A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

Abstract

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order $0<\alpha <1,$ and $1<\beta <2$. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

1. Introduction

Fractional order calculus (FOC) has not been explored in engineering and other sciences, due to its inherent complexity, as well as the fact that it lacks a completely valid geometric or physical interpretation [1]. However, some natural behaviors related to different fields of engineering are more accurately represented by using it; now, it is a promising tool used in physics [2], fluid mechanics [3], signal processing [4], thermal diffusion phenomenon [5], botanical electrical impedances [6], robotics [7], etc. Various analytical and numerical methods have been developed for fractional order differential equations (FODEs). For instance, in [8], the authors derived the analytic solution of the fractional diffusion wave equation using the Wright function. Liu et al. [9] obtained the complete solution of the time fractional advection–dispersion equation by employing Mellin and Laplace transforms as well as the properties of H functions. Mainardi [10] obtained the fundamental solution of the fractional diffusion-wave equation using an auxiliary function. Wyss [11] obtained the solution of the fractional diffusion equation using Fox functions. Some notable works have been done on the theoretical analysis of FODEs (see [12,13,14] and references therein).

Fractional partial differential equations (FPDEs) in mathematical models have experienced significant growth in recent years, establishing them as one of the most crucial topics in the field of fractional order calculus (FOC). Several FPDE-based models have been proposed [15,16,17], and various analytical and numerical schemes have been developed to solve them. Numerous authors have studied the numerical solution of FPDEs. For instance, Liu et al. [18] employed a first-order finite difference scheme for both time and space derivatives, deriving stability conditions in the process. Zhao et al. [19] used the finite element method to solve the multi-term time fractional mixed diffusion and diffusion-wave equation with a variable coefficient. In [20], an ADI spectral method based on the Legendre spectral approximation in space and finite difference method in time was developed to obtain the solution of wave-diffusion equations. Agrawal [21] obtained the general solution of wave-diffusion equations using the Laplace transform and finite sine transform. Tadjeran et al. [22] obtained the solution of the fractional diffusion equation using the Crank–Nicholson technique combined with the spatial extrapolation method. Li and Chen [23] employed three methods: finite difference methods, Galerkin finite element methods, and spectral methods, for numerical simulations of FPDEs. Lie et al. [24] utilized the fully discrete spectral method to solve multi-term fractional mixed diffusion and diffusion wave equations. Dehgan et al. [25] applied the homotopy analysis method to study the numerical solutions of various nonlinear FPDEs.

Different types of meshless methods are rapidly gaining popularity due to their meshless nature and ease of implementation for multi-dimensional problems on a set of uniform or non-uniform nodes in regular and irregular domains. Numerous meshless methods have been developed for the numerical modeling of partial differential equations (PDEs). For instance, Liu et al. [26] proposed an implicit meshless approach based on radial basis functions for solving fractional diffusion equations. Chen et al. [27] solved the time fractional diffusion equation using the Kansa method. Shivanian and Jafarbadi [28] employed the spectral meshless radial point interpolation technique coupled with finite difference discretization for the numerical simulation of fractional diffusion equations. Meshless methods can be categorized into two types: global meshless methods (GMMs) and local meshless methods (LMMs). A GMM has two major disadvantages, namely a dense ill-conditioned interpolation matrix and sensitivity to shape parameters. To overcome these issues, the LMM was developed in [29].

In an LMM, collocation is performed over small overlapping domains of influence. This approach allows for solving small systems of linear equations, with dimensions equal to the number of nodes in each domain of influence, for each node individually. LMMs have found applications in various scientific and engineering problems. For instance, in [30], the authors proposed an efficient local meshless upwind method based on radial basis functions (RBFs) for the numerical simulation of the time fractional advection–diffusion equation. Kumar and Bhardwaj [31] developed a local meshless method for the numerical solution of the nonlinear time fractional diffusion wave equation. Bhardwaj and Kumar [32] introduced an efficient local RBF-based meshless method for the numerical simulation of Tricomi-type equations of fractional order. The authors of [33,34] obtained the numerical solutions of linear and nonlinear time fractional diffusion waves and the telegraph equation using local RBFs. Hosseini et al. [35] obtained the approximate solution of the time fractional diffusion wave equation with damping via an efficient meshless local radial point interpolation method. Salehi [36] used a meshless point collocation method to solve the multi-term fractional diffusion wave equation. An RBF-based meshless technique was proposed in [37] for the approximate solution of the time fractional diffusion-wave equation. Yang et al. [38] employed the moving least squares approximation method for the numerical approximation of fractional diffusion wave equations. Other works on the applications of RBF-based local meshless methods can be found in the references (see [39,40,41,42] and the references therein).

Many researchers have investigated the time fractional mixed diffusion and wave diffusion equation. Sun et al. [43] obtained the solution of the fractional mixed diffusion and wave diffusion equation using an L-type difference scheme. The authors of [19] proposed a linear triangle finite element method for the numerical simulation of the multi-term fractional mixed diffusion and wave diffusion equation. Feng et al. [44] obtained the solution of the 2D multi-term fractional mixed sub-diffusion and wave-diffusion equation using finite difference and finite element methods. Liu et al. [24] proposed an efficient fully discrete spectral method for the numerical modeling of the fractional mixed diffusion and diffusion wave equations. Bhardwaj and Kumar [45] obtained the numerical solution of the fractional mixed diffusion and wave diffusion equation using an efficient local RBF-based meshless method. The authors used the finite difference method for time discretization; for space discretization, an RBF-based local collocation method was used.

In this work, our aim is to approximate the solution of fractional mixed diffusion and wave diffusion equations using an RBF-based local meshless method coupled with Laplace transform. The RBF-based local meshless methods are easy to implement for solving problems in irregular and complex geometries and offer high accuracy at a low computational cost. Additionally, the Laplace transform is employed to overcome stability limitations associated with finite difference temporal discretization techniques. To the best of our knowledge, this is the first attempt to utilize the RBF-based local meshless method coupled with Laplace transform for the numerical modeling of time fractional mixed diffusion and wave diffusion problems. In this study, we consider a fractional mixed diffusion and wave diffusion problem of the following form:

$$\left\{\begin{array}{c}{}_0^c{D}_t^{\alpha }u(\overline{\theta },t)+{}_0^c{D}_t^{\beta }u(\overline{\theta },t)=\Delta u(\overline{\theta },t)+f(\overline{\theta },t),(\overline{\theta },t)\in \mathsf{\Omega }\times [0,T],\hfill \\ u(\overline{\theta },0)=\xi \left(\overline{\theta }\right),\frac{\partial u(\overline{\theta },0)}{\partial t}=\eta \left(\overline{\theta }\right),\overline{\theta }\in \mathsf{\Omega },\hfill \\ {\mathcal{L}}_{\mathbf{b}}u(\overline{\theta },t)=\beta (\overline{\theta },t),\overline{\theta }\in \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

where 0 < $\alpha$ $<1$, 1 < $\beta$ $<2$ and $\xi \left(\overline{\theta }\right)$, $\eta \left(\overline{\theta }\right)$, and $f(\overline{\theta },t)$ are sufficiently smooth functions on domain $\mathsf{\Omega }$, with its boundary $\partial \mathsf{\Omega },$ and the Caputo derivatives ${}_0^c{D}_t^{\alpha }u(\overline{\theta },t)$ and ${}_0^c{D}_t^{\beta }u(\overline{\theta },t)$ of order $\alpha$ and $\beta$ are defined by

$${}_0^c{D}_t^{\alpha }u(\overline{\theta },t)=\frac{1}{\Gamma (1-\alpha )}{\int }_0^t\frac{\partial u(\overline{\theta },\tau )}{\partial \tau }{(t-\tau )}^{-\alpha }d\tau ,0<\alpha <1,$$

and

$${}_0^c{D}_t^{\beta }u(\overline{\theta },t)=\frac{1}{\Gamma (2-\beta )}{\int }_0^t\frac{{\partial }^{2}u(\overline{\theta },\tau )}{\partial {\tau }^2}{(t-\tau )}^{1-\beta }d\tau ,1<\beta <2.$$

The rest of the paper is organized as follows: Section 2.1 discusses the time discretization through Laplace transform. Section 2.2 discusses spatial discretization using the local RBF method. Section 2.3 explains the Stehfest method for the numerical inversion of the Laplace transform. Section 2.4 focuses on the error analysis of the method. Section 3 addresses the stability of the method. Section 4 presents numerical examples to demonstrate the effectiveness of the proposed method. Finally, Section 5 provides the conclusions.

2. Proposed Numerical Method

In our proposed numerical method, there are three main steps: (i) time discretization via the Laplace transform, (ii) spatial discretization via the local RBF method, and (iii) a numerical inversion of the Laplace transform via the Stehfest method.

2.1. (i) Time Discretization via the Laplace Transform

In this section, we use the Laplace transform to transform the problem into an equivalent time-independent problem and avoid the time stepping technique. The Laplace transform of a $u(\overline{\theta },t)$ is defined as

$$\mathcal{L}\left\{u(\overline{\theta },t)\right\}=\widehat{u}(\overline{\theta },z)={\int }_0^{\infty }{e}^{-zt}u(\overline{\theta },t)dt,$$

the Laplace transform of the Caputo derivative ${}_0^c{D}_t^{\alpha }u(\overline{\theta },t)$ is defined as

$$\mathcal{L}\left\{{}_0^c{D}_t^{\alpha }u(\overline{\theta },t)\right\}=z^{\alpha }\widehat{u}(\overline{\theta },z)-z^{\alpha -1}u(\overline{\theta },0),$$

and the Laplace transform of the Caputo derivative ${}_0^c{D}_t^{\beta }u(\overline{\theta },t)$ is defined as

$$\mathcal{L}\left\{{}_0^c{D}_t^{\beta }u(\overline{\theta },t)\right\}=z^{\beta }\widehat{u}(\overline{\theta },z)-z^{\beta -1}u(\overline{\theta },0)-z^{\beta -2}{u}_t(\overline{\theta },0).$$

Applying the Laplace transform to the model defined in (1), we have

$$\mathcal{L}\left[{}_0^c{D}_t^{\alpha }u(\overline{\theta },t)+{}_0^c{D}_t^{\beta }u(\overline{\theta },t)\right]=\mathcal{L}\left[\Delta u(\overline{\theta },t)+f(\overline{\theta },t)\right],$$

and

$$\mathcal{L}\left[{\mathcal{L}}_{\mathit{b}}u(\overline{\theta },t)\right]=\mathcal{L}\left[\beta (\overline{\theta },t)\right],$$

which implies

$$z^{\alpha }\widehat{u}(\overline{\theta },z)-z^{\alpha -1}u(\overline{\theta },0)+z^{\beta }\widehat{u}(\overline{\theta },z)-z^{\beta -1}u(\overline{\theta },0)-z^{\beta -2}{u}_t(\overline{\theta },0)=\Delta \widehat{u}(\overline{\theta },z)+\widehat{f}(\overline{\theta },z),$$

and

$${\mathcal{L}}_{\mathit{b}}\widehat{u}(\overline{\theta },z)=\beta (\overline{\theta },z),$$

simplifying the above equations, we obtain the following system

$$(z^{\alpha }I+z^{\beta }I-{\mathcal{L}}_{\mathit{g}})\widehat{u}(\overline{\theta },z)=\widehat{F}(\overline{\theta },z),$$

(2)

$${\mathcal{L}}_{\mathit{b}}\widehat{u}(\overline{\theta },z)=\beta (\overline{\theta },z),$$

(3)

where $\widehat{F}(\overline{\theta },z)=z^{\alpha -1}\xi \left(\overline{\theta }\right)+z^{\beta -1}\xi \left(\overline{\theta }\right)-z^{\beta -2}\eta \left(\overline{\theta }\right)+\widehat{f}(\overline{\theta },z)$ and ${\mathcal{L}}_{\mathit{g}}=\Delta ,$ where I denotes the identity operator, ${\mathcal{L}}_{\mathit{g}}$ is the linear differential operator, and ${\mathcal{L}}_{\mathit{b}}$ is the boundary differential operator. In order to obtain the solutions of the systems in (2) and (3), first, we use the local RBF method to discretize the operators ${\mathcal{L}}_{\mathit{g}}$ and ${\mathcal{L}}_{\mathit{b}}$, then systems (2) and (3) are solved in parallel for each point z in the complex plane (see, e.g., [46]). Finally, the solution of the original problem (1) is obtained using the inverse Laplace transform. The local RBF method is described in the next section.

2.2. (ii) Spatial Discretization via Local RBF Method

In the local RBF method, the problem and its corresponding boundary are interpolated on $n_g$ nodes that can be distributed regularly or irregularly. For each node ${\overline{\theta }}_j(j=1,2,3,\dots ,n_g)$, there is a local domain ${\mathsf{\Omega }}_j={\left\{{\overline{\theta }}_i^j\right\}}_{j=1}^{n_l}(i=1,2,3,\dots ,n_g),$ where $n_l$ denotes the number of nodes in the local domain ${\mathsf{\Omega }}_j.$ Thus, the local RBF interpolant of $\widehat{u}\left(\overline{\theta }\right)$ is of the form,

$$\widehat{u}\left({\overline{\theta }}_j\right)=\sum _{i=1}^{n_l}{\lambda }_i^j\phi (\parallel {\overline{\theta }}_j-{\overline{\theta }}_i^j\parallel ),$$

(4)

where the vector ${\lambda }^j={\left\{{\lambda }_i^j\right\}}_{i=1}^{n_g}$ is a vector of unknown expansion coefficients, $\phi \left(r\right)$ is a RBF and $r=\parallel {\overline{\theta }}_j-{\overline{\theta }}_i^j\parallel$ denotes the distance between ${\overline{\theta }}_j$ and ${\overline{\theta }}_i^j$. In the literature, there are several RBFs. Some commonly used RBFs are listed in

Table 1. Different types of finite and infinite smooth RBFs.

Multiquadric (MQ)	$\varphi \left(r\right)=\sqrt{1+{\epsilon }^2{r}^2}$
Inverse multiquadric (IMQ)	$\varphi \left(r\right)=\frac{1}{\sqrt{1+{\epsilon }^2{r}^2}}$
Gaussian (GA)	$\varphi \left(r\right)=e^{-{\left(\epsilon r\right)}^2}$
Wendlan	$\varphi \left(r\right)={(1-r)}_{+}^{k}p\left(r\right)$, $p\left(r\right)$ is a polynomial, $k\in \mathbb{N}$
Duchon spline	$\varphi \left(r\right)=r^{2k}log\left(r\right),k\in \mathbb{N}$
Thin Plate Spline (TPS)	$\varphi \left(r\right)=r^{2}log\left(r\right)$,

and the corresponding graphs are shown in Figure 1.

In this study, we select MQ RBF and GA RBF, which are defined as

$$\phi (\parallel {\overline{\theta }}_j-{\overline{\theta }}_i^j\parallel )=\sqrt{1+{\epsilon }^2{\parallel {\overline{\theta }}_j-{\overline{\theta }}_i^j\parallel }^2},$$

and

$$\phi (\parallel {\overline{\theta }}_j-{\overline{\theta }}_i^j\parallel )=exp(-{\epsilon }^2\parallel {\overline{\theta }}_j-{\overline{\theta }}_i^j{\parallel }^2),$$

where $\epsilon$ denotes the shape parameter. Each node ${\overline{\theta }}_i^j$ and its $n_l-1$ neighbors are referred to as a stencil. A typical stencil is shown in Figure 2.

At each stencil, we obtain the $n_l\times n_l$ matrix, given as

$${\widehat{\mathit{u}}}^j={\mathit{B}}^j{\mathit{\lambda }}^j,$$

(5)

where ${\widehat{\mathit{u}}}^j={[u\left({\overline{\theta }}_1^j\right),u\left({\overline{\theta }}_2^j\right),\dots ,u\left({\overline{\theta }}_{n_l}^j\right)]}^T,$${\mathit{B}}^j=\left[\varphi \right(\parallel {\overline{\theta }}_k^j-{\overline{\theta }}_i^j{\parallel \left)\right]}_{1\le k,i\le n_l},$ and ${\mathit{\lambda }}^j=[{\lambda }_1^j,{\lambda }_2^j,\dots ,$${\lambda }_{n_l}^j{]}^T.$ The invertibility of the system (5) requires the following definitions

Definition 1

[47]. A function $\zeta \left(r\right)$ is completely monotone on $[0,\infty ),$ if

1. 

$\zeta \left(r\right)\in C[0,\infty ).$

2. 

$\zeta \left(r\right)\in C^{\infty }[0,\infty ).$

3. 

${(-1)}^m{\zeta }^{\left(m\right)}\left(r\right)\ge 0,$ for $m=1,2,3,\dots$

Theorem 1

[47]. If $\zeta \left(r\right)=\phi \left(\sqrt{r}\right)\in C[0,\infty )$ and $\zeta \left(r\right)>0$ for $r>0$. Let ${\zeta }^{\prime }\left(r\right)$ be completely monotone and non-constant on $(0,\infty )$, then for any set of $n_g$ distinct centers ${\overline{\theta }}_j(j=1,2,3,\dots ,n_g)$, the interpolation matrix ${\mathit{B}}^j$ is invertible.

The unknown coefficients ${\mathit{\lambda }}^j$ in Equation (5) can be computed as

$${\mathit{\lambda }}^j={\left({\mathit{B}}^j\right)}^{-1}{\widehat{\mathit{u}}}^j,$$

(6)

whose entries are

$$\left(\begin{array}{c}{\lambda }_1^j\\ {\lambda }_2^j\\ .\\ .\\ .\\ {\lambda }_{n_l}^j\end{array}\right)={\left(\begin{array}{cccccc}\phi (\parallel {\overline{\theta }}_1^j-{\overline{\theta }}_1^j\parallel )& \phi (\parallel {\overline{\theta }}_1^j-{\overline{\theta }}_2^i\parallel )& .& .& .& \phi (\parallel {\overline{\theta }}_1^j-{\overline{\theta }}_{n_l}^j\parallel )\\ \phi (\parallel {\overline{\theta }}_2^j-{\overline{\theta }}_1^j\parallel )& \phi (\parallel {\overline{\theta }}_2^j-{\overline{\theta }}_2^i\parallel )& .& .& .& \phi (\parallel {\overline{\theta }}_2^j-{\overline{\theta }}_{n_l}^j\parallel )\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ \phi (\parallel {\overline{\theta }}_{n_l}^j-{\overline{\theta }}_1^j\parallel )& \phi (\parallel {\overline{\theta }}_{n_l}^j-{\overline{\theta }}_2^j\parallel )& .& .& .& \phi (\parallel {\overline{\theta }}_{n_l}^j-{\overline{\theta }}_{n_l}^j\parallel )\end{array}\right)}^{-1}\left(\begin{array}{c}\widehat{u}\left({\overline{\theta }}_1^j\right)\\ \widehat{u}\left({\overline{\theta }}_2^j\right)\\ .\\ .\\ .\\ \widehat{u}\left({\overline{\theta }}_{n_l}^j\right)\end{array}\right).$$

The differential operator ${\mathcal{L}}_{\mathit{g}}$ can be approximated as

$${\mathcal{L}}_{\mathit{g}}\widehat{u}=\sum _{i=1}^{n_l}{\lambda }_i^j\mathcal{L}\phi (\parallel {\overline{\theta }}_i-{\overline{\theta }}_i^j\parallel ),$$

which can be written as

$${\mathcal{L}}_{\mathit{g}}\widehat{u}={\mathbf{\psi }}^j{\left({\mathit{B}}^j\right)}^{-1}{\widehat{\mathit{u}}}^j,$$

where ${\mathbf{\psi }}^j=\left[\mathcal{L}\varphi \right(\parallel {\overline{\theta }}_k^j-{\overline{\theta }}_i^j{\parallel \left)\right]}_{1\le k,i\le n_l}.$ Hence, we have

$${\mathcal{L}}_{\mathit{g}}\widehat{u}={\mathsf{\Psi }}^j{\widehat{\mathit{u}}}^j,$$

(7)

where ${\mathsf{\Psi }}^j={\mathbf{\psi }}^j{\left({\mathit{B}}^j\right)}^{-1}.$ Equation (7), is the local form; the global form can be obtained by expanding ${\mathsf{\Psi }}^j$ to $\mathbf{Y}$ by inserting zeros into each row in corresponding positions. Therefore, the approximation of the differential operator $\mathcal{L}$ at each ${\overline{\theta }}_j$ is given as

$${\mathcal{L}}_{\mathit{g}}\widehat{u}=\mathbf{Y}\widehat{u}.$$

(8)

The matrix $\mathbf{Y}$ is sparse, having $n_g-n_l$ zero entries and $n_l$ non-zero entries. The boundary differential operator ${\mathcal{L}}_{\mathit{b}}$ can be approximated in a similar way

$${\mathcal{L}}_{\mathit{b}}\widehat{u}=\mathbf{\Phi }\widehat{u},$$

(9)

using Equations (8) and (9) in Equations (2) and (3), we obtained the following fully discrete system

$$(z^{\alpha }I+z^{\beta }I-\mathbf{Y})\widehat{u}(\overline{\theta },z)=\widehat{F}(\overline{\theta },z),$$

(10)

$$\mathbf{\Phi }\widehat{u}(\overline{\theta },z)=\beta (\overline{\theta },z).$$

(11)

Now, solving the system defined in Equations (10) and (11) for each point z in the complex plan will give us the solution $\widehat{u}(\overline{\theta },z).$

Selection: Good Value of the Shape Parameter

We utilized MQ RBF and GA RBF in our numerical experiments. These RBFs contain parameter $\epsilon$; for optimal accuracy of the numerical solution, the value of $\epsilon$ varies. The condition number ${\mathit{C}}_d$ of the matrix ${\mathit{B}}^j$ is used to quantify the sensitivity to perturbation of the linear system. In this work, the uncertainty principle [48] is used for the best shape parameter $\epsilon$.

The necessary steps for obtaining the optimal value of shape parameter $\epsilon$ are presented in the following Algorithm 1.

Algorithm 1: Optimal Shape Parameter

1: Input: ${{\mathit{C}}_d}_{min},{{\mathit{C}}_d}_{max},{\epsilon }_{Increment}$.  2: Step 1: ${\mathit{C}}_d=1$  3: Step 2: select ${10}^{+12}<{\mathit{C}}_d<{10}^{+16}$  4: Step 3: $\mathrm{while}{\mathit{C}}_d>{{\mathit{C}}_d}_{max}\mathrm{and}{\mathit{C}}_d<{{\mathit{C}}_d}_{min}$  5: Step 4: Construct the matrix ${\mathit{B}}^j$   6: Step 5: $\mathbf{E},\mathbf{S},\mathbf{P}=svd\left({\mathit{B}}^j\right)$  7: Step 6: ${\mathit{C}}_d=\frac{{\rho }_{max}}{{\rho }_{min}}$   8: Step 7: $\mathrm{if}{\mathit{C}}_d<{{\mathit{C}}_d}_{min},\epsilon =\epsilon -{\epsilon }_{Increment}$   9: Step 8: $\mathrm{if}{\mathit{C}}_d>{{\mathit{C}}_d}_{max},\epsilon =\epsilon +{\epsilon }_{Increment}$   10: Output: $\epsilon \left(\mathrm{optimal}\right)=\epsilon$.

After obtaining the best value of the shape parameter, we compute ${\left({\mathit{B}}^j\right)}^{-1}$ using svd as ${\left({\mathit{B}}^j\right)}^{-1}={\left({\mathbf{ESP}}^{\mathbf{T}}\right)}^{-1}={\mathbf{PS}}^{-\mathbf{1}}{\mathbf{E}}^{\mathbf{T}}$ (see [49]). Hence, we can calculate ${\mathsf{\Psi }}^j$ in (7).

2.3. (iii) Stehfest Method

In this section, we utilize the Stehfest algorithm for numerically inverting the Laplace transform. It is one of the most important algorithms used for the numerical inversion of the Laplace transform. This method was developed in the late 1960s [50,51]. Due to its simplicity and performance, it is becoming popular in many areas, such as economics, geophysics, financial mathematics, chemistry, and computational physics. This method is fast and gives good results for smooth functions. The solution $u(\overline{\theta },t)$ can be retrieved via the Stehfest method, as

$$u(\overline{\theta },t)=\frac{ln2}{t}\sum _{i=1}^N{\rho }_i\widehat{u}(\overline{\theta },z),$$

(12)

where the weights ${\rho }_i$ are given by

$${\rho }_i={(-1)}^{\frac{N}{2}+i}\sum _{p=\lfloor \frac{i+1}{2}\rfloor }^{min(i,\frac{N}{2})}\frac{p^{\frac{N}{2}}(2p!)}{(\frac{N}{2}-p)!p!(p-1)!(i-p)!(2p-i)!}.$$

(13)

Here, $N,$ refers to the Stehfest number, which should be even. Solving (10) and (11) for parameters $z=\frac{ln2}{t}i,i=1,2,3,\dots ,N$ in the LT space. Using (12), the original solution can be obtained.

The Gaver–Stehfest algorithm possesses several attractive properties. Firstly, the approximations of any function using this algorithm are linear in the values of the transform. Secondly, it only requires the values of the transform function for real z. Thirdly, the computation of the coefficients is straightforward. Lastly, the approximations obtained using this algorithm are exact for constant functions. The Gaver–Stehfest algorithm has been extensively studied by various authors, as documented in [52,53]. These studies have demonstrated that this method has very fast convergence for non-oscillatory functions.

2.4. Error Analysis

Here, we discuss the error analysis of the proposed method. In step (a), we employ the LT, which is analytic; thus, no error occurs in this step. The error of the method arises in steps (b) and (c). In step (b), we utilize the RBF-based local meshless method, which has an error estimate of order $O({\sigma }^{\frac{1}{\epsilon H_d}}),0<\sigma <1,$ where $H_d$ denotes the fill distance and $\epsilon$ is the shape parameter [47]. In step (c), the Stehfest method is employed for the numerical inversion of the LT. The error analysis of this method has been studied in [54,55], where the authors conducted various numerical experiments to investigate the effects of the involved parameters on the accuracy and efficiency of the method. They noted that “If $\beta$ significant digits are required, suppose $N=\lceil 2.2\beta \rceil$ positive integer. Set the precision of the system at $\nu =\lceil 1.1N\rceil$. Then for the argument t and $\widehat{u}(\overline{\theta },z)$, calculate $u(\overline{\theta },t)$ in (12).” According to these observations, the following error estimation is obtained.

Remark 1

[56]. If the error of input data satisfies ${10}^{-(\nu +1)}\le \frac{\parallel \widehat{u}-u\parallel }{\parallel u\parallel }\le {10}^{-\nu }$, where N is a positive even integer via $\nu =\lceil 1.1N\rceil$, then the error is ${10}^{-(\beta +1)}\le \frac{\parallel \widehat{u}-u\parallel }{\parallel u\parallel }\le {10}^{-\beta }$, where $N=\lceil 1.1\beta \rceil .$

3. Stability

Here, we discuss the stability of the fully discrete systems (10) and (11). We express the system in the following form

$$\mathcal{C}\widehat{\mathit{u}}=\mathit{R},$$

(14)

the coefficient matrix $\mathcal{C}$ is obtained using the local meshless method. The constant for the stability of the system (14) is defined as

$$\delta =\underset{\widehat{\mathit{u}}\ne 0}{sup}\frac{\parallel \widehat{\mathit{u}}\parallel }{\parallel \mathcal{C}\widehat{\mathit{u}}\parallel },$$

(15)

$\delta$ has a finite value for any norm $\parallel .\parallel$ defined in ${\mathbb{R}}^N$. From Equation (15), we obtain

$${\parallel \mathcal{C}\parallel }^{-1}\le \frac{\parallel \widehat{\mathit{u}}\parallel }{\parallel \mathcal{C}\widehat{\mathit{u}}\parallel }\le \delta .$$

(16)

Moreover, we can write

$$\parallel {\mathcal{C}}^{†}\parallel =\underset{\mathit{v}\ne 0}{sup}\frac{\parallel {\mathcal{C}}^{†}\mathit{v}\parallel }{\parallel \mathit{v}\parallel }.$$

(17)

where ${\mathcal{C}}^{†}$ is the pseudoinverse of $\mathcal{C}.$ Therefore,

$$\parallel {\mathcal{C}}^{†}\parallel \ge \underset{\mathit{v}=\mathcal{C}\widehat{\mathit{u}}\ne 0}{sup}\frac{\parallel {\mathcal{C}}^{†}\mathcal{C}\widehat{\mathit{u}}\parallel }{\parallel \mathcal{C}\widehat{\mathit{u}}\parallel }=\underset{\widehat{\mathit{u}}\ne 0}{sup}\frac{\parallel \widehat{\mathit{u}}\parallel }{\parallel \mathcal{C}\widehat{\mathit{u}}\parallel }=\delta .$$

(18)

Equations (16) and (17) give the bounds for the constant $\delta$. For system (14), the pseudoinverse may be hard to compute, but by using it, stability is guaranteed. The MATLAB condest function can be used for evaluating $\parallel {\mathcal{C}}^{-1}{\parallel }_{\infty }$ as

$$\delta =\frac{condest\left({\mathcal{C}}^{\prime }\right)}{{\parallel \mathcal{C}\parallel }_{\infty }}$$

(19)

4. Applications

This section focuses on presenting numerical results and discussions. The performance, efficiency, accuracy, and convergence of the proposed method are tested and validated using five test problems in one and two dimensions. For two-dimensional problems, six different sets of regular and irregular domains with uniform and non-uniform points are selected. Three error norms are utilized: the maximum absolute error, the relative error, and the root mean square error, which are denoted and defined as follows:

$$Erro{r}_1=\underset{1\le j\le n_g}{max}|u({\mathit{\zeta }}_j,t)-u_k({\mathit{\zeta }}_j,t)|,$$

$$Erro{r}_2=\sqrt{\frac{{\sum }_{j=1}^{n_g}{(u({\overline{\mathit{\zeta }}}_j,t)-u_k({\overline{\mathit{\zeta }}}_j,t))}^2}{n_g}},$$

$$Erro{r}_3=\sqrt{\frac{{\sum }_{j=1}^{n_g}{(u({\mathit{\zeta }}_j,t)-u_k({\mathit{\zeta }}_j,t))}^2}{{\sum }_{j=1}^{n_g}{\left(u({\mathit{\zeta }}_j,t)\right)}^2}},$$

where $n_g$ represents the total number of points, $n_l$ represents the points in the sub-domain, and N represents the points used in the Stehfest method. $u(\mathit{\zeta }j,t)$ denotes the analytic solution, and $uk({\mathit{\zeta }}_j,t)$ denotes the numerical solution. The numerical experiments are conducted using MQ and GA RBFs. In all experiments, the initial conditions, boundary conditions, and source terms are calculated based on the analytic solutions. All of the numerical experiments are performed with a final time of $T=1$.

Problem 1.

Consider the following one-dimensional test problem

$${}_0^c{D}_t^{\alpha }u(x,t)+{}_0^c{D}_t^{\beta }u(x,t)=\Delta u(x,t)+f(x,t),0\le x\le 1,$$

(20)

with the exact solution

$$u(x,t)=t^{3}sin\left(\pi x\right).$$

The errors of the proposed method for problem 1 with $\alpha =0.5$, $\beta =1.1$, and different values of $n_g$, $n_l$, and N using MQ RBF are presented in

Table 2. The errors obtained using the proposed method for problem 1 using MQ RBF.

$\mathit{\alpha }=0.5,$ $\mathit{\beta }=1.1$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	95	10	16	2.72 × ${10}^{-4}$	2.79 × ${10}^{-5}$	3.97 × ${10}^{-5}$	0.132263
		15		8.99 × ${10}^{-5}$	9.22 × ${10}^{-6}$	1.31 × ${10}^{-5}$	0.183286
		20		7.34 × ${10}^{-5}$	7.53 × ${10}^{-6}$	1.07 × ${10}^{-5}$	0.228481
		22		6.32 × ${10}^{-5}$	6.48 × ${10}^{-6}$	9.22 × ${10}^{-6}$	0.258556
	120	21	12	9.12 × ${10}^{-4}$	8.32 × ${10}^{-5}$	1.18 × ${10}^{-4}$	0.242945
			14	4.59 × ${10}^{-4}$	4.19 × ${10}^{-5}$	5.95 × ${10}^{-5}$	0.248445
			16	1.74 × ${10}^{-4}$	1.59 × ${10}^{-5}$	2.26 × ${10}^{-5}$	0.238210
			18	8.95 × ${10}^{-5}$	8.17 × ${10}^{-6}$	1.16 × ${10}^{-5}$	0.237966
	82	21	16	9.20 × ${10}^{-5}$	1.01 × ${10}^{-5}$	1.44 × ${10}^{-5}$	0.226662
	85			8.70 × ${10}^{-5}$	9.44 × ${10}^{-6}$	1.34 × ${10}^{-5}$	0.227972
	88			7.08 × ${10}^{-5}$	7.55 × ${10}^{-6}$	1.07 × ${10}^{-5}$	0.234824
	95			6.88 × ${10}^{-5}$	7.06 × ${10}^{-6}$	1.00 × ${10}^{-5}$	0.239582
[32]				1.69 × ${10}^{-4}$

, while the errors using GA RBF are presented in

Table 3. The errors obtained using the proposed method for problem 1 using GA RBF.

$\mathit{\alpha }=0.5,$ $\mathit{\beta }=1.6$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	115	10	12	9.86 × ${10}^{-4}$	9.19 × ${10}^{-5}$	1.30 × ${10}^{-4}$	0.188929
	125			6.69 × ${10}^{-4}$	5.98 × ${10}^{-5}$	8.50 × ${10}^{-5}$	0.185436
	130			5.69 × ${10}^{-4}$	4.99 × ${10}^{-5}$	7.09 × ${10}^{-5}$	0.223009
	135			4.92 × ${10}^{-5}$	4.23 × ${10}^{-5}$	6.01 × ${10}^{-5}$	0.213365
	95	8	12	7.69 × ${10}^{-4}$	7.89 × ${10}^{-5}$	1.12 × ${10}^{-5}$	0.173427
		13		3.06 × ${10}^{-4}$	3.14 × ${10}^{-5}$	4.47 × ${10}^{-5}$	0.220220
		14		2.92 × ${10}^{-4}$	3.00 × ${10}^{-5}$	4.27 × ${10}^{-5}$	0.202532
		15		1.08 × ${10}^{-4}$	1.11 × ${10}^{-5}$	1.58 × ${10}^{-5}$	0.196352
	74	10	12	6.84 × ${10}^{-4}$	7.95 × ${10}^{-5}$	1.13 × ${10}^{-4}$	0.102373
			14	4.09 × ${10}^{-4}$	4.75 × ${10}^{-5}$	6.77 × ${10}^{-5}$	0.136801
			16	2.01 × ${10}^{-4}$	2.33 × ${10}^{-5}$	3.33 × ${10}^{-5}$	0.129105
			18	1.73 × ${10}^{-4}$	2.01 × ${10}^{-5}$	2.87 × ${10}^{-5}$	0.148823
[32]				9.47 × ${10}^{-4}$

. It is observed that the performance of MQ RBF is better than GA RBF. The results demonstrate that the proposed numerical method is efficient and reliable. Figure 3a,b depict the numerical and analytic solutions of problem 1 with $\alpha =0.5$ and $\beta =1.6$. The surface plots of $Erro{r}_1$ and $Erro{r}_2$ with $n_g=70,n_l=10,N=16,\alpha =0.5$, and $\beta =1.9$ using MQ RBF are presented in Figure 4a,b; using GA RBF with $n_g=70,n_l=10,N=16,\alpha =0.5$, and $\beta =1.7$, they are presented in Figure 5a,b. The surface plots of $Erro{r}_3$ of the proposed method corresponding to problem 1 with $n_g=90,n_l=6,N=18,\alpha =0.5,\beta =1.9$ using MQ RBF and GA RBF are presented in Figure 6a,b. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ using MQ RBF for different values of $\alpha$ with $\beta =1.6,n_g=120,n_l=5,N=16$ are depicted in Figure 7a and for different values of $\beta$ with $\alpha =0.5,n_g=120,n_l=5,N=16$ are presented in Figure 7b. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ for various values of $\alpha$ and $\beta$ with the same parameters using GA RBF are shown in Figure 8a,b, respectively. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ for different values of N with $\alpha =0.5,\beta =1.9,n_g=120,n_l=5$ using MQ RBF are shown in Figure 9a; using GA RBF, they are presented in Figure 9b. Similarly, the plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ for different values of t with $N=16,\alpha =0.7,\beta =1.4,n_g=120,n_l=5$ using MQ RBF are shown in Figure 10a; using GA RBF, they are shown in Figure 10b. From the data reported in the figures and tables, it is evident that the numerical solution is in good agreement with the analytic solution.

Problem 2.

Consider the following one-dimensional test problem

$${}_0^c{D}_t^{\alpha }u(x,t)+{}_0^c{D}_t^{\beta }u(x,t)=\Delta u(x,t)+f(x,t),0\le x\le 1,$$

(21)

with the exact solution

$$u(x,t)=t^{(3+\alpha +\beta )}exp(-x^2).$$

The errors of the proposed method for problem 2 with $\alpha =0.5,\beta =1.5$, and different values of $n_g,n_l$, and N using MQ RBF are presented in

Table 4. The errors obtained using the proposed method for problem 2 using MQ RBF.

$\mathit{\alpha }=0.5,$ $\mathit{\beta }=1.5$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	120	15	18	3.76 × ${10}^{-4}$	3.44 × ${10}^{-5}$	4.45 × ${10}^{-5}$	0.203494
		16		3.34 × ${10}^{-4}$	3.05 × ${10}^{-5}$	3.95 × ${10}^{-5}$	0.216352
		17		1.65 × ${10}^{-4}$	1.51 × ${10}^{-5}$	1.95 × ${10}^{-5}$	0.198863
		18		1.15 × ${10}^{-4}$	1.05 × ${10}^{-5}$	1.36 × ${10}^{-5}$	0.259377
	90	16	14	9.20 × ${10}^{-3}$	9.64 × ${10}^{-4}$	1.20 × ${10}^{-3}$	0.193569
			16	2.10 × ${10}^{-3}$	2.22 × ${10}^{-4}$	2.87 × ${10}^{-4}$	0.199498
			18	2.65 × ${10}^{-4}$	2.79 × ${10}^{-5}$	3.62 × ${10}^{-5}$	0.188744
			20	1.98 × ${10}^{-4}$	2.08 × ${10}^{-5}$	2.70 × ${10}^{-5}$	0.202167
	110	18	18	2.38 × ${10}^{-4}$	2.27 × ${10}^{-5}$	2.94 × ${10}^{-5}$	0.232646
	112			9.76 × ${10}^{-5}$	9.22 × ${10}^{-6}$	1.19 × ${10}^{-5}$	0.228243
	115			9.75 × ${10}^{-5}$	9.09 × ${10}^{-6}$	1.17 × ${10}^{-5}$	0.223190
	116			4.45 × ${10}^{-5}$	4.13 × ${10}^{-6}$	5.34 × ${10}^{-5}$	0.233154
[32]				4.38 × ${10}^{-5}$

; using the GA RBF, they are presented in

Table 5. The errors obtained using the proposed method for problem 2 using GA RBF.

$\mathit{\alpha }=0.5,$ $\mathit{\beta }=1.5$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	80	11	18	3.42 × ${10}^{-4}$	3.83 × ${10}^{-5}$	4.95 × ${10}^{-5}$	0.147621
	90			2.53 × ${10}^{-4}$	2.66 × ${10}^{-5}$	3.44 × ${10}^{-5}$	0.148642
	100			1.26 × ${10}^{-4}$	1.26 × ${10}^{-5}$	1.63 × ${10}^{-5}$	0.161187
	101			8.43 × ${10}^{-5}$	8.39 × ${10}^{-6}$	1.08 × ${10}^{-5}$	0.156145
	99	10	14	9.20 × ${10}^{-3}$	9.87 × ${10}^{-4}$	1.30 × ${10}^{-3}$	0.206573
			16	2.40 × ${10}^{-3}$	2.46 × ${10}^{-4}$	3.18 × ${10}^{-4}$	0.272171
			18	5.02 × ${10}^{-4}$	5.05 × ${10}^{-5}$	6.53 × ${10}^{-5}$	0.225353
			20	4.87 × ${10}^{-4}$	4.90 × ${10}^{-5}$	6.33 × ${10}^{-5}$	0.247079
	95	10	18	4.94 × ${10}^{-4}$	5.07 × ${10}^{-5}$	6.56 × ${10}^{-5}$	0.205846
		11		4.22 × ${10}^{-4}$	4.33 × ${10}^{-5}$	5.60 × ${10}^{-5}$	0.200952
		12		3.22 × ${10}^{-4}$	3.30 × ${10}^{-5}$	4.27 × ${10}^{-5}$	0.216622
		15		2.33 × ${10}^{-4}$	2.39 × ${10}^{-5}$	3.09 × ${10}^{-5}$	0.201578
[32]				5.72 × ${10}^{-6}$

. We can see that the method has produced accurate and stable results. The numerical and analytic solutions of problem 2 with $\alpha =0.5$ and $\beta =1.1$ are depicted in Figure 11a,b; a nice agreement between the numerical and analytic solutions is observed. The surface plots of $Erro{r}_1$ and $Erro{r}_2$ with $n_g=90,n_l=20,N=18,\alpha =0.9$ and $\beta =1.3$ using MQ RBF are presented in Figure 12a,b. The surface plots of $Erro{r}_1$ and $Erro{r}_2$ with $n_g=70,n_l=10,N=16,\alpha =0.2$, and $\beta =1.6$ using GA RBF are presented in Figure 13a,b. The surface plot of $Erro{r}_3$ with $n_g=90,n_l=10,N=18,\alpha =0.2$ and $\beta =1.6$ using MQ RBF is presented in Figure 14a and surface plot of $Erro{r}_3$ with $n_g=90,n_l=10,N=18,\alpha =0.2$ and $\beta =1.6$ using GA RBF is presented in Figure 14b. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ using MQ RBF for different values of $\alpha$ with $\beta =1.1,n_g=120,n_l=6,N=18$ are depicted in Figure 15a, and for different values of $\beta$ with $\alpha =0.9,n_g=120,n_l=5,N=18$ are presented in Figure 15b. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ using GA RBF for different values of $\alpha$ with $\beta =1.6,n_g=450,n_l=5,N=18$ are depicted in Figure 16a, and for different values of $\beta$ with $\alpha =0.2,n_g=450,n_l=5,N=18$, are presented in Figure 16b. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ using MQ RBF for different values of N with $\alpha =0.9,\beta =1.6,n_g=450,n_l=6$ are shown in Figure 17a; using GA RBF, they are presented in Figure 17b. The plots of $Erro{r}_1$, $Erro{r}_2$, and $Erro{r}_3$ using MQ RBF for different values of t with $N=18,\alpha =0.9,\beta =1.1,n_g=450,n_l=5$ are depicted in Figure 18a; using GA RBF, they are presented in Figure 18b. From the data presented in the tables and figures, it is observed that the proposed method has produced accurate and stable results, and can handle such types of problems efficiently.

Problem 3.

Consider the following two-dimensional test problem

$${}_0^c{D}_t^{\alpha }u(x,y,t)+{}_0^c{D}_t^{\beta }u(x,y,t)=\Delta u(x,y,t)+f(x,y,t),0\le x,y\le \pi ,$$

(22)

with analytic solution

$$u(x,y,t)=t^{4}sin\left(\pi x\right)sin\left(\pi y\right).$$

(23)

This problem is considered for validating the numerical scheme on domains ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_6$ as shown in Figure 19a and Figure 19f, respectively. The errors of the proposed method for problem 3 using the MQ RBF on domain ${\mathsf{\Omega }}_1$ are presented in

Table 6. The errors obtained using the proposed method for problem 3 on domain ${\mathsf{\Omega }}_1$ using MQ RBF.

$\mathit{\alpha }=0.50,$ $\mathit{\beta }=1.50$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	4624	13	14	1.27 × ${10}^{-2}$	1.86 × ${10}^{-4}$	3.78 × ${10}^{-4}$	28.244165
			16	9.32 × ${10}^{-4}$	1.37 × ${10}^{-5}$	2.78 × ${10}^{-5}$	21.367193
			18	3.91 × ${10}^{-4}$	5.75 × ${10}^{-6}$	1.16 × ${10}^{-5}$	25.048697
			20	3.62 × ${10}^{-4}$	5.33 × ${10}^{-6}$	1.08 × ${10}^{-5}$	25.466929
	4225	13	16	1.20 × ${10}^{-3}$	1.82 × ${10}^{-5}$	3.71 × ${10}^{-5}$	24.562205
		14		9.56 × ${10}^{-4}$	1.47 × ${10}^{-5}$	2.99 × ${10}^{-5}$	25.343475
		15		4.42 × ${10}^{-4}$	6.80 × ${10}^{-6}$	1.38 × ${10}^{-5}$	25.994300
		17		4.16 × ${10}^{-4}$	6.40 × ${10}^{-6}$	1.30 × ${10}^{-5}$	26.284367
	4096	16	16	5.86 × ${10}^{-4}$	9.16 × ${10}^{-6}$	1.86 × ${10}^{-5}$	26.098623
	4356			5.84 × ${10}^{-4}$	8.85 × ${10}^{-6}$	1.79 × ${10}^{-5}$	27.673088
	4761			5.51 × ${10}^{-4}$	7.99 × ${10}^{-6}$	1.62 × ${10}^{-5}$	31.860493
	4900			5.30 × ${10}^{-4}$	7.58 × ${10}^{-6}$	1.53 × ${10}^{-5}$	37.681474
[32]				1.44 × ${10}^{-4}$

; using the GA RBF on domain ${\mathsf{\Omega }}_6$, they are presented in

Table 7. The errors obtained using the proposed method for problem 3 on domain ${\mathsf{\Omega }}_6$ using GA RBF.

$\mathit{\alpha }=0.50,$ $\mathit{\beta }=1.50$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	1600	15	16	6.10 × ${10}^{-3}$	1.51 × ${10}^{-4}$	3.98 × ${10}^{-4}$	10.247910
			18	5.90 × ${10}^{-3}$	1.47 × ${10}^{-4}$	3.87 × ${10}^{-4}$	10.275675
			20	5.80 × ${10}^{-3}$	1.45 × ${10}^{-4}$	3.83 × ${10}^{-4}$	10.530108
			22	5.70 × ${10}^{-3}$	1.43 × ${10}^{-4}$	3.78 × ${10}^{-4}$	13.210822
	1521	13	16	8.00 × ${10}^{-3}$	2.04 × ${10}^{-4}$	5.38 × ${10}^{-4}$	6.366293
		14		6.30 × ${10}^{-3}$	1.61 × ${10}^{-4}$	4.25 × ${10}^{-4}$	6.503034
		15		6.20 × ${10}^{-3}$	1.59 × ${10}^{-4}$	4.18 × ${10}^{-4}$	7.186865
		18		6.18 × ${10}^{-3}$	1.58 × ${10}^{-4}$	4.17 × ${10}^{-4}$	8.509750
	1600	13	18	8.30 × ${10}^{-3}$	2.08 × ${10}^{-4}$	5.48 × ${10}^{-4}$	7.300554
	1764			6.80 × ${10}^{-3}$	1.62 × ${10}^{-4}$	4.26 × ${10}^{-4}$	8.319459
	1936			6.60 × ${10}^{-3}$	1.49 × ${10}^{-4}$	3.91 × ${10}^{-4}$	9.356419
	2401			6.30 × ${10}^{-3}$	1.28 × ${10}^{-4}$	3.36 × ${10}^{-4}$	12.664379
[32]				1.98 × ${10}^{-3}$

. It is observed that the method has produced results with acceptable accuracy. The numerical and analytic solutions of problem 3 on domain ${\mathsf{\Omega }}_1$ are depicted in Figure 20a,b, a good agreement between the numerical and analytic solutions is observed. The mesh plots of $Erro{r}_1$ and $Erro{r}_2$ on domain ${\mathsf{\Omega }}_1$ using MQ RBF with $n_g=6400,n_l=35,N=18,\alpha =0.4,\beta =1.7$ are presented in Figure 21a,b. The mesh plots of $Erro{r}_1$ and $Erro{r}_2$ on domain ${\mathsf{\Omega }}_6$ using MQ RBF with $n_g=1225,n_l=35,N=18,\alpha =0.4,\beta =1.7$ are presented in Figure 22a,b. The mesh plots of $Erro{r}_1$ and $Erro{r}_2$ on domain ${\mathsf{\Omega }}_1$ using GA RBF with $n_g=2500,n_l=35,N=18,\alpha =0.9,\beta =1.1$ are presented in Figure 23a,b. The mesh plots of $Erro{r}_1$ and $Erro{r}_2$ on domain ${\mathsf{\Omega }}_6$ using GA RBF with $n_g=1600,n_l=35,N=18,\alpha =0.5,\beta =1.5$ are depicted in Figure 24a,b. The surface plots of $Erro{r}_3$ on domain ${\mathsf{\Omega }}_1$ using MQ RBF with $n_g=1600,n_l=30,N=16,\alpha =0.5,\beta =1.5$ are depicted in Figure 25a and the surface plots of $Erro{r}_3$ on domain ${\mathsf{\Omega }}_6$ using GA RBF with $n_g=1600,n_l=30,N=16,\alpha =0.5,\beta =1.5$ are depicted in Figure 20b. From the obtained results, it is evident that the proposed method can efficiently solve problems with both uniform and non-uniform node distributions.

Problem 4.

Consider the following two-dimensional test problem

$${}_0^c{D}_t^{\alpha }u(x,y,t)+{}_0^c{D}_t^{\beta }u(x,y,t)=\Delta u(x,y,t)+f(x,y,t),0\le x,y\le 1,$$

(24)

with analytic solution

$$u(x,y,t)=(1+t^3)(1-x^2-y^2)$$

(25)

This problem serves to validate the numerical scheme on domains ${\mathsf{\Omega }}_1$, ${\mathsf{\Omega }}_3$, and ${\mathsf{\Omega }}_5$ as illustrated in Figure 19a, Figure 19c and Figure 19e, respectively. The errors of the proposed method for problem 4 in domain ${\mathsf{\Omega }}_1$ using MQ RBF are presented in

Table 8. The errors obtained using the proposed method for problem 4 on domain ${\mathsf{\Omega }}_1$ using MQ RBF.

$\mathit{\alpha }=0.7,$ $\mathit{\beta }=1.6$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	625	40	20	9.75 × ${10}^{-4}$	3.90 × ${10}^{-5}$	2.29 × ${10}^{-4}$	5.780576
	676			8.36 × ${10}^{-4}$	3.21 × ${10}^{-5}$	1.88 × ${10}^{-4}$	5.887121
	729			7.78 × ${10}^{-4}$	2.88 × ${10}^{-5}$	1.69 × ${10}^{-4}$	6.109537
	784			7.69 × ${10}^{-4}$	2.74 × ${10}^{-5}$	1.60 × ${10}^{-4}$	6.602348
	1444	55	12	8.18 × ${10}^{-4}$	2.15 × ${10}^{-5}$	1.24 × ${10}^{-4}$	18.340002
			14	7.57 × ${10}^{-4}$	1.99 × ${10}^{-5}$	1.15 × ${10}^{-4}$	18.271623
			16	6.07 × ${10}^{-4}$	1.59 × ${10}^{-5}$	9.26 × ${10}^{-5}$	18.396910
			18	5.87 × ${10}^{-4}$	1.54 × ${10}^{-5}$	8.96 × ${10}^{-5}$	19.094877
	841	50	22	1.10 × ${10}^{-3}$	3.84 × ${10}^{-5}$	2.24 × ${10}^{-4}$	9.939883
		51		1.00 × ${10}^{-3}$	3.55 × ${10}^{-5}$	2.07 × ${10}^{-4}$	10.480804
		52		6.16 × ${10}^{-4}$	2.12 × ${10}^{-5}$	1.24 × ${10}^{-4}$	10.572760
		53		5.45 × ${10}^{-4}$	1.88 × ${10}^{-5}$	1.09 × ${10}^{-4}$	10.763141
[32]				9.54 × ${10}^{-6}$

, while the errors using GA RBF are presented in

Table 9. The errors obtained using the proposed method for problem 4 on domain ${\mathsf{\Omega }}_1$ using GA RBF.

$\mathit{\alpha }=0.7,$ $\mathit{\beta }=1.90$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	1444	10	16	9.00 × ${10}^{-3}$	2.36 × ${10}^{-4}$	2.18 × ${10}^{-4}$	3.095921
	1521			8.70 × ${10}^{-3}$	2.22 × ${10}^{-4}$	2.05 × ${10}^{-4}$	3.330961
	1600			8.60 × ${10}^{-3}$	2.14 × ${10}^{-4}$	1.98 × ${10}^{-4}$	3.742602
	1764			8.50 × ${10}^{-3}$	2.03 × ${10}^{-4}$	1.87 × ${10}^{-4}$	4.456763
	2025	11	18	7.90 × ${10}^{-3}$	1.75 × ${10}^{-4}$	1.62 × ${10}^{-4}$	5.512489
		17		7.60 × ${10}^{-3}$	1.69 × ${10}^{-4}$	1.56 × ${10}^{-4}$	6.991000
		18		5.60 × ${10}^{-3}$	1.24 × ${10}^{-4}$	1.15 × ${10}^{-4}$	6.989841
		19		3.30 × ${10}^{-3}$	7.36 × ${10}^{-5}$	6.81 × ${10}^{-5}$	6.974949
	676	19	14	3.40 × ${10}^{-3}$	1.29 × ${10}^{-4}$	1.19 × ${10}^{-4}$	2.241376
		51	16	3.10 × ${10}^{-3}$	1.17 × ${10}^{-4}$	1.07 × ${10}^{-4}$	3.233400
		52	18	3.00 × ${10}^{-3}$	1.15 × ${10}^{-4}$	1.06 × ${10}^{-4}$	2.891570
		53	20	2.80 × ${10}^{-3}$	1.08 × ${10}^{-4}$	9.96 × ${10}^{-5}$	2.693629
[32]				8.90 × ${10}^{-5}$

. In

Table 10. The errors obtained using the proposed method for problem 4 on domain ${\mathsf{\Omega }}_3$ using MQ RBF.

$\mathit{\alpha }=0.5,$ $\mathit{\beta }=1.50$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	560	23	18	9.91 × ${10}^{-4}$	4.19 × ${10}^{-5}$	3.59 × ${10}^{-5}$	2.342120
		30		8.62 × ${10}^{-4}$	3.64 × ${10}^{-5}$	3.13 × ${10}^{-5}$	3.562138
		31		8.49 × ${10}^{-4}$	3.58 × ${10}^{-5}$	3.08 × ${10}^{-5}$	3.845987
		32		8.27 × ${10}^{-4}$	3.49 × ${10}^{-5}$	3.00 × ${10}^{-5}$	4.054985
	481	65	12	1.60 × ${10}^{-3}$	7.47 × ${10}^{-5}$	6.41 × ${10}^{-5}$	10.473426
			14	1.00 × ${10}^{-3}$	4.71 × ${10}^{-5}$	4.04 × ${10}^{-5}$	10.964542
			16	7.50 × ${10}^{-4}$	3.42 × ${10}^{-5}$	2.93 × ${10}^{-5}$	10.637479
			18	7.48 × ${10}^{-4}$	3.41 × ${10}^{-5}$	2.93 × ${10}^{-5}$	10.676450
	481	74	18	8.04 × ${10}^{-4}$	3.66 × ${10}^{-5}$	3.15 × ${10}^{-5}$	13.894779
	560			7.56 × ${10}^{-4}$	3.19 × ${10}^{-5}$	2.74 × ${10}^{-5}$	15.420818
	645			7.28 × ${10}^{-4}$	2.87 × ${10}^{-5}$	2.46 × ${10}^{-5}$	16.253816
	736			7.11 × ${10}^{-4}$	2.62 × ${10}^{-5}$	2.25 × ${10}^{-5}$	17.652880
[32]				6.22 × ${10}^{-6}$

the errors of the proposed method on domain ${\mathsf{\Omega }}_3$ using MQ RBF are presented; in

Table 11. The errors obtained using the proposed method for problem 4 on domain ${\mathsf{\Omega }}_5$ using GA RBF.

$\mathit{\alpha }=0.7,$ $\mathit{\beta }=1.90$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	800	70	18	4.70 × ${10}^{-3}$	1.65 × ${10}^{-4}$	1.59 × ${10}^{-4}$	16.835572
	805			3.60 × ${10}^{-3}$	1.26 × ${10}^{-4}$	1.21 × ${10}^{-4}$	18.014559
	819			1.90 × ${10}^{-3}$	6.54 × ${10}^{-5}$	6.31$\times {10}^{-5}$	17.489709
	839			1.70 × ${10}^{-3}$	5.80$\times {10}^{-5}$	5.59 × ${10}^{-5}$	17.416943
	905	38	16	4.70 × ${10}^{-3}$	1.56 × ${10}^{-4}$	1.50 × ${10}^{-3}$	8.011904
		39		3.40 × ${10}^{-3}$	1.13 × ${10}^{-4}$	1.09 × ${10}^{-4}$	8.525006
		40		2.80 × ${10}^{-3}$	9.44 × ${10}^{-5}$	9.08 × ${10}^{-5}$	6.758216
		41		2.20 × ${10}^{-3}$	7.23 × ${10}^{-5}$	6.95 × ${10}^{-5}$	6.854242
	824	58	10	8.91 × ${10}^{-2}$	3.10 × ${10}^{-3}$	3.00 × ${10}^{-3}$	8.248187
			12	3.10 × ${10}^{-3}$	1.07 × ${10}^{-4}$	1.03 × ${10}^{-4}$	9.049039
			14	2.40 × ${10}^{-3}$	8.23 × ${10}^{-5}$	7.93 × ${10}^{-5}$	9.104889
			16	2.30 × ${10}^{-3}$	7.95 × ${10}^{-5}$	7.67 × ${10}^{-5}$	9.238572
[32]				3.19 × ${10}^{-5}$

, the errors on domain ${\mathsf{\Omega }}_5$ using GA RBF are shown. We can see that the proposed method has efficiently solved the considered problem of regular and irregular domains. Figure 26a,b shows the numerical and exact solutions of problem 4 on domain ${\mathsf{\Omega }}_1.$ It is observed that the numerical solution agrees well with the analytic solution. In Figure 27a,b the surface plots of $Erro{r}_1$ and $Erro{r}_2$ using MQ RBF on domain ${\mathsf{\Omega }}_1$ with $n_g=4225,n_l=40,N=16,\alpha =0.7,\beta =1.9$ are depicted. The surface plots of $Erro{r}_1$ and $Erro{r}_2$ using GA RBF on domain ${\mathsf{\Omega }}_1$ with $n_g=6400,n_l=30,N=16,\alpha =0.2,\beta =1.8$ are shown in Figure 28a,b. The scatter plots of $Erro{r}_1$ and $Erro{r}_2$ using MQ RBF on domain ${\mathsf{\Omega }}_3$ with $n_g=1281,n_l=40,N=16,\alpha =0.7,\beta =1.9$ are depicted in Figure 29a,b; using GA RBF, they are presented in Figure 30a,b. The scatter plots of $Erro{r}_1$ and $Erro{r}_2$ using MQ RBF on domain ${\mathsf{\Omega }}_5$ with $n_g=3069,n_l=80,N=16,\alpha =0.3,\beta =1.6$ are shown in Figure 31a,b; using GA RBF with $n_g=3069,n_l=30,N=16,\alpha =0.3,\beta =1.6$, they are displayed in Figure 32a,b. The scatter plots of $Erro{r}_3$ of the proposed method corresponding to problem 4 using MQ RBF on ${\mathsf{\Omega }}_3$ with $n_g=1875,n_l=30,N=18,\alpha =0.7,\beta =1.9$ are shown in Figure 33a. Similarly, the scatter plots of $Erro{r}_3$ using GA RBF on ${\mathsf{\Omega }}_5$ with $n_g=2259,n_l=30,N=18,\alpha =0.7,\beta =1.9$ are shown in Figure 33b. It can be seen that the performance of the method is excellent for irregular domains as well.

Problem 5.

$${}_0^c{D}_t^{\alpha }u(x,y,t)+{}_0^c{D}_t^{\beta }u(x,y,t)=\Delta u(x,y,t)+f(x,y,t),0\le x,y\Leftarrow \Rightarrow 1,$$

(26)

with analytic solution

$$u(x,y,t)=t^3\left(\frac{{(x-0.5)}^2}{\gamma }+\frac{{(y-0.5)}^2}{\gamma }\right).$$

This problem is considered for validating the numerical scheme on domains ${\mathsf{\Omega }}_1$, ${\mathsf{\Omega }}_2$ and ${\mathsf{\Omega }}_4$, as shown in Figure 19a, Figure 19b and Figure 19d, respectively. The errors of the proposed method on domain ${\mathsf{\Omega }}_1$ for problem 5 using MQ RBF are presented in

Table 12. The errors obtained using the proposed method for problem 5 on domain ${\mathsf{\Omega }}_1$ using MQ RBF.

$\mathit{\alpha }=0.60,$ $\mathit{\beta }=1.50$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	1296	18	20	7.11 × ${10}^{-4}$	1.97 × ${10}^{-5}$	1.14 × ${10}^{-4}$	4.254578
	1369			6.30 × ${10}^{-4}$	1.70 × ${10}^{-5}$	9.87 × ${10}^{-5}$	4.517336
	1444			6.03 × ${10}^{-4}$	1.58 × ${10}^{-5}$	9.20 × ${10}^{-5}$	5.330350
	1600			5.92 × ${10}^{-4}$	1.48 × ${10}^{-5}$	8.57 × ${10}^{-5}$	5.933110
	2304	15	12	1.10 × ${10}^{-3}$	2.38 × ${10}^{-5}$	1.37 × ${10}^{-4}$	6.226840
			14	4.60 × ${10}^{-4}$	9.58 × ${10}^{-6}$	5.52 × ${10}^{-5}$	6.486711
			16	2.86 × ${10}^{-4}$	5.96 × ${10}^{-6}$	3.43 × ${10}^{-5}$	7.310871
			18	2.81 × ${10}^{-4}$	5.85 × ${10}^{-6}$	3.37 × ${10}^{-5}$	7.713956
	1600	13	18	6.47 × ${10}^{-4}$	1.61 × ${10}^{-5}$	9.36 × ${10}^{-5}$	3.810299
		14		5.98 × ${10}^{-4}$	1.49 × ${10}^{-5}$	8.66 × ${10}^{-5}$	4.302964
		15		5.94 × ${10}^{-4}$	1.48 × ${10}^{-5}$	8.59 × ${10}^{-5}$	4.404455
		16		5.03 × ${10}^{-4}$	1.25 × ${10}^{-5}$	7.28 × ${10}^{-5}$	4.543246
[32]				1.01 × ${10}^{-4}$

; using GA RBF, they are presented in

Table 13. The errors obtained using the proposed method for problem 5 on domain ${\mathsf{\Omega }}_1$ using GA RBF.

$\mathit{\alpha }=0.70,$ $\mathit{\beta }=1.60$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	2025	16	18	3.33 × ${10}^{-4}$	7.42 × ${10}^{-6}$	4.28 × ${10}^{-5}$	6.454437
	2500			1.98 × ${10}^{-4}$	3.97 × ${10}^{-6}$	2.28 × ${10}^{-5}$	8.824310
	3025			1.25 × ${10}^{-4}$	2.28 × ${10}^{-6}$	1.31 × ${10}^{-5}$	11.457521
	3600			1.06 × ${10}^{-4}$	1.77 × ${10}^{-6}$	1.02 × ${10}^{-5}$	15.700804
	3969	12	18	9.70 × ${10}^{-3}$	1.53 × ${10}^{-4}$	8.80 × ${10}^{-4}$	21.993975
		13		1.49 × ${10}^{-4}$	2.37 × ${10}^{-6}$	1.36 × ${10}^{-5}$	24.161273
		14		1.48 × ${10}^{-4}$	2.35 × ${10}^{-6}$	1.34 × ${10}^{-5}$	23.322577
		16		1.16 × ${10}^{-4}$	1.85 × ${10}^{-6}$	1.06 × ${10}^{-5}$	24.757581
	3721	16	12	1.40 × ${10}^{-3}$	2.35 × ${10}^{-5}$	1.35 × ${10}^{-4}$	16.911628
			14	4.58 × ${10}^{-4}$	7.52 × ${10}^{-6}$	4.31 × ${10}^{-5}$	18.235145
			16	1.10 × ${10}^{-4}$	1.80 × ${10}^{-6}$	1.03 × ${10}^{-5}$	20.645716
			18	9.52 × ${10}^{-5}$	1.56 × ${10}^{-6}$	8.95 × ${10}^{-6}$	21.862306
[32]				1.01 × ${10}^{-4}$

. The errors obtained using MQ RBF on domain ${\mathsf{\Omega }}_2$ are presented in

Table 14. The errors obtained using the proposed method for problem 5 on domain ${\mathsf{\Omega }}_2$ using MQ RBF.

$\mathit{\alpha }=0.70,$ $\mathit{\beta }=1.60$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	718	53	23	3.90 × ${10}^{-4}$	1.45 × ${10}^{-5}$	7.60 × ${10}^{-5}$	9.462843
	822			2.99 × ${10}^{-4}$	1.04 × ${10}^{-5}$	5.44 × ${10}^{-5}$	10.868986
	893			2.62 × ${10}^{-4}$	8.77 × ${10}^{-6}$	4.62 × ${10}^{-5}$	11.586178
	1115			2.61 × ${10}^{-4}$	7.83 × ${10}^{-6}$	4.10 × ${10}^{-5}$	13.733598
	1286	50	12	7.92 × ${10}^{-4}$	2.21 × ${10}^{-5}$	1.14 × ${10}^{-4}$	13.692401
			14	6.78 × ${10}^{-4}$	1.89 × ${10}^{-5}$	9.81 × ${10}^{-5}$	13.796613
			16	4.95 × ${10}^{-4}$	1.38 × ${10}^{-5}$	7.16 × ${10}^{-5}$	14.059077
			18	4.70 × ${10}^{-4}$	1.31 × ${10}^{-5}$	6.81 × ${10}^{-5}$	14.525208
	1357	45	18	5.19 × ${10}^{-4}$	1.41 × ${10}^{-5}$	7.33 × ${10}^{-5}$	14.082632
		46		4.68 × ${10}^{-4}$	1.27 × ${10}^{-5}$	6.61 × ${10}^{-5}$	13.687417
		47		4.03 × ${10}^{-4}$	1.09 × ${10}^{-5}$	5.68 × ${10}^{-5}$	14.141039
		48		1.96 × ${10}^{-5}$	5.33 × ${10}^{-6}$	2.77 × ${10}^{-5}$	14.555474
[32]				1.01 × ${10}^{-4}$

; using GA RBF on domain ${\mathsf{\Omega }}_4$, they are presented in

Table 15. The errors obtained using the proposed method for problem 5 on domain ${\mathsf{\Omega }}_4$ using GA RBF.

$\mathit{\alpha }=0.70,$ $\mathit{\beta }=1.60$	${\mathit{n}}_{\mathit{g}}$	${\mathit{n}}_{\mathit{l}}$	N	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_1$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_2$	${\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}}_3$	C.Time(s)
	1765	28	18	2.70 × ${10}^{-3}$	6.32 × ${10}^{-5}$	3.48 × ${10}^{-4}$	12.738675
		29		7.78 × ${10}^{-4}$	1.85 × ${10}^{-5}$	1.02 × ${10}^{-4}$	12.319389
		30		7.03 × ${10}^{-4}$	1.67 × ${10}^{-5}$	9.23 × ${10}^{-5}$	12.649178
		32		5.20 × ${10}^{-4}$	1.23 × ${10}^{-5}$	6.82 × ${10}^{-5}$	13.501318
	2031	15	16	8.69 × ${10}^{-4}$	1.93 × ${10}^{-5}$	1.06 × ${10}^{-4}$	10.713900
	2108			7.38 × ${10}^{-4}$	1.60 × ${10}^{-5}$	8.80 × ${10}^{-5}$	6.853962
	2292			5.97 × ${10}^{-4}$	1.24 × ${10}^{-5}$	6.81 × ${10}^{-5}$	7.950902
	2367			3.93 × ${10}^{-4}$	8.09 × ${10}^{-6}$	4.39 × ${10}^{-5}$	8.201278
	2460	15	12	9.49 × ${10}^{-4}$	1.91 × ${10}^{-5}$	1.03 × ${10}^{-4}$	7.867139
			14	8.75 × ${10}^{-4}$	1.76 × ${10}^{-5}$	9.56 × ${10}^{-5}$	7.951785
			16	5.96 × ${10}^{-4}$	1.20 × ${10}^{-5}$	6.52 × ${10}^{-5}$	8.449387
			18	5.57 × ${10}^{-4}$	1.12 × ${10}^{-5}$	6.09 × ${10}^{-5}$	8.862747
[32]				1.01 × ${10}^{-4}$

. The proposed method produced accurate and stable results on regular as well as irregular domains. The numerical and exact solutions of problem 5 on domain ${\mathsf{\Omega }}_1$ are shown in Figure 34a,b; a nice agreement is observed. The surface plots of $Erro{r}_1$ and $Erro{r}_2$ on domain ${\mathsf{\Omega }}_1$ using MQ RBF with $n_g=4900,n_l=35,N=18,\alpha =0.7,\beta =1.6$ are shown in Figure 35a,b. The surface plots of $Erro{r}_1$ and $Erro{r}_2$ using GA RBF on ${\mathsf{\Omega }}_1$ with $n_g=8100,n_l=35,N=18,\alpha =0.7,\beta =1.6$ are shown in Figure 36a,b. The mesh plots of $Erro{r}_1$ and $Erro{r}_2$ using MQ RBF on domain ${\mathsf{\Omega }}_2$ with $n_g=5064,n_l=30,N=18,\alpha =0.9,\beta =1.3$ are depicted in Figure 37a,b; using GA RBF, they are presented in Figure 38a,b. The scatter plots of $Erro{r}_1$ and $Erro{r}_2$ using MQ RBF on domain ${\mathsf{\Omega }}_4$ with $n_g=6393,n_l=35,N=16,\alpha =0.7,\beta =1.6$ are displayed in Figure 39a,b; using GA RBF, they are presented in Figure 40a,b. The mesh plot of $Erro{r}_3$ using MQ RBF on domain ${\mathsf{\Omega }}_2$ with $n_g=5064,n_l=35,N=18,\alpha =0.7,\beta =1.6$ is presented in Figure 41a; the scatter plot of $Erro{r}_3$ using GA RBF on domain ${\mathsf{\Omega }}_4$ with $n_g=6393,n_l=35,N=18,\alpha =0.7,\beta =1.6$ is presented in Figure 41b. It is observed that the method can solve these types of problems efficiently on regular as well as irregular domains.

5. Conclusions

In this article, an efficient numerical method based on Laplace transform and local radial basis functions for the numerical modeling of fractional mixed diffusion and wave diffusion equations with two Caputo derivatives of order $0<\alpha <1$ and $1<\beta <2$ is proposed. Numerical experiments were conducted in regular and irregular domains with uniform and non-uniform nodes. It is observed that the proposed method effectively solves problems defined on regular and irregular domains with a large number of nodes, requiring less computational time without experiencing any time instability. The results lead to the conclusion that the proposed method is an efficient and reliable technique for the considered problem in all types of domains. However, implementing this computational framework in coupled multi-component systems of partial differential equations remains an open problem. Successful implementation of such a framework could pave the way for further mathematical and biological applications. In future work, we will aim to extend our numerical method to include multiple components of the differential operator of order 2, as introduced in [57]. This will allow us to compare the impact of the errors of the proposed method and the validity of the model with clinical outcomes, thereby facilitating the understanding of complex phenomena.