFCKdVEs. Through the analysis of convergence order and stability, it is evident that the CHNPSM offers a reliable computational framework for tackling complex phenomena such as wave propagation and shallow water waves on viscous fluids. The application of the CHNPSM to two case studies highlights its superior performance compared to existing methods like the meshless spectral method and Petrov–Galerkin method. The comparison, conducted using error norms, unequivocally demonstrates the efficacy of the CHNPSM in providing more accurate and reliable results. Overall, the CHNPSM emerges as a promising tool for researchers and practitioners engaged in numerical computations involving TFCKdVEs. Its effectiveness in capturing the intricate dynamics of real-world phenomena underscores its potential for further advancements in computational science and engineering. Further research may focus on extending the applicability of the CHNPSM to other nonlinear partial differential equations and exploring its performance in multidimensional scenarios.