An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

Abstract

In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.

1. Introduction

In recent decades, scientists and researchers have paid much attention to the expression of the physical models and chemical processes in the form of fractional derivative equations. Over time, the development of fractional calculus theory has provided an advantageous tool for modelling many natural processes that often have complex and anomalous modeling and cannot be easily expressed with classical derivative calculus. Thus, the popularity of fractional calculus theory increased because they can satisfactorily model such problems, which especially have been occurred in physics, chemistry, quantum mechanics, and viscoelasticity [1,2]. From about 20 years, radial basis functions (RBFs) are a primary mesh-free method for numerically solving PDEs on with collocation approaches (see, for example [3,4,5]). The versatility of scattered data interpolation techniques is confirmed by a lot of applications, e.g., surface reconstruction, image restoration and inpainting, meshless/Lagrangian methods for fluid dynamics, surface deformation or motion capture systems allowing the recording of sparse motions from deformable objects such as human faces and bodies [6]. The numerical solution of partial differential equations by a global collocation approach based on RBF, is also referred to as a strong form solution in the PDE literature [7,8,9,10]. An alternative interesting approach in collocation methods is to use other bases as for example Hermite exponential spline defined in [11]. The main drawback in the global approach, though spectrally accurate, consists in solving large, ill-conditioned, dense linear systems and many attempts are known to deal with it [12,13]. Recently, a direct meshless method based on Gaussian radial basis functions has been developed to solve one-dimensional linear and nonlinear convection-diffusion problems [14]. Local methods are so preferred, giving up spectral accuracy for a sparse, better-conditioned linear system and more flexibility for handling non-linearities. A wide literature concerns local RBF schemes by partitioning the domain, referred to as Partition of Unity (PU) [15,16,17]. Moreover, the two-term time-fractional PDE model, one of the interesting models in mathematical physics, is investigated and solved by employing a local meshless method [18]. More recently, numerically, a local meshless collocation method has been applied to simulate the time-fractional coupled Korteweg-de Vries and Klein-Gordon equations [19].

This paper applies an efficient local meshless approach based on one of the most applicable and popular RBFs, the so-called Gaussian RBF, to solve the space fractional derivative equations. The proposed method, due to its local property, which creates the sparse coefficient matrix and accelerates the execution of the algorithm as well as uses the space–time RBFs, can be a suitable tool for solving practical problems in physics and engineering with space fraction derivatives. More in detail, one of the most popular fractional models is the space fractional wave equation created from the classical wave equation by replacing the integer space derivatives of order 2 with some fractional derivatives of order $\gamma$, $1<\gamma \le 2$. Finally, the numerical results demonstrate the power and efficiency of the suggested method to apply for more complex fractional partial differential equations with applications in high-dimensional computational finance.

In this work, we consider a space fractional damped wave equation as follows:

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial t^2}u(\mathbf{x},t)+A_1\frac{\partial }{\partial t}u(\mathbf{x},t)+A_{2}u(\mathbf{x},t)& =& K_1·{}_{a_1}{D}_{x_1}^{{\alpha }_1}u(\mathbf{x},t)+K_2·{}_{a_2}{D}_{x_2}^{{\alpha }_2}u(\mathbf{x},t)+\hfill \\ & \cdots & +K_n·{}_{a_n}{D}_{x_n}^{{\alpha }_n}u(\mathbf{x},t)+f(\mathbf{x},t),\hfill \\ & \mathrm{with},& \mathbf{x}=(x_1,x_2,\cdots ,x_n)\in \Omega \subset {\mathbb{R}}^n,0<t\le T,\hfill \end{array}$$

(1)

with the initial and boundary conditions

$$u(\mathbf{x},0)=\varphi (x,y),\frac{\partial u}{\partial t}(\mathbf{x},0)=\psi (x,y),(x,y)\in \Omega ,$$

(2)

$$u(\mathbf{x},t)=0,(x,y)\in \partial \Omega ,0<t\le T,$$

(3)

where u is the unknown function, f is the source function, $\overline{\Omega }=\Omega \cup \partial \Omega ={[0,1]}^n$, the coefficients $K_1,K_2,\cdots ,K_n>0$. Moreover, $1<{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\le 2$ are the order of the left Riemann–Liouville space fractional derivatives ${}_{a_1}{D}_{x1}^{{\alpha }_1}u{,}_{a_2}{D}_{x2}^{{\alpha }_2}u,\cdots {,}_{a_n}{D}_{x_n}^{{\alpha }_n}u$ respect to $x_1,x_2,\cdots ,x_n$, respectively. The left and right Riemann–Liouville fractional derivative is defined by

$${}_{a_i}{D}_{x_i}^{{\alpha }_i}u(\mathbf{x},t)=\frac{1}{\Gamma (2-{\alpha }_i)}\frac{d^2}{d{x}_i^2}{\int }_{a_i}^{x_i}{(x_i-\xi )}^{1-{\alpha }_i}u(x_1,\cdots ,\xi ,\cdots ,x_n,t)d\xi ,$$

(4)

and

$${}_{x_i}{D}_{b_i}^{{\alpha }_i}u(\mathbf{x},t)=\frac{1}{\Gamma (2-{\alpha }_i)}\frac{d^2}{d{x}_i^2}{\int }_{x_i}^{b_i}{(\xi -x_i)}^{1-{\alpha }_i}u(x_1,\cdots ,\xi ,\cdots ,x_n,t)d\xi ,$$

(5)

where $\Gamma$ denotes the gamma function.

In the governing Equation (1), if $A_1=A_2=0$, we have the space fractional wave equation. In this work, both wave and damped wave equation are investigated.

The paper is organized as follow. The Section 2 is devoted to the derivation of the numerical method for wave and damped equations. In the Section 3, numerical results in terms of accuracy and efficiency have been reported. Finally, Conclusions close the paper.

2. The Numerical Scheme for Wave and Damped Models

In this section, an impressive numerical method in the meshless literature with local features based on the space–time Gaussian radial basis function is investigated to solve the left Riemann–Liouville space fractional derivatives wave and damped wave equation in high-dimensional space. Due to using the space–time radial basis function, the suggested method does not need to discretize the problem in the time direction; this capability leads to $(n+1)$-dimensional spatial-temporal computational domain for n-dimensional problems. Hence, by employing the local meshless technique as a significant feature [20], the resulting coefficient matrix will be sparse. This remarkable ability reduces computational costs and increases the speed-up of the numerical process for high-dimensional problems. Therefore the local feature, as a remarkable and efficient property, of the suggested numerical procedure leads to the sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. For this purpose, we need to evaluate and approximate the differential operators that appear in governing problem (1) based on the space–time Gaussian radial basis function. Then the localized meshless method is applied to solve the high-dimensional wave and damped wave equations.

2.1. Evaluate Derivative Operators with Space–Time Gaussian RBF

Radial Basis Functions (RBFs) are a class of real-valued functions whose values at a specified data point depending on the distance from two points in multi-dimensional space. Kansa first used RBFs in 1991 to solve fluid dynamics problems [7]. Then, due to its ease of use, RBF was given special attention by scientists in science and engineering, and several meshless methods using RBFs were proposed. In recent decades, meshless numerical techniques attracted a lot of attention due to their high flexibility and good performance to deal with practical high dimensional models with complicated and irregular domains applied problems in science and engineering. Therefore, it is necessary to first provide the general definition of the function as follows

Definition 1.

Let ${\mathbb{R}}^n$ be n-dimensional Euclidean space. Let $\Phi :{\mathbb{R}}^n\to \mathbb{R}$ be an invariant function whose value at any point $x\in {\mathbb{R}}^n$ depends only on the distance from the fixed point $\xi \in {\mathbb{R}}^n$, and can be written

$$\Phi (\parallel x-\xi \parallel ).$$

(6)

Then the function Φ is a radial basis function (RBF), where ξ is the center of the RBF Φ.

Some popular and commonly used kinds of the RBFs include

• Gaussian: $\Phi \left(r\right)=e^{-c^2{r}^2}$,
• Multiquadric: $\Phi \left(r\right)=\sqrt{r^2+c^2}$,
• Inverse multiquadric: $\Phi \left(r\right)=\frac{1}{r^2+c^2}$’
• Thin plate spline: $r^{2}logr$,

where $r=\parallel x-\xi \parallel$ denotes the Euclidean distance between x and $\xi$. Moreover, c is the shape parameter which has an important rule to control the accuracy and stability of the numerical method. In the current work, the Gaussian RBF is used in the numerical experiments. Gaussian RBF is strictly positive definite function which causes the Gaussian RBF to be the most commonly used RBF. Aslo, Gaussian RBF is a representative member of the class of infinitely differentiable functions with global support.

As mentioned earlier, we use space–time dependent radial basis functions. In the current work, the Gaussian RBF is used to approximate the numerical solution u in relation (21), which is defined as

$${\Phi }_i(\mathbf{x},t)=e^{-c^2{\parallel (\mathbf{x},t)-({\mathbf{x}}_i,t_i)\parallel }_2^2},\mathbf{x}\in {\mathbb{R}}^n,0<t\le T.$$

(7)

Moreover, the Gaussian RBF could be described by the following form

$${\Phi }_i\left(r\right)=e^{-c^2{r}^2},$$

(8)

where

$$r=\sqrt{{(x_1-x_{1i})}^2+{(x_2-x_{2i})}^2+\cdots +{(x_n-x_{ni})}^2+ϵ{(t-t_i)}^2}.$$

(9)

with $(x_1,x_2,\cdots ,x_n)=\mathbf{x}\in {\mathbb{R}}^n,0<t\le T$. In the two above definitions of the Gaussian function, the c and $ϵ$ constants are known as shape parameters. They play an important role in the accuracy and stability of computation. Finding the optimal value of shape parameters is a fundamental issue of meshless methods based on RBFs. We have used trial and error techniques to find the best value of shape parameters in the numerical procedure. As can be seen, Gaussian functions, due to their mathematical structure, have a special ability and ease of use to be applied to approximate fractional and integer derivative operators. Therefore the second integer derivative of $u(\mathbf{x},t)$ respect to t could be approximated as follow

$$\frac{{\partial }^2}{\partial t^2}u(\mathbf{x},t)=2c^2{ϵ}^2\sum _{i=1}^N(2c^2{ϵ}^2{(t-t_i)}^2-1){\Phi }_i(\mathbf{x},t).$$

(10)

The approximation of Riemann–Liouville spatial fractional derivatives using RBFs is a significant and major issue. Estimating and calculating such derivatives is not easy for most RBFs; however, selecting the Gaussian RBF allows us to use the Taylor expansion of this function to approximate Riemann–Liouville fractional derivatives in the x and y directions. The approximation of fractional derivatives due to the expansion of the gaussian in a MacLaurin series is investigated in [21]. In this work, we follow a similar way to approximate the space fractional derivative as follows

$$\begin{array}{c}\hfill D_{x_j}^{\alpha }{\Phi }_i(x_1,\cdots ,x_j,\cdots ,x_n,t)=D_{x_j}^{\alpha }{e}^{c^2[{(x_1-x_{1i})}^2+\cdots +{(x_j-x_{ji})}^2+\cdots +{(x_n-x_{ni})}^2+{ϵ}^2{(t-t_i)}^2]}\\ \hfill =e^{-c^2[{(x_1-x_{1i})}^2,\cdots ,{(x_{j-1}-x_{j-1i})}^2+{(x_{j+1}-x_{j+1i})}^2+\cdots +{(x_n-x_{ni})}^2+{ϵ}^2{(t-t_i)}^2]}{·}_a{D}_{x_j}^{\alpha }{e}^{-c^2{(x_j-x_{ji})}^2},\end{array}$$

(11)

where

$$D_{x_j}^{\alpha }{e}^{-c^2{(x_j-x_{ji})}^2}=\sum _{p=0}^{\infty }\frac{{(-1)}^p{c}^{2p}}{p!}\left(\sum _{q=0}^{2p}\frac{\left(2p\right)!{(-1)}^q{x}_{ji}^{2p-q}}{q!(2p-q)!}{D}_{x_j}^{\alpha }{x}_j^p\right)$$

(12)

Moreover, if the left Riemann–Liouville fractional derivative operator is considered, i.e., $D_{x_j}^{\alpha }:={}_a{D}_{x_j}^{\alpha }$ then the left Riemann–Liouville fractional derivative of polynomials are defined by

$${}_a{D}_{x_j}^{\alpha }{x}_j^q=\frac{q!{(x_j-a)}^{-\alpha }\Gamma (\alpha +1)}{\Gamma (q+1-\alpha )}\sum _{k=0}^q\frac{x^{q-k}{(-a)}^k}{k!\Gamma (\alpha -k+1)}$$

(13)

Moreover, for right Riemann–Liouville fractional derivative operator, i.e., $D_{x_j}^{\alpha }:={}_{x_j}{D}_b^{\alpha }$ we have

$${}_{x_j}{D}_b^{\alpha }{x}_j^q=\frac{q!{(b-x_j)}^{-\alpha }\Gamma (\alpha +1)}{\Gamma (q+1-\alpha )}\sum _{k=0}^q\frac{x^{q-k}{(-b)}^k}{k!\Gamma (\alpha -k+1)}.$$

(14)

2.2. Meshless Localized Space–Time RBF Collocation Method

One of the most efficient meshless methods to deal with time-dependent problems is the localized space–time radial basis function (RBF) collocation method [22]. This technique as a generalization of radial basis function collocation method is defined by considering both the spatial and time variable to construct the radial basis functions. To better clarify, before introducing the localized RBF collocation method, the main idea of the global RBF collocation method is briefly reviewed. In general, we consider the following boundary value problem on global bounded domain $\overline{\Omega }=\Omega \cup \partial \Omega$:

$$\mathcal{L}u(\mathbf{x},t)=\mathfrak{f}(\mathbf{x},t),(\mathbf{x},t)\in \Omega ,$$

(15)

$$\mathcal{B}u(\mathbf{x},t)=\mathfrak{g}(\mathbf{x},t),(\mathbf{x},t)\in \partial \Omega ,$$

(16)

where $\mathcal{L}$ is the given differential operator in problem (1), which is defined as follows:

$$\mathcal{L}:=\frac{{\partial }^2}{\partial t^2}+A_1\frac{\partial }{\partial t}+A_2-K_1·{}_{a_1}{D}_{x_1}^{{\alpha }_1}-K_2·{}_{a_2}{D}_{x_2}^{{\alpha }_2}-\cdots -K_n·{}_{a_n}{D}_{x_n}^{{\alpha }_n},$$

(17)

and $\mathcal{B}$ is the boundary operator in Equation (3).

According to the global RBF collocation method, assume a set of scattered center points $X_I={\left\{({\mathbf{x}}_i,t_i)\right\}}_{i=1}^{N_i}$ in $\Omega$ and $X_B={\left\{({\mathbf{x}}_i,t_i)\right\}}_{i=N_{i+1}}^N$ on $\partial \Omega$ cover the entire global domain. Then the unknown solution of the boundary value problem (15) and (16) is represented in the following form:

$$u(\mathbf{x},t)\simeq \sum _{i=1}^N{\lambda }_i\Phi (\parallel (\mathbf{x},t)-({\mathbf{x}}_i,t_i)\parallel ),(\mathbf{x},t)\in \overline{\Omega },$$

(18)

where $\Phi :\overline{\Omega }\times \overline{\Omega }⟶\mathbb{R}$ is a selected radial basis function, $\parallel ·\parallel$ denotes the Euclidean norm and ${\left\{{\lambda }_i\right\}}_{i=1}^N$ are unknown coefficients to be determined. By Substituting the sugggested solution (18) in the Equation (15) with boundary condition (16), the collocation procedure conclude the algebraic system under the form:

$$\sum _{i=1}^N{\lambda }_i\mathcal{L}\Phi (\parallel ({\mathbf{x}}_j,t_j)-({\mathbf{x}}_i,t_i)\parallel )=\mathfrak{f}({\mathbf{x}}_j,t_j),j=1,\cdots ,N_i,$$

(19)

$$\sum _{i=1}^N{\lambda }_i\mathcal{B}\Phi (\parallel ({\mathbf{x}}_j,t_j)-({\mathbf{x}}_i,t_i)\parallel )=\mathfrak{g}({\mathbf{x}}_j,t_j),j=N_{i+1},\cdots ,N,$$

(20)

By solving the system of Equations (19) and (20) the unknown vector ${\left\{{\lambda }_i\right\}}_{i=1}^N$ is determined. Then the solution u at any point of the entire computational domain can be determined by substituting the vector ${\left\{{\lambda }_i\right\}}_{i=1}^N$ in the Equation (18). The meshless global RBF collocation method, also known in the literature as the Kansa method, was introduced by Kansa in two famous references [7] and got some attention from researchers. The global collocation method, in turn, has been proposed as an efficient technique in dealing with high-dimensional problems and solving problems in complex computational domains. Despite all the advantages of the global collocation method, the coefficient matrix is dense and imposing high computational costs. Moreover, the coefficients matrix is generally ill-conditioned and leading to ill-conditioning behaviour in the numerical method. In addition, the numerical solution is seriously affected by the shape parameter in the radial basis functions dependent on the shape parameter such as Gaussian and multiquadric RBFs, so selecting the appropriate shape parameter is difficult and sensitive. Due to the popularity of meshless methods, various techniques have been proposed to overcome these difficulties and reduce their destructive effects. One of the most popular numerical techniques is the local collocation method developed by Lee et al. They demonstrated that the local collocation method is less sensitive to the selection of the shape parameter and the distribution of scattered points in the computational domain, but the computational accuracy is slightly reduced compared to the global method. Moreover, the resulting final coefficients matrix in the local collocation method is sparse and considerably well-conditioned. In the local collocation method, for any center point $({\mathbf{x}}_c,t_c)$ in the computational domain $\Omega$, a local sub-domain ${\Omega }_c$ consist of $n_c$ nearest neighbor points is considered. The set of points in the local sub-domain is called a stencil. The approximate solution $u({\mathbf{x}}_c,t_c)$ using the RBF collocation method on the stencil ${\Omega }_c$ can be obtained by a linear combination of the radial basis functions at $n_c$ nearest neighbouring points of $({\mathbf{x}}_c,t_c)$ in the following form:

$$u({\mathbf{x}}_c,t_c)\simeq \widehat{u}({\mathbf{x}}_c,t_c)=\sum _{k=1}^{n_c}{\lambda }_k\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_k,t_k)\parallel ),({\mathbf{x}}_k,t_k)\in {\Omega }_c,$$

(21)

where $\Phi$ is a radial basis function and ${\left\{{\lambda }_k\right\}}_{k=1}^{n_c}$ are unknown coefficients to be determined. Since ${\left\{({\mathbf{x}}_k,t_k)\right\}}_{k=1}^{n_c}\subset {\Omega }_c$ from Equation (21) the following linear system is obtained

$${\widehat{\mathbf{U}}}_c={\mathbf{\Phi }}_c{\mathbf{\Lambda }}_c,$$

(22)

where

$$\begin{array}{ccc}\hfill {\mathbf{\Phi }}_c& =& [\Phi (\parallel ({\mathbf{x}}_i,t_i)-({\mathbf{x}}_j,t_j)\parallel \left)\right],1\le i,j\le n_c,\hfill \\ \hfill {\widehat{\mathbf{U}}}_c& =& [\widehat{u}({\mathbf{x}}_1,t_1),\widehat{u}({\mathbf{x}}_2,t_2),\cdots ,\widehat{u}({\mathbf{x}}_{n_c},t_{n_c})],\hfill \\ \hfill {\mathbf{\Lambda }}_c& =& {[{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{n_c}]}^T.\hfill \end{array}$$

Therefore, solving the linear system (22) yields the unknown coefficients as follows:

$${\mathbf{\Lambda }}_c={\mathbf{\Phi }}_c^{-1}{\widehat{\mathbf{U}}}_c.$$

(23)

Evaluating the operator $\mathcal{L}$ of the interpolant (21) and using the relation (23) give

$$\begin{array}{}\hfill \mathrm{(24)}& \hfill \mathcal{L}\widehat{u}({\mathbf{x}}_c,t_c)& =& \sum _{k=1}^{n_c}{\lambda }_k\mathcal{L}\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_k,t_k)\parallel )\hfill \hfill \mathrm{(25)}& & =& \left[\mathcal{L}\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_1,t_1)\parallel ),\cdots ,\mathcal{L}\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_k,t_k)\parallel )\right]\Lambda \hfill \hfill \mathrm{(26)}& & =& \left(\mathcal{L}{\mathbf{\Phi }}_c\right){\mathbf{\Phi }}_c^{-1}{\widehat{\mathbf{U}}}_c.\hfill \end{array}$$

Now, by substituting the derivative operator $\mathcal{L}$ in relation (17), the following equation is obtained

$$\begin{array}{ccc}\hfill & & [\frac{{\partial }^2}{\partial t^2}+A_1\frac{\partial }{\partial t}+A_2-K_1·{}_{a_1}{D}_{x_1}^{{\alpha }_1}-K_2·{}_{a_2}{D}_{x_2}^{{\alpha }_2}-\cdots -K_n·{}_{a_n}{D}_{x_n}^{{\alpha }_n}]\widehat{u}({\mathbf{x}}_c,t_c)\hfill \\ & =& \underset{{\mathbf{W}}_{c\mathcal{L}}}{\underbrace{\left([\frac{{\partial }^2}{\partial t^2}+A_1\frac{\partial }{\partial t}+A_2-K_1·{}_{a_1}{D}_{x_1}^{{\alpha }_1}-K_2·{}_{a_2}{D}_{x_2}^{{\alpha }_2}-\cdots -K_n·{}_{a_n}{D}_{x_n}^{{\alpha }_n}]{\mathbf{\Phi }}_c\right){\mathbf{\Phi }}_c^{-1}}}{\widehat{\mathbf{U}}}_c\hfill \\ & =& {\mathbf{W}}_{c\mathcal{L}}{\widehat{\mathbf{U}}}_c.\hfill \end{array}$$

(27)

As the same way, for all $({\mathbf{x}}_j,t_j)\in \Omega$ the weight vector ${\mathbf{W}}_{\mathcal{L}}={[{\mathbf{W}}_{1\mathcal{L}},\cdots ,{\mathbf{W}}_{N_i\mathcal{L}}]}^T$ is computed such that

$$\mathcal{L}\widehat{u}(\mathbf{x},t)={\mathbf{W}}_{\mathcal{L}}{\widehat{\mathbf{U}}}_{\mathcal{L}},(\mathbf{x},t)\in \Omega ,$$

(28)

where ${\widehat{\mathbf{U}}}_{\mathcal{L}}=[\widehat{u}({\mathbf{x}}_1,t_1),\cdots ,\widehat{u}({\mathbf{x}}_{N_i},t_{N_i})]$. By substituting the relation (28) in Equation (15), we have

$${\mathbf{W}}_{\mathcal{L}}{\widehat{\mathbf{U}}}_{\mathcal{L}}=\mathfrak{f}(\mathbf{x},t),(\mathbf{x},t)\in \Omega .$$

(29)

We also consider a local computational domain for each point on the boundary of computational domain similar to that discussed at points within the computational domain. Therefore, for each point $({\mathbf{x}}_c,t_c)\in \partial \Omega$, according to the collocation method, the effect of the operator $\mathcal{B}$ on the function u is approximated as follows:

$$\begin{array}{}\hfill \mathrm{(30)}& \hfill \mathcal{B}\widehat{u}({\mathbf{x}}_c,t_c)& =& \sum _{k=1}^{n_c}{\lambda }_k\mathcal{B}\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_k,t_k)\parallel )\hfill \hfill \mathrm{(31)}& & =& \left[\mathcal{B}\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_1,t_1)\parallel )\cdots \mathcal{B}\Phi (\parallel ({\mathbf{x}}_c,t_c)-({\mathbf{x}}_k,t_k)\parallel )\right]\Lambda \hfill \hfill \mathrm{(32)}& & =& \left(\mathcal{B}{\mathbf{\Phi }}_c\right){\mathbf{\Phi }}_c^{-1}{\widehat{\mathbf{U}}}_c.\hfill \end{array}$$

Given that the boundary condition’ operator is the identity operator, the following equation has resulted

$$\begin{array}{ccc}\hfill & & \widehat{u}({\mathbf{x}}_c,t_c)=\underset{{\mathbf{W}}_{c\mathcal{B}}}{\underbrace{{\mathbf{\Phi }}_c{\mathbf{\Phi }}_c^{-1}}}{\widehat{\mathbf{U}}}_c\hfill \\ & =& {\mathbf{W}}_{c\mathcal{B}}{\widehat{\mathbf{U}}}_c.\hfill \end{array}$$

(33)

Moreover, for all $({\mathbf{x}}_j,t_j)\in \partial \Omega$ the weight vector ${\mathbf{W}}_{\mathcal{B}}={[{\mathbf{W}}_{N_{i+1}\mathcal{B}},\cdots ,{\mathbf{W}}_{N\mathcal{L}}]}^T$ is calculated as follows:

$$\widehat{u}(\mathbf{x},t)={\mathbf{W}}_{\mathcal{B}}{\widehat{\mathbf{U}}}_{\mathcal{B}},(\mathbf{x},t)\in \partial \Omega ,$$

(34)

where ${\widehat{\mathbf{U}}}_{\mathcal{B}}=[\widehat{u}({\mathbf{x}}_{N_{i+1}},t_{N_{i+1}}),\cdots ,\widehat{u}({\mathbf{x}}_N,t_N)]$, By substituting the Equation (34) in Equation (16), the following relation is obtained

$${\mathbf{W}}_{\mathcal{B}}{\widehat{\mathbf{U}}}_{\mathcal{B}}=\mathfrak{g}(\mathbf{x},t),(\mathbf{x},t)\in \partial \Omega .$$

(35)

Then, the following linear system of equations with $N\times N$ global sparse matrix $\mathbf{W}=[W_{\mathbf{L}};W_{\mathbf{B}}]$ for all center points ${\left\{({\mathbf{x}}_j,t_j)\right\}}_{j=1}^N$, could be assembled such that

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\mathbf{W}}_{\mathcal{L}}\\ {\mathbf{W}}_{\mathcal{B}}\end{array}\right)\left(\begin{array}{c}{\widehat{\mathbf{U}}}_{\mathcal{L}}\\ {\widehat{\mathbf{U}}}_{\mathcal{B}}\end{array}\right)=\left(\begin{array}{c}\mathfrak{f}({\mathbf{x}}_i,t_i)\\ \mathfrak{g}({\mathbf{x}}_j,t_j)\end{array}\right),1\le i\le N_i,N_{i+1}\le j\le N.\end{array}$$

(36)

Then the approximate solution ${\left\{u({\mathbf{x}}_i,t_i)\right\}}_{i=1}^N$ can be obtained by solving the above sparse linear system of equations.

3. Numerical Results

In this section, some test problems in one, two, and three spatial dimensional for investigating the accuracy and efficiency of the presented method for both wave and damped wave equations are considered. As mentioned in the suggested technique the space–time RBFs are considered, therefore in the computational process for n-dimensional spatial model, we consider the $(n+1)$-dimensional spatial-temporal computational domain with uniform distributed points. The scheme of computational domain $\Omega ={[0,1]}^2$ with scattered data points for one and two-dimensional are demonstrated in Figure 1. The three-dimensional model leads to the four-dimensional computational domain and it could not be shown. Moreover, two numerical criteria to show the accuracy, convergence, and stability of the method as absolute error $\left({ϵ}_{\infty }\right)$ and the root mean square error $\left({ϵ}_r\right)$ are considered as follows:

$${ϵ}_{\infty }={\parallel u(\mathbf{x},t)-\widehat{u}(\mathbf{x},t)\parallel }_{\infty },{ϵ}_r=\frac{\parallel u(\mathbf{x},t)-\widehat{u}{(\mathbf{x},t)\parallel }_2}{\sqrt{N}}.$$

(37)

Moreover, to investigate the convergence rates of the presented discretization scheme the following rate is estimated:

$${\mathcal{R}}_{\infty }=\frac{{log}_2({ϵ}_{\infty }\left(h_1\right)/{ϵ}_{\infty }\left(h_2\right))}{{log}_2(h_1/h_2)},$$

where $h=1/N$. Moreover, in numerical implementation for constructing the local sub-domains, the k-dimensional tree (k-d tree) algorithm is used [23]. To improve search performance, the search tree structure is first created. Then for each computational point in the spatial-temporal domain, the k-d tree algorithm search $n_c$ nearest neighbors. Moreover, to investigate the numerical stability of the suggested method to deal with the mentioned models, the noise efficacy on computational error estimates is perused. To this end, we assume that the initial solution $u_0$ in the numerical procedure is perturbed to ${\widehat{u}}_0=(1+\sigma )u_0$. Thus, the impression of the input noise $\sigma$ on error estimates is studied for perturbation solution $\widehat{u}$.

Example 1.

As the first example, we consider the one-dimensional space fractional derivatives problem (1) with corresponding to wave model by coefficients $K_1=10$, $A_1=A_2=0$ and damped wave model by coefficients $K_1=10$, $A_1=A_2=0.2$. The exact solution of the problem for both models is $u(x,t)=sin\left(\pi (t+1)\right)(x-x^2)$. The source functions for both models are calculated as follows

$$\begin{array}{ccc}\hfill f(x,t)& =& \left(-{\pi }^{2}sin\left(\pi (t+1)\right)+A_1\pi cos\left(\pi (t+1)\right)+A_{2}sin\left(\pi (t+1)\right)\right)(x-x^2)\hfill \\ & -& K_{1}sin\left(\pi (t+1)\right)\left(\frac{x^{1-{\alpha }_1}}{\Gamma (2-{\alpha }_1)}-\frac{2x^{2-{\alpha }_1}}{\Gamma (3-{\alpha }_1)}\right).\hfill \end{array}$$

In one-dimensional problem the computational domain is $\Omega \times T$ where $\Omega =[0,1]$ with uniform distribution data points and $T=[0,1]$. Moreover, the local sub-domains are determined withk-d treealgorithm with size $n_c=5$.

Table 1. Error estimates, condition numbers, CPU times and convergence rate of Example 1 by letting $n_c=5$ and different values of N and several fractional orders ${\alpha }_1$.

${\mathit{\alpha }}_1$	N	Wave Model							Damped Wave Model
		c	${\mathit{ϵ}}_{\mathit{\infty }}$	${\mathit{ϵ}}_{\mathit{r}}$	$\mathbf{CN}$	Time	${\mathcal{R}}_{\mathit{\infty }}$		c	${\mathit{ϵ}}_{\mathit{\infty }}$	${\mathit{ϵ}}_{\mathit{r}}$	$\mathbf{CN}$	Time	${\mathcal{R}}_{\mathit{\infty }}$
$1.80$	$5^2$	0.006	$2.2269\times {10}^{-3}$	$9.0263\times {10}^{-4}$	$5.1233\times {10}^3$	$0.11$	−		0.006	$2.2934\times {10}^{-3}$	$9.0038\times {10}^{-4}$	$5.0633\times {10}^3$	$0.14$	−
	${10}^2$	0.007	$4.0344\times {10}^{-4}$	$1.9577\times {10}^{-4}$	$1.2380\times {10}^5$	$0.16$	$2.46$		0.007	$4.0829\times {10}^{-4}$	$1.9889\times {10}^{-4}$	$1.2371\times {10}^5$	$0.18$	$2.48$
	${15}^2$	0.008	$1.5062\times {10}^{-4}$	$7.7824\times {10}^{-5}$	$1.5415\times {10}^6$	$0.24$	$2.42$		0.008	$1.5571\times {10}^{-4}$	$7.9075\times {10}^{-5}$	$1.4739\times {10}^6$	$0.28$	$2.37$
	${20}^2$	0.009	$6.9551\times {10}^{-5}$	$3.7304\times {10}^{-5}$	$2.8412\times {10}^6$	$0.38$	$2.68$		0.009	$7.3215\times {10}^{-5}$	$3.8292\times {10}^{-5}$	$3.2369\times {10}^6$	$0.40$	$2.62$
	${25}^2$	0.010	$3.6964\times {10}^{-5}$	$1.8750\times {10}^{-5}$	$1.8328\times {10}^7$	$0.53$	$2.83$		0.010	$3.8836\times {10}^{-5}$	$1.9777\times {10}^{-5}$	$1.5684\times {10}^6$	$0.58$	$2.84$
	${30}^2$	0.015	$2.4838\times {10}^{-5}$	$1.1746\times {10}^{-5}$	$1.0483\times {10}^7$	$0.73$	$2.18$		0.012	$2.2484\times {10}^{-5}$	$1.1545\times {10}^{-5}$	$1.1188\times {10}^7$	$0.80$	$2.99$
$1.60$	$5^2$	0.006	$3.3795\times {10}^{-3}$	$1.4492\times {10}^{-3}$	$4.5845\times {10}^3$	$0.11$	−		0.006	$3.1493\times {10}^{-3}$	$1.4469\times {10}^{-3}$	$3.9339\times {10}^3$	$0.12$	−
	${10}^2$	0.007	$6.5764\times {10}^{-4}$	$3.1243\times {10}^{-4}$	$1.5830\times {10}^5$	$0.16$	$2.36$		0.007	$7.0697\times {10}^{-4}$	$3.1627\times {10}^{-4}$	$1.5540\times {10}^5$	$0.17$	$2.15$
	${15}^2$	0.008	$2.4993\times {10}^{-4}$	$1.2304\times {10}^{-4}$	$1.5025\times {10}^6$	$0.25$	$2.38$		0.008	$2.5861\times {10}^{-4}$	$1.2397\times {10}^{-4}$	$1.4145\times {10}^6$	$0.26$	$2.48$
	${20}^2$	0.009	$1.1825\times {10}^{-4}$	$5.8391\times {10}^{-5}$	$2.1201\times {10}^7$	$0.37$	$2.60$		0.009	$1.1111\times {10}^{-4}$	$5.7728\times {10}^{-5}$	$1.8260\times {10}^6$	$0.41$	$2.93$
	${25}^2$	0.010	$6.0763\times {10}^{-5}$	$2.9338\times {10}^{-5}$	$7.0838\times {10}^7$	$0.54$	$2.98$		0.010	$4.9088\times {10}^{-5}$	$2.7473\times {10}^{-5}$	$6.9984\times {10}^6$	$0.55$	$3.66$
	${30}^2$	0.015	$5.6002\times {10}^{-5}$	$2.1354\times {10}^{-5}$	$9.0419\times {10}^7$	$0.73$	$2.18$		0.012	$2.9940\times {10}^{-5}$	$1.4093\times {10}^{-5}$	$5.8078\times {10}^7$	$0.77$	$2.99$
$1.90$	$5^2$	0.006	$1.8605\times {10}^{-3}$	$7.5008\times {10}^{-4}$	$3.6377\times {10}^3$	$0.11$	−		0.005	$1.8677\times {10}^{-3}$	$7.4356\times {10}^{-4}$	$3.6238\times {10}^3$	$0.12$	−
	${10}^2$	0.007	$4.7886\times {10}^{-4}$	$1.7566\times {10}^{-4}$	$1.1151\times {10}^6$	$0.16$	$1.95$		0.006	$7.0447\times {10}^{-4}$	$2.4132\times {10}^{-4}$	$9.7984\times {10}^5$	$0.17$	$1.40$
	${15}^2$	0.008	$1.5025\times {10}^{-4}$	$7.4091\times {10}^{-5}$	$1.9605\times {10}^6$	$0.24$	$2.85$		0.007	$2.7777\times {10}^{-4}$	$1.1673\times {10}^{-4}$	$3.1303\times {10}^6$	$0.27$	$2.29$
	${20}^2$	0.009	$7.8533\times {10}^{-5}$	$3.8566\times {10}^{-5}$	$8.9269\times {10}^6$	$0.36$	$2.25$		0.008	$1.2280\times {10}^{-4}$	$5.2598\times {10}^{-5}$	$9.5888\times {10}^6$	$0.38$	$2.67$
	${25}^2$	0.010	$5.0342\times {10}^{-5}$	$2.3522\times {10}^{-5}$	$1.4118\times {10}^7$	$0.53$	$1.99$		0.009	$7.6035\times {10}^{-5}$	$3.2475\times {10}^{-5}$	$1.0856\times {10}^7$	$0.55$	$2.14$
	${30}^2$	0.015	$4.1866\times {10}^{-5}$	$1.9279\times {10}^{-5}$	$2.7048\times {10}^7$	$0.74$	$1.01$		0.010	$5.2990\times {10}^{-5}$	$2.2894\times {10}^{-5}$	$3.1228\times {10}^7$	$0.76$	$1.98$

is demonstrated the computational errors, condition numbers, computational process times, and convergence rates concerning the different numbers of data points N and several values of fractional orders ${\alpha }_1$ for both wave and damped wave models. The numerical results verify that the presented method is accurate, convergent and, well-posed due to the increasing number of data points in the computational domain. The best value of the shape parameter c is determined by trial and error so that there is a balance between the obtained accuracy and the conditional number of the coefficient matrix. Therefore, the plot of absolute errors and condition numbers concerning the shape parameter c by letting $N={20}^2$ and ${\alpha }_1=1.80$ is demonstrated in Figure 2 for both wave and damped wave models. Moreover, the estimated errors for different values of $N_t$ and various fractional orders ${\alpha }_1$ by taking $N_x=30$ and shape parameter $c=0.015$ for wave model and $c=0.010$ for damped wave model are reported in the

Table 2. Error estimates for $N_x=30$ of Example 1 by letting $c=0.015$ in wave model and $c=0.010$ in damped wave model, $n_c=5$, different values of $N_t$ and several fractional orders ${\alpha }_1$.

${\mathit{\alpha }}_1$	${\mathit{N}}_{\mathit{t}}$	Wave Model			Damped Wave Model
		${\mathbf{ϵ}}_{\mathbf{\infty }}$	${\mathbf{ϵ}}_{\mathbf{r}}$		${\mathbf{ϵ}}_{\mathbf{\infty }}$	${\mathbf{ϵ}}_{\mathbf{r}}$
$1.55$	15	$3.8035\times {10}^{-2}$	$1.3685\times {10}^{-2}$		$5.2703\times {10}^{-2}$	$2.5408\times {10}^{-2}$
	20	$1.4421\times {10}^{-4}$	$6.3442\times {10}^{-5}$		$1.2401\times {10}^{-4}$	$6.3900\times {10}^{-5}$
	25	$8.4107\times {10}^{-5}$	$3.8606\times {10}^{-5}$		$5.5679\times {10}^{-5}$	$3.2129\times {10}^{-5}$
	30	$6.3251\times {10}^{-5}$	$3.0383\times {10}^{-5}$		$3.3992\times {10}^{-5}$	$1.7063\times {10}^{-5}$
$1.65$	15	$2.7988\times {10}^{-2}$	$1.2182\times {10}^{-2}$		$4.4133\times {10}^{-2}$	$2.1276\times {10}^{-2}$
	20	$8.1961\times {10}^{-5}$	$3.9625\times {10}^{-5}$		$8.5992\times {10}^{-5}$	$4.7289\times {10}^{-5}$
	25	$4.6205\times {10}^{-5}$	$2.0393\times {10}^{-5}$		$4.1815\times {10}^{-5}$	$2.3432\times {10}^{-5}$
	30	$4.3104\times {10}^{-5}$	$1.7328\times {10}^{-5}$		$2.6376\times {10}^{-5}$	$1.2837\times {10}^{-5}$
$1.75$	15	$3.4041\times {10}^{-2}$	$1.5951\times {10}^{-2}$		$3.7402\times {10}^{-2}$	$1.8022\times {10}^{-2}$
	20	$8.0199\times {10}^{-5}$	$3.3220\times {10}^{-5}$		$8.2582\times {10}^{-5}$	$3.9420\times {10}^{-5}$
	25	$3.5874\times {10}^{-5}$	$1.6133\times {10}^{-5}$		$3.7671\times {10}^{-5}$	$1.9878\times {10}^{-5}$
	30	$3.2914\times {10}^{-5}$	$1.3236\times {10}^{-5}$		$2.2795\times {10}^{-5}$	$1.1375\times {10}^{-5}$

. The demonstrated results in this table show the temporal convergence of the presented procedure. The stability of the suggested method is verified numerically in

Table 3. The effect of noise on error estimates of Example 1 for different values of N and letting ${\alpha }_1=1.75$.

N	$\mathit{\sigma }$	Wave Model				Damped Wave Model
		$\mathbf{c}$	${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$		$\mathbf{c}$	${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$
${10}^2$	0	$0.007$	$4.5573\times {10}^{-4}$	$2.1751\times {10}^{-4}$		$0.008$	$4.5430\times {10}^{-4}$	$2.1807\times {10}^{-4}$
	$0.001$		$6.9889\times {10}^{-4}$	$3.3333\times {10}^{-4}$			$6.6901\times {10}^{-4}$	$3.3199\times {10}^{-4}$
	$0.01$		$2.8873\times {10}^{-3}$	$1.3784\times {10}^{-3}$			$2.8574\times {10}^{-3}$	$1.3744\times {10}^{-3}$
	$0.1$		$2.4771\times {10}^{-2}$	$1.1834\times {10}^{-2}$			$2.4742\times {10}^{-2}$	$1.1829\times {10}^{-2}$
${15}^2$	0	$0.008$	$1.6829\times {10}^{-4}$	$8.5675\times {10}^{-5}$		$0.009$	$1.6508\times {10}^{-4}$	$8.5114\times {10}^{-5}$
	$0.001$		$4.1572\times {10}^{-4}$	$2.0592\times {10}^{-4}$			$3.9821\times {10}^{-4}$	$2.0389\times {10}^{-4}$
	$0.01$		$2.6657\times {10}^{-3}$	$1.2901\times {10}^{-3}$			$2.6428\times {10}^{-3}$	$1.2873\times {10}^{-3}$
	$0.1$		$2.5165\times {10}^{-2}$	$1.2134\times {10}^{-2}$			$2.5142\times {10}^{-2}$	$1.2131\times {10}^{-2}$
${20}^2$	0	$0.009$	$8.2212\times {10}^{-5}$	$4.0938\times {10}^{-5}$		$0.010$	$8.4842\times {10}^{-5}$	$4.0896\times {10}^{-5}$
	$0.001$		$3.2011\times {10}^{-4}$	$1.6265\times {10}^{-4}$			$3.0582\times {10}^{-4}$	$1.6079\times {10}^{-4}$
	$0.01$		$2.5562\times {10}^{-3}$	$1.2661\times {10}^{-3}$			$2.5419\times {10}^{-3}$	$1.2638\times {10}^{-3}$
	$0.1$		$2.4917\times {10}^{-2}$	$1.2304\times {10}^{-2}$			$2.4902\times {10}^{-2}$	$1.2301\times {10}^{-2}$
${25}^2$	0	$0.010$	$3.9177\times {10}^{-5}$	$1.9537\times {10}^{-5}$		$0.011$	$3.7677\times {10}^{-5}$	$1.9476\times {10}^{-5}$
	$0.001$		$2.7922\times {10}^{-4}$	$1.4207\times {10}^{-4}$			$2.7153\times {10}^{-4}$	$1.4024\times {10}^{-4}$
	$0.01$		$2.5277\times {10}^{-3}$	$1.2572\times {10}^{-3}$			$2.5185\times {10}^{-3}$	$1.2552\times {10}^{-3}$
	$0.1$		$2.5027\times {10}^{-2}$	$1.2411\times {10}^{-2}$			$2.5018\times {10}^{-2}$	$1.2409\times {10}^{-2}$

by considering the different input noise values $\sigma =0$, $\sigma =0.001$, $\sigma =0.01$, and $\sigma =0.1$. This table shows the influence of noise σ on computational errors and presents that the proposed method has a reasonable and stable behavior against input noise. The Figure 3 and Figure 4 are shown the absolute errors for different time levels $t=0.25$, $t=0.50$, and $t=0.75$ by letting $N_x=41$, shape parameter $c=0.007$, and ${\alpha }_1=1.65$ for both wave and damped wave model, respectively. Moreover, Figure 5 and Figure 6 are the plots of the exact and approximated solutions for wave and damped wave problems in the spatial-temporal computational domain, respectively. The sparsity patterns of the coefficient matrix related to both wave and damped wave models are plotted in Figure 7. To investigate the method’s stability numerically, the impression of the input noise on the initial solution is evaluated. For this purpose, effect of the several input noise levels $\sigma =0$, $\sigma =0.001$, $\sigma =0.01$, and $\sigma =0.1$ on error estimates are considered. These results are reported in

Table 4. The effect of noise on error estimates of Example 4 for different values of N and letting ${\alpha }_1=1.85$, ${\alpha }_2=1.65$ and ${\alpha }_3=1.45$.

N	$\mathit{\sigma }$	Wave Model				Damped Wave Model
		$\mathbf{c}$	${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$		$\mathbf{c}$	${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$
$5^4$	0	$0.030$	$6.9620\times {10}^{-5}$	$1.2229\times {10}^{-5}$		$0.010$	$5.9004\times {10}^{-5}$	$1.0412\times {10}^{-5}$
	$0.001$		$8.5245\times {10}^{-5}$	$1.4963\times {10}^{-5}$			$7.4629\times {10}^{-5}$	$1.3117\times {10}^{-5}$
	$0.01$		$2.2587\times {10}^{-4}$	$3.9592\times {10}^{-5}$			$2.1525\times {10}^{-4}$	$3.7679\times {10}^{-5}$
	$0.1$		$1.6321\times {10}^{-3}$	$2.8600\times {10}^{-4}$			$1.6215\times {10}^{-3}$	$2.8406\times {10}^{-4}$
$7^4$	0	$0.035$	$2.7000\times {10}^{-5}$	$5.7760\times {10}^{-6}$		$0.014$	$2.4669\times {10}^{-5}$	$5.2211\times {10}^{-6}$
	$0.001$		$4.2625\times {10}^{-5}$	$8.9208\times {10}^{-6}$			$4.0294\times {10}^{-5}$	$8.3482\times {10}^{-6}$
	$0.01$		$1.8325\times {10}^{-4}$	$3.7324\times {10}^{-5}$			$1.8091\times {10}^{-4}$	$3.6730\times {10}^{-5}$
	$0.1$		$1.5895\times {10}^{-3}$	$3.2153\times {10}^{-4}$			$1.5871\times {10}^{-3}$	$3.2093\times {10}^{-4}$
$9^4$	0	$0.040$	$1.6106\times {10}^{-5}$	$3.3205\times {10}^{-6}$		$0.018$	$1.7590\times {10}^{-5}$	$3.1765\times {10}^{-6}$
	$0.001$		$3.1488\times {10}^{-5}$	$6.4778\times {10}^{-6}$			$3.2283\times {10}^{-5}$	$6.4089\times {10}^{-6}$
	$0.01$		$1.7211\times {10}^{-4}$	$3.6879\times {10}^{-5}$			$1.7290\times {10}^{-4}$	$3.6873\times {10}^{-5}$
	$0.1$		$1.5783\times {10}^{-3}$	$3.4274\times {10}^{-4}$			$1.5791\times {10}^{-3}$	$3.4275\times {10}^{-4}$
${11}^4$	0	$0.045$	$8.8645\times {10}^{-6}$	$1.9753\times {10}^{-6}$		$0.020$	$7.6517\times {10}^{-6}$	$1.7709\times {10}^{-6}$
	$0.001$		$2.1782\times {10}^{-5}$	$5.0290\times {10}^{-6}$			$2.3160\times {10}^{-5}$	$5.2187\times {10}^{-6}$
	$0.01$		$1.6139\times {10}^{-4}$	$3.6821\times {10}^{-5}$			$1.6378\times {10}^{-4}$	$3.7177\times {10}^{-5}$
	$0.1$		$1.5676\times {10}^{-3}$	$3.5682\times {10}^{-4}$			$1.5700\times {10}^{-3}$	$3.5720\times {10}^{-4}$

and illustrated the acceptable and stable behavior of the numerical procedure against the input noise.

Example 2.

As the second example (see Figure 8), the two-dimensional space fractional derivatives problem (1) related to wave model by coefficients $K_1=K_2=10$, $A_1=A_2=0$ and damped wave model by by coefficients $K_1=K_2=10$, $A_1=A_2=0.2$. The exact solution of the problem for either wave and damped wave model is $u(x,y,t)=sin\left(\pi (t+1)\right)(x-x^2)(y-y^2)$. The source functions for both models are obtained as follows

$$\begin{array}{ccc}\hfill f(x,y,t)& =& \left(-{\pi }^{2}sin\left(\pi (t+1)\right)+A_1\pi cos\left(\pi (t+1)\right)+A_{2}sin\left(\pi (t+1)\right)\right)\hfill \\ & ·& (x-x^2)(y-y^2)\hfill \\ & -& K_{1}sin\left(\pi (t+1)\right)(y-y^2)\left(\frac{x^{1-{\alpha }_1}}{\Gamma (2-{\alpha }_1)}-\frac{2x^{2-{\alpha }_1}}{\Gamma (3-{\alpha }_1)}\right)\hfill \\ & -& K_{2}sin\left(\pi (t+1)\right)(x-x^2)\left(\frac{y^{1-{\alpha }_2}}{\Gamma (2-{\alpha }_2)}-\frac{2y^{2-{\alpha }_2}}{\Gamma (3-{\alpha }_2)}\right).\hfill \end{array}$$

In two-dimensional case the computational domain is $\Omega \times T$ where $\Omega ={[0,1]}^2$ with uniform distributed points and $T=[0,1]$. Moreover, thek-d treeprogram constructs the local sub-domains with size $n_c=10$.

The accuracy, convergence, and wellposedness of the presented procedure for solving both wave and damped wave models have shown in

Table 5. Error estimates, condition numbers, CPU times convergence rate of Example 2 by letting $n_c=10$ and different values of N and several fractional orders ${\alpha }_1$ and ${\alpha }_2$.

$({\mathit{\alpha }}_1,{\mathit{\alpha }}_2)$	N	Wave Model							Damped Wave Model
		c	${\mathit{ϵ}}_{\mathit{\infty }}$	${\mathit{ϵ}}_{\mathit{r}}$	$\mathbf{CN}$	Time	${\mathcal{R}}_{\mathit{\infty }}$		c	${\mathit{ϵ}}_{\mathit{\infty }}$	${\mathit{ϵ}}_{\mathit{r}}$	$\mathbf{CN}$	Time	${\mathcal{R}}_{\mathit{\infty }}$
$(1.5,1.5)$	$5^3$	0.006	$4.8054\times {10}^{-4}$	$1.3983\times {10}^{-4}$	$2.4132\times {10}^3$	$0.14$	−		0.008	$4.8850\times {10}^{-4}$	$1.3968\times {10}^{-4}$	$2.9502\times {10}^3$	$0.17$	−
	${10}^3$	0.009	$1.5822\times {10}^{-4}$	$3.8986\times {10}^{-5}$	$1.1431\times {10}^6$	$1.08$	$1.60$		0.009	$2.4934\times {10}^{-4}$	$6.3957\times {10}^{-5}$	$1.0003\times {10}^6$	$1.24$	$0.97$
	${15}^3$	0.011	$5.6943\times {10}^{-5}$	$1.7840\times {10}^{-5}$	$9.0793\times {10}^6$	$3.83$	$2.52$		0.011	$1.3405\times {10}^{-4}$	$4.1633\times {10}^{-5}$	$1.0522\times {10}^7$	$3.93$	$1.53$
	${20}^3$	0.016	$3.9691\times {10}^{-5}$	$1.1626\times {10}^{-5}$	$2.4709\times {10}^8$	$9.95$	$1.25$		0.016	$6.7846\times {10}^{-5}$	$2.0873\times {10}^{-5}$	$3.2388\times {10}^8$	$10.36$	$2.36$
$(1.4,1.8)$	$5^3$	0.008	$3.7221\times {10}^{-4}$	$1.0343\times {10}^{-4}$	$2.4408\times {10}^3$	$0.17$	−		0.009	$3.7560\times {10}^{-4}$	$1.0388\times {10}^{-4}$	$2.8029\times {10}^3$	$0.19$	−
	${10}^3$	0.010	$7.1562\times {10}^{-5}$	$2.2577\times {10}^{-5}$	$1.0923\times {10}^6$	$1.02$	$2.37$		0.010	$7.8547\times {10}^{-5}$	$2.2794\times {10}^{-5}$	$1.6189\times {10}^6$	$1.03$	$2.25$
	${15}^3$	0.013	$3.1481\times {10}^{-5}$	$8.5817\times {10}^{-6}$	$2.3637\times {10}^7$	$3.80$	$2.18$		0.013	$3.2378\times {10}^{-5}$	$8.8020\times {10}^{-6}$	$2.6136\times {10}^7$	$3.86$	$2.23$
	${20}^3$	0.020	$2.4336\times {10}^{-5}$	$4.4087\times {10}^{-6}$	$3.3440\times {10}^8$	$10.03$	$1.41$		0.020	$2.1563\times {10}^{-5}$	$4.4523\times {10}^{-6}$	$4.5045\times {10}^8$	$10.20$	$2.08$
$(1.9,1.5)$	$5^3$	0.008	$3.1363\times {10}^{-4}$	$8.4538\times {10}^{-5}$	$2.5236\times {10}^3$	$0.17$	−		0.009	$3.1664\times {10}^{-4}$	$8.5260\times {10}^{-5}$	$2.8063\times {10}^3$	$0.18$	−
	${10}^3$	0.010	$6.1541\times {10}^{-5}$	$1.8819\times {10}^{-5}$	$9.4970\times {10}^5$	$1.03$	$2.34$		0.010	$6.6609\times {10}^{-5}$	$1.8938\times {10}^{-5}$	$9.1129\times {10}^5$	$1.04$	$2.24$
	${15}^3$	0.013	$2.3758\times {10}^{-5}$	$7.1676\times {10}^{-6}$	$8.5077\times {10}^6$	$3.58$	$2.34$		0.013	$2.4072\times {10}^{-5}$	$7.2265\times {10}^{-6}$	$8.3153\times {10}^6$	$3.84$	$2.51$
	${20}^3$	0.020	$1.2159\times {10}^{-5}$	$3.3100\times {10}^{-6}$	$3.9961\times {10}^7$	$10.08$	$2.32$		0.020	$1.2487\times {10}^{-5}$	$3.3356\times {10}^{-6}$	$3.7771\times {10}^7$	$10.21$	$2.28$

by computing the estimated errors, conditional numbers, CPU times, and convergence rates for several values of N and various fractional orders ${\alpha }_1$ and ${\alpha }_2$. The optimal value of the shape parameter c is specified by trial and error such that there is an equivalence between the accuracy of the numerical method and the conditional number of the coefficient matrix. Thus, the plot of absolute errors and condition numbers with respect to the shape parameter c by taking $N={10}^3$ and ${\alpha }_1={\alpha }_2=1.50$ is shown in Figure 9 for both wave and damped wave models. Moreover, the computational errors for various $N_t$ and different fractional orders ${\alpha }_1$ and ${\alpha }_1$ by letting $N_x={20}^2$ and shape parameter $c=0.017$ for wave model and $c=0.020$ for damped wave model are shown in the

Table 6. Error estimates for $N_x={20}^2$ of Example 2 by letting $c=0.017$ for wave model and $c=0.020$ for damped wave model, $n_c=10$, different values of $N_t$ and several fractional orders ${\alpha }_1$ and ${\alpha }_2$.

$({\mathit{\alpha }}_1,{\mathit{\alpha }}_2)$	${\mathit{N}}_{\mathit{t}}$	Wave Model			Damped Wave Model
		${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$		${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$
$(1.80,1.70)$	5	$4.7343\times {10}^{-3}$	$1.4830\times {10}^{-3}$		$8.3832\times {10}^{-2}$	$1.6320\times {10}^{-3}$
	10	$4.7001\times {10}^{-3}$	$1.5718\times {10}^{-3}$		$4.7465\times {10}^{-3}$	$1.5801\times {10}^{-3}$
	15	$1.6565\times {10}^{-5}$	$4.9223\times {10}^{-6}$		$1.6170\times {10}^{-5}$	$4.7275\times {10}^{-6}$
	20	$9.2334\times {10}^{-6}$	$2.8096\times {10}^{-6}$		$9.9772\times {10}^{-6}$	$3.3610\times {10}^{-6}$
$(1.60,1.90)$	5	$4.5883\times {10}^{-3}$	$1.4206\times {10}^{-3}$		$5.7820\times {10}^{-3}$	$1.4226\times {10}^{-3}$
	10	$4.5549\times {10}^{-3}$	$1.4870\times {10}^{-3}$		$4.6549\times {10}^{-3}$	$1.5537\times {10}^{-3}$
	15	$1.7864\times {10}^{-5}$	$5.5979\times {10}^{-6}$		$1.6792\times {10}^{-5}$	$5.2939\times {10}^{-6}$
	20	$1.0684\times {10}^{-5}$	$2.9493\times {10}^{-6}$		$1.0694\times {10}^{-5}$	$3.2222\times {10}^{-6}$
$(1.90,1.90)$	5	$3.5657\times {10}^{-3}$	$1.0914\times {10}^{-3}$		$3.7033\times {10}^{-3}$	$1.2397\times {10}^{-3}$
	10	$3.5159\times {10}^{-3}$	$1.1562\times {10}^{-3}$		$3.6942\times {10}^{-3}$	$1.2102\times {10}^{-3}$
	15	$1.7359\times {10}^{-5}$	$4.9861\times {10}^{-6}$		$1.6265\times {10}^{-5}$	$4.7185\times {10}^{-6}$
	20	$1.0300\times {10}^{-5}$	$2.7453\times {10}^{-6}$		$9.8631\times {10}^{-6}$	$2.5639\times {10}^{-6}$

. The reported numerical results in this table demonstrate the temporal convergence of the proposed method in deal with both mentioned models. The behavior of the numerical method for either wave and damped wave model against the input noise is demonstrated in

Table 7. The effect of noise on error estimates of Example 2 for different values of N and letting ${\alpha }_1=1.85$ and ${\alpha }_2=1.45$.

N	$\mathit{\sigma }$	Wave Model				Damped Wave Model
		$\mathbf{c}$	${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$		$\mathbf{c}$	${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$
$5^3$	0	$0.008$	$3.4082\times {10}^{-4}$	$9.3084\times {10}^{-5}$		$0.009$	$3.4392\times {10}^{-4}$	$9.3683\times {10}^{-5}$
	$0.001$		$4.0332\times {10}^{-4}$	$1.0973\times {10}^{-4}$			$4.0642\times {10}^{-4}$	$1.1020\times {10}^{-4}$
	$0.01$		$9.6582\times {10}^{-4}$	$2.6043\times {10}^{-4}$			$9.6892\times {10}^{-4}$	$2.6049\times {10}^{-4}$
	$0.1$		$6.5908\times {10}^{-3}$	$1.7720\times {10}^{-3}$			$6.5939\times {10}^{-3}$	$1.7718\times {10}^{-3}$
${10}^3$	0	$0.010$	$6.4696\times {10}^{-5}$	$2.0480\times {10}^{-5}$		$0.010$	$6.9680\times {10}^{-5}$	$2.1079\times {10}^{-5}$
	$0.001$		$1.2473\times {10}^{-5}$	$4.0433\times {10}^{-5}$			$1.2972\times {10}^{-4}$	$4.0640\times {10}^{-5}$
	$0.01$		$6.6509\times {10}^{-4}$	$2.2138\times {10}^{-4}$			$6.7008\times {10}^{-4}$	$2.2125\times {10}^{-4}$
	$0.1$		$6.0687\times {10}^{-3}$	$2.0322\times {10}^{-3}$			$6.0736\times {10}^{-3}$	$2.0321\times {10}^{-3}$
${15}^3$	0	$0.015$	$3.1420\times {10}^{-5}$	$7.6784\times {10}^{-6}$		$0.015$	$6.8234\times {10}^{-5}$	$1.0182\times {10}^{-5}$
	$0.001$		$8.2470\times {10}^{-5}$	$2.8254\times {10}^{-5}$			$8.4454\times {10}^{-5}$	$2.8972\times {10}^{-5}$
	$0.01$		$6.4478\times {10}^{-4}$	$2.1932\times {10}^{-4}$			$6.4691\times {10}^{-4}$	$2.1934\times {10}^{-4}$
	$0.1$		$6.2697\times {10}^{-3}$	$2.1320\times {10}^{-3}$			$6.2719\times {10}^{-3}$	$2.1319\times {10}^{-3}$
${20}^3$	0	$0.020$	$1.2032\times {10}^{-5}$	$3.5863\times {10}^{-6}$		$0.020$	$1.2386\times {10}^{-5}$	$3.6963\times {10}^{-6}$
	$0.001$		$6.7756\times {10}^{-5}$	$2.3326\times {10}^{-5}$			$6.8420\times {10}^{-5}$	$2.3303\times {10}^{-5}$
	$0.01$		$6.2523\times {10}^{-4}$	$2.1953\times {10}^{-4}$			$6.2589\times {10}^{-4}$	$2.1948\times {10}^{-4}$
	$0.1$		$6.2000\times {10}^{-3}$	$2.1837\times {10}^{-3}$			$6.2006\times {10}^{-3}$	$2.1836\times {10}^{-3}$

. The computed results show the stability of the presented method to deal with different noise levels. The absolute errors for several time levels by considering $N_x={21}^2$, ${\alpha }_1=1.65$, ${\alpha }_2=1.85$ for wave model with shape parameter $c=0.022$ and damped wave model with shape parameter $c=0.020$ are plotted in Figure 10 and Figure 11, respectively. The sparsity patterns of the coefficient matrix for both wave and damped wave models are demonstrated in Figure 7.

Example 3.

As the third example, the two-dimensional space fractional derivatives problem (1) related to wave model by coefficients $K_1=K_2={10}^3$, $A_1=A_2=0$ and damped wave model by coefficients $K_1=K_2={10}^3$, $A_1=10$ and $A_2=20$. The exact solution of the problem for either wave and damped wave model is $u(x,y,t)=t^6(x-x^3)(y^3-2y^2+y)$. The source functions for both models are obtained as follows    

$$\begin{array}{ccc}\hfill f(x,y,t)& =& \left(30t^4+6A_1{t}^5+A_2{t}^6\right)(x-x^3)(y^3-2y^2+y)\hfill \\ & -& K_1{t}^6(y^3-2y^2+y)\left(\frac{x^{-{\alpha }_1}}{\Gamma (1-{\alpha }_1)}-\frac{6x^{2-{\alpha }_1}}{\Gamma (3-{\alpha }_1)}\right)\hfill \\ & -& K_2{t}^6(x-x^3)\left(\frac{6y^{2-{\alpha }_2}}{\Gamma (3-{\alpha }_2)}-\frac{4y^{1-{\alpha }_2}}{\Gamma (2-{\alpha }_2)}+\frac{y^{-{\alpha }_2}}{\Gamma (1-{\alpha }_2)}\right).\hfill \end{array}$$

In this case the computational domain is $\Omega \times T$ where $\Omega ={[0,1]}^2$ with uniform and irregular distributed points and $T=[0,1]$. Moreover, thek-d treeprogram constructs the local sub-domains with size $n_c=10$.

In the current example, the accuracy and efficiency of the proposed method in the two modes of uniform point distribution and irregular point distribution in the computational domain are compared for both wave and damped wave models. The irregular distribution points scheme is demonstrated in Figure 12.

Figure 13 shows the plot of absolute estimated error concerning the number of uniform and irregular distribution data points for both uniform and irregular distributed data points.As can be seen, the accuracy of the suggested method in confronting the irregular distribution points is relatively good, but in comparison with the uniform distribution points, it Significantly decreases. Moreover, the condition numbers versus the number of uniform and irregular distributed data points for both wave and damped wave models have demonstrated in Figure 14. These Figures show that the present numerical method for irregular distribution points is not as good as uniform distribution points according to the condition number of the coefficient matrix. The sparsity pattern of the coefficient matrix for wave and damped wave models respect to the uniform and irregular distribution points are displayed in Figure 15 and Figure 16, respectively. These figures demonstrate that the arising coefficient matrix is sparsity for both types of point distributions. Therefore, the presented method causes to decrease in the computational cost and accelerates the algorithm for irregular distribution points as good as uniform distribution points.

Example 4.

As the last example, the three-dimensional space fractional derivatives problem (1) for wave model by coefficients $K_1=K_2=K_3=10$, $A_1=A_2=0$ and damped wave model by coefficients $K_1=K_2=K_3=10$, $A_1=A_2=0.2$. The exact solution of the problem is $u(x,y,t)=sin\left(\pi (t+1)\right)(x-x^2)(y-y^2)(z-z^2)$ The source functions for both models are concluded as follows

$$\begin{array}{ccc}\hfill f(x,y,z,t)& =& \left(-{\pi }^{2}sin\left(\pi (t+1)\right)+A_1\pi cos\left(\pi (t+1)\right)+A_{2}sin\left(\pi (t+1)\right)\right)\hfill \\ & ·& (x-x^2)(y-y^2)(z-z^2)\hfill \\ & -& K_{1}sin\left(\pi (t+1)\right)(y-y^2)(z-z^2)\left(\frac{x^{1-{\alpha }_1}}{\Gamma (2-{\alpha }_1)}-\frac{2x^{2-{\alpha }_1}}{\Gamma (3-{\alpha }_1)}\right)\hfill \\ & -& K_{2}sin\left(\pi (t+1)\right)(x-x^2)(z-z^2)\left(\frac{y^{1-{\alpha }_2}}{\Gamma (2-{\alpha }_2)}-\frac{2y^{2-{\alpha }_2}}{\Gamma (3-{\alpha }_2)}\right)\hfill \\ & -& K_{3}sin\left(\pi (t+1)\right)(x-x^2)(y-y^2)\left(\frac{z^{1-{\alpha }_3}}{\Gamma (2-{\alpha }_3)}-\frac{2z^{2-{\alpha }_3}}{\Gamma (3-{\alpha }_3)}\right).\hfill \end{array}$$

In three-dimensional case the computational domain is $\Omega \times T$ where $\Omega ={[0,1]}^3$ with uniform distributed points and $T=[0,1]$. Further, in the computational procedure the local sub-domains with size $nc=15$ are built byk-d treealgorithm.

The error estimates, condition numbers, computational times, and convergence rates vs. N are reported in

Table 8. Error estimates, condition numbers, CPU times and convergence rate of Example 4 by letting $n_c=15$ and different values of N and several fractional orders ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$.

$({\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3)$	N	Wave Model							Damped Wave Model
		c	${\mathit{ϵ}}_{\mathit{\infty }}$	${\mathit{ϵ}}_{\mathit{r}}$	$\mathbf{CN}$	Time	${\mathcal{R}}_{\mathit{\infty }}$		c	${\mathit{ϵ}}_{\mathit{\infty }}$	${\mathit{ϵ}}_{\mathit{r}}$	$\mathbf{CN}$	Time	${\mathcal{R}}_{\mathit{\infty }}$
$(1.5,1.5,1.5)$	$5^4$	0.030	$3.3885\times {10}^{-4}$	$1.9901\times {10}^{-5}$	$1.9289\times {10}^3$	$0.38$	−		0.009	$7.9669\times {10}^{-5}$	$1.4917\times {10}^{-5}$	$2.1248\times {10}^3$	$0.389$	−
	$7^4$	0.035	$3.4788\times {10}^{-5}$	$7.4475\times {10}^{-6}$	$6.5765\times {10}^4$	$2.00$	$6.76$		0.011	$3.7856\times {10}^{-5}$	$7.4366\times {10}^{-6}$	$6.4930\times {10}^4$	$2.02$	$2.21$
	$9^4$	0.074	$1.8830\times {10}^{-5}$	$4.3033\times {10}^{-6}$	$5.5365\times {10}^5$	$6.91$	$2.44$		0.014	$2.0860\times {10}^{-5}$	$4.2544\times {10}^{-6}$	$5.9485\times {10}^5$	$7.01$	$2.37$
	${11}^4$	0.099	$1.1783\times {10}^{-5}$	$2.7394\times {10}^{-6}$	$3.0414\times {10}^6$	$19.03$	$2.33$		0.019	$1.8689\times {10}^{-5}$	$3.0590\times {10}^{-6}$	$2.5062\times {10}^6$	$19.42$	$0.54$
$(1.8,1.7,1.6)$	$5^4$	0.030	$6.5181\times {10}^{-5}$	$1.1376\times {10}^{-5}$	$2.5678\times {10}^3$	$0.35$	−		0.010	$5.4917\times {10}^{-5}$	$9.5723\times {10}^{-6}$	$2.7288\times {10}^3$	$0.36$	−
	$7^4$	0.037	$2.5089\times {10}^{-5}$	$5.3738\times {10}^{-6}$	$2.6462\times {10}^4$	$1.79$	$2.83$		0.014	$2.2968\times {10}^{-5}$	$4.7842\times {10}^{-6}$	$2.6368\times {10}^4$	$1.81$	$2.59$
	$9^4$	0.041	$1.4295\times {10}^{-5}$	$3.0710\times {10}^{-6}$	$6.9574\times {10}^5$	$6.42$	$2.23$		0.017	$1.9604\times {10}^{-5}$	$3.3110\times {10}^{-6}$	$7.5516\times {10}^5$	$6.55$	$0.63$
	${11}^4$	0.050	$9.4011\times {10}^{-6}$	$1.9582\times {10}^{-6}$	$9.1183\times {10}^6$	$18.47$	$2.08$		0.020	$7.6139\times {10}^{-6}$	$1.5645\times {10}^{-6}$	$7.1603\times {10}^6$	$20.01$	$4.71$
$(1.5,1.9,1.7)$	$5^4$	0.020	$6.0536\times {10}^{-5}$	$1.0547\times {10}^{-5}$	$2.7232\times {10}^3$	$0.35$	−		0.011	$5.5235\times {10}^{-5}$	$9.6439\times {10}^{-6}$	$2.8963\times {10}^3$	$0.36$	−
	$7^4$	0.030	$2.5729\times {10}^{-5}$	$5.3827\times {10}^{-6}$	$2.6185\times {10}^4$	$1.75$	$2.54$		0.015	$2.3448\times {10}^{-5}$	$4.8658\times {10}^{-6}$	$2.6109\times {10}^4$	$1.90$	$2.54$
	$9^4$	0.035	$1.6602\times {10}^{-5}$	$3.9895\times {10}^{-6}$	$4.9197\times {10}^5$	$6.51$	$1.74$		0.018	$1.7139\times {10}^{-5}$	$3.8854\times {10}^{-6}$	$9.4595\times {10}^5$	$7.23$	$1.24$
	${11}^4$	0.035	$1.1607\times {10}^{-5}$	$2.3294\times {10}^{-6}$	$4.3370\times {10}^6$	$18.11$	$1.78$		0.021	$8.3998\times {10}^{-6}$	$1.7479\times {10}^{-6}$	$3.8080\times {10}^6$	$22.24$	$3.55$

. The numerical results in this table are obtained by considering different fractional orders ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ for either wave and damped wave models. These results show the convergence, accuracy, wellposedness, and speeds of the method to deal with the high-dimensional wave and damped wave models well. To obtain the best value for the shape parameter, a trial and error process is used such that there is an equilibrium between the accuracy of the suggested computational procedure and obtained condition number of the coefficient matrix. Hence, the plot of absolute errors and condition numbers for different shape parameters by considering $N=9^4$ and ${\alpha }_1={\alpha }_2={\alpha }_3=1.50$ are displayed in Figure 17 for both wave and damped wave models. Furthermore,

Table 9. Error estimates for $N_x={12}^3$ of Example 4 by letting $c=0.060$ for wave model and $c=0.025$, $n_c=15$, different values of $N_t$ and several fractional orders ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$.

$({\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\alpha }}_3)$	${\mathit{N}}_{\mathit{t}}$	Wave Model			Damped Wave Model
		${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$		${\mathbf{ϵ}}_{\infty }$	${\mathbf{ϵ}}_{\mathbf{r}}$
$(1.7,1.7,1.7)$	3	$9.2461\times {10}^{-4}$	$1.8013\times {10}^{-4}$		$9.4060\times {10}^{-4}$	$1.8327\times {10}^{-4}$
	6	$8.7935\times {10}^{-4}$	$2.0139\times {10}^{-4}$		$9.1323\times {10}^{-4}$	$2.0482\times {10}^{-4}$
	9	$1.4375\times {10}^{-5}$	$3.2206\times {10}^{-6}$		$1.6470\times {10}^{-5}$	$2.9951\times {10}^{-6}$
	12	$1.2491\times {10}^{-5}$	$2.1729\times {10}^{-6}$		$5.6840\times {10}^{-6}$	$1.2599\times {10}^{-6}$
$(1.5,1.7,1.9)$	3	$8.9200\times {10}^{-4}$	$1.7432\times {10}^{-4}$		$9.0557\times {10}^{-4}$	$1.7705\times {10}^{-4}$
	6	$8.4834\times {10}^{-4}$	$1.9489\times {10}^{-4}$		$8.7932\times {10}^{-4}$	$1.9787\times {10}^{-4}$
	9	$2.0456\times {10}^{-5}$	$4.8243\times {10}^{-6}$		$1.5055\times {10}^{-5}$	$3.8044\times {10}^{-6}$
	12	$9.2618\times {10}^{-6}$	$2.0153\times {10}^{-6}$		$7.6079\times {10}^{-6}$	$1.5858\times {10}^{-6}$
$(1.8,1.7,1.6)$	3	$9.1571\times {10}^{-4}$	$1.7857\times {10}^{-4}$		$9.3117\times {10}^{-4}$	$1.8162\times {10}^{-4}$
	6	$8.7089\times {10}^{-4}$	$1.9965\times {10}^{-4}$		$9.0402\times {10}^{-4}$	$2.0297\times {10}^{-4}$
	9	$1.9037\times {10}^{-5}$	$4.5341\times {10}^{-6}$		$2.2248\times {10}^{-5}$	$3.9412\times {10}^{-6}$
	12	$1.1909\times {10}^{-5}$	$2.1102\times {10}^{-6}$		$6.2373\times {10}^{-6}$	$1.3494\times {10}^{-6}$

indicates the temporal convergence of the suggested numerical method to deal with mentioned models. The results are obtained vs. $N_t$ by considering $N_x={12}^3$ and shape parameter $c=0.060$ for wave model and $c=0.060$ for damped wave model. To investigate the method’s stability numerically, the impression of the input noise on the initial solution is evaluated. For this purpose, the effects of the several input noise levels $\sigma =0$, $\sigma =0.001$, $\sigma =0.01$, and $\sigma =0.1$ on error estimates are considered. These results are reported in Table 4 and the acceptable and stable behavior of the numerical procedure against the input noise are illustrated. The coefficient matrix sparsity patterns for both models are shown in Figure 18.

4. Conclusions

Local Mashless numerical methods have widely proved their ability to solve efficiently partial differential equations. In this work for solving the wave and damped wave equation with Riemann–Liouville fractional derivatives, we focus on the space–time Gaussian radial basis function in a high-dimensional setting. The proposed technique has some noticeable features, such as constructing a sparse matrix that reduces the time execution of the implementation of algorithms. This feature allows the method to be applied to high-dimensional problems that occur in nature and engineering as well as the using the RBFs makes it much easier to work on high-dimensional spaces with irregular computational domains. Moreover, employing the space–time Gaussian RBF eliminates the need for time discretization of the model. Numerical experiments verify that the local features lead to the sparse coefficient matrix and reduce the computational costs in high-dimensional problems. Improvements in terms of accuracy and efficiency on the computed solution have been confirmed by the proposed numerical scheme. Finally, we aim to apply this method to more complex fractional partial differential equations with applications in computational finance.