Theory and Modern Applications

# A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation

## Abstract

In this paper, we focus on the development and study of the finite difference/pseudo-spectral method to obtain an approximate solution for the time-fractional diffusion-wave equation in a reproducing kernel Hilbert space. Moreover, we make use of the theory of reproducing kernels to establish certain reproducing kernel functions in the aforementioned reproducing kernel Hilbert space. Furthermore, we give an approximation to the time-fractional derivative term by applying the finite difference scheme by our proposed method. Over and above, we present an appropriate technique to derive the numerical solution of the given equation by utilizing a pseudo-spectral method based on the reproducing kernel. Then, we provide two numerical examples to support the accuracy and efficiency of our proposed method. Finally, we apply numerical experiments to calculate the quality of our approximation by employing discrete error norms.

## Introduction

The use of the fractional-order derivative has become popular due to its nonlocality property, which is an intrinsic property of many complex systems. The fractional-order derivative has recently been applied in modeling different phenomena including viscoelasticity, financial modeling, nanotechnology, control theory of dynamical systems, random walk, anomalous transport, biological modeling, and anomalous diffusion as well. For further applications of the fractional order derivative in the fields of engineering, physical sciences, we may refer to  and  and the references cited therein.

The fractional partial differential equations (FPDEs) have been deployed in recent years as a powerful tool in nonlocality and spatial heterogeneity modelling. Many applications of the fractional models can be found in . The fractional diffusion equation assumed in this paper covers not only the classic state of the heat equation but also the kernel of many other FPDEs. This equation describes the anomalous diffusion of particles. For applications of this equation, we refer to the one presented in , which describes the transfer processes with a long memory and the water transport in the soil model . Many mathematical models are usually based on the diffusion model such as the diffusion equation on fractals , Fischer information theory , and so on. Moreover, the implication of the fractional in time (sub-diffusion) and fractional in space (super-diffusion) has been observed in the solution profiles in many fractional models [26, 27]. The superior capabilities of fractional differential equations to accurately model such processes have raised significant interest in assaying numerical methods for obtaining the solutions to such problems . The analysis we present in this paper depends on the following time-fractional diffusion-wave problem:

\begin{aligned}& \partial ^{\alpha}_{t} u(x,t)=\Delta u(x,t)-u(x,t)+f(x,t),\quad (x,t) \in \Omega \times J, \end{aligned}
(1.1)
\begin{aligned}& u(0,t)=0, \qquad u(b,t)=0,\quad t \in \overline{J}, \end{aligned}
(1.2)
\begin{aligned}& u(x,0)=\phi _{1}(x), \qquad \partial _{t}u(x,0)=\phi _{2}(x),\quad x \in \overline{\Omega}, \end{aligned}
(1.3)

where $$\Omega =[0,b]$$ is a bounded domain in R; $$J=(0,T]$$ is the time interval satisfying $$0< T<+\infty$$; $$u:\overline{\Omega} \times \overline{J} \rightarrow \mathbf{R}$$ is a sufficiently differentiable function, and $$1< \alpha < 2$$ is the order of the fractional derivative, and the time-fractional derivative $$\partial ^{\alpha}_{t}$$ is the Caputo fractional derivative of order $$1<\alpha <2$$ defined by

\begin{aligned} \partial _{t}^{\alpha}u(x,t)=\frac{1}{\Gamma (2-\alpha )} \int _{0}^{t} \frac{\partial ^{2} u(x,s)}{\partial s^{2}} \frac{ds}{(t-s)^{\alpha -1}}. \end{aligned}

The reproducing theory was investigated by Mercer in 1909 . He named functions satisfying the reproducing property as “positive definite kernels”. Around 1948, Aronszajn  systematized the concept of reproducing kernels. From 1980, Cui and co-workers [33, 34] have been pioneers and beginners in the numerical analysis of linear and nonlinear problems using the “reproducing kernel Hilbert space method”. Recently, a lot of research has been done to solve several linear and nonlinear problems using the theory of reproducing kernel .

The aim of this paper is to introduce a finite difference/pseudo-spectral method based on reproducing kernel (RK) for solving the time-fractional diffusion-wave equation (1.1)–(1.3). This paper spreads over four sections, including the introduction. In Sect. 2, we present a finite difference/pseudo-spectral method based on a reproducing kernel (RK) for solving the time-fractional diffusion-wave equation. In Sect. 2, we solve some test problems and derive several results. In Sect. 4, we present some concluding remarks.

## Implementation of the method

### Discretization of Caputo derivative and semi-discrete scheme

First, we obtain the semi-discrete scheme for (1.1)–(1.3). The discretization of is performed with a constant time step $$\tau =\frac{T}{N}$$, where $$N \in \mathcal{N}^{*}$$. Denote $$t_{n}=n \tau$$ for $$N=0:N$$. Let $$u^{n}=u(x,t_{n})$$. For a discrete function $$\{u^{n}\}_{n=0}^{N+1}$$, we provide some preliminaries concerning the approximation of the time fractional derivative $$\partial _{t}^{\alpha} u(x,t)$$ with $$1<\alpha <2$$. A Caputo derivative approximation formula (CDAF) for $$\partial _{t}^{\alpha} u(x,t_{n+1})$$ with $$1<\alpha <2$$ can be defined as a linear combination of the discrete second time derivatives $$\{\partial ^{2} u^{j}\}_{j=1}^{n+1}$$ 

\begin{aligned} \partial _{t}^{\alpha}u^{n+1}= \frac{\tau ^{2-\alpha}}{\Gamma (3-\alpha )} \sum _{j=0}^{n}b_{j} \partial ^{2}u^{n+1-j}+\mathcal{R}_{1}^{n+1}(u), \end{aligned}
(2.1)

where

$$b_{j}=(j+1)^{2-\alpha}-j^{2-\alpha},$$

and $$\mathcal{R}_{1}^{n+1}(u)$$ is the local truncation error such that

$$\bigl\vert \mathcal{R}_{1}^{n+1}(u) \bigr\vert \leq C_{u} \tau ^{3-\alpha} \text{or} \mathcal{R}_{1}^{n+1}(u)=O \bigl( \tau ^{3-\alpha}\bigr).$$

The following lemma summarizes some properties of the coefficients $$b_{j}$$ which will be used in this paper.

### Lemma 2.1

(See . Properties of the coefficients $$b_{j}^{n+1}$$)

For any $$1 < \alpha <2$$, the coefficients of $$b_{j}^{n+1}$$ satisfy the following properties:

• $$b_{j}>0$$, $$j=0,1,\ldots ,n$$,

• $$1=b_{0}>b_{1}>\cdots >b_{n}$$ and $$b_{n} \rightarrow 0$$ as $$n\rightarrow \infty$$,

• $$\sum_{j=0}^{n}(b_{j}-b_{j+1})=1$$.

Substituting (2.1) into (1.1) gives

\begin{aligned}& a(\alpha ,\tau ) \bigl(u^{n+1}-2u^{n}+u^{n-1} \bigr)+a(\alpha ,\tau )\sum_{j=1}^{n} b_{j}\bigl(u^{n+1-j}-2u^{n-j}+u^{n-1-j}\bigr) \\& \quad =\Delta u^{n+1}(x)-u^{n+1}(x)+ f^{n+1}(x)+R^{n+1}(u), \end{aligned}
(2.2)

where $$a(\alpha ,\tau )=\frac{1}{\tau ^{\alpha}\Gamma (3-\alpha )}$$ and $$f^{n+1}(x)=f(x,t_{n+1})$$.

Replacing $$u^{n+1}$$ by the approximate solution $$U^{n+1}$$, we can obtain the following semi-discrete problem.

### Scheme I

Given $$U^{0}=\phi _{1}(x)$$, $$U^{-1}=U^{1}-2\tau \phi _{2}(x)$$ and find $$U^{n+1}$$ ($$n=0,1,2,\ldots ,N-1$$) such that

\begin{aligned} \textstyle\begin{cases} a(\alpha ,\tau )(U^{n+1}-2U^{n}+U^{n-1})+a(\alpha ,\tau )\sum_{j=1}^{n}b_{j}(U^{n+1-j}-2U^{n-j}+U^{n-1-j}) \\ \quad =\Delta U^{n+1}(x)-U^{n+1}(x)+ f^{n+1}(x), \\ U^{n+1}|_{x \in \partial \Omega}=0,\quad -1\leq n \leq N-1. \end{cases}\displaystyle \end{aligned}

For the convenience of discussion, define the linear operator L as follows:

\begin{aligned} \textrm{L} (*)=\textstyle\begin{cases} ((2a(\alpha ,\tau )+1)-\Delta ) (*),& n=0, \\ ((a(\alpha ,\tau )+1)-\Delta )(*),& 1\leq n \leq N-2. \end{cases}\displaystyle \end{aligned}

Therefore, a semi-discrete problem can be converted into the following equivalent:

\begin{aligned} \textrm{L} U^{n+1}(x)=F^{n+1}(x),\quad 0\leq n \leq N-1, \end{aligned}
(2.3)

where

$$F^{n+1}(x)=\textstyle\begin{cases} -a(\alpha ,\tau )(-2\phi _{1}(x)-2\tau \phi _{2}(x))+ f^{1}(x),\quad n=0, \\ -a(\alpha ,\tau )(-2U^{n}+U^{n-1})-a(\alpha ,\tau )\sum_{j=1}^{n}b_{j}(U^{n+1-j} \\ \quad {}-2U^{n-j}+U^{n-1-j})+ f^{n+1}(x),\quad 1\leq n \leq N-2. \end{cases}$$

### A pseudo-spectral kernel-based method

Now, we employ a pseudo-spectral method based on RK to discrete the space direction and obtain a full-discrete scheme of (2.3). To obtain this, we need some notations and preliminaries.

We now give background material and preliminaries, which are used in the following sections. Recall that a real reproducing kernel Hilbert space (RKHS) on a nonempty abstract set Ω is a particular type of a real Hilbert space H of functions that satisfies the following additional property (called reproducing kernel property): for each $$x\in \Omega$$, there exists $$K(x,\cdot)\in \mathsf{H}$$ ($$R:\Omega \times \Omega \longrightarrow \mathbf{R}$$) such that, for every $$u \in \mathsf{H}$$, one has

\begin{aligned} u(x)=\bigl(u(\cdot),K(x,\cdot)\bigr)_{\mathsf{H}},\quad \forall u \in \mathsf{H}, \forall x \in \Omega . \end{aligned}
(2.4)

### Definition 2.2

(See )

A Hilbert space H of real functions on a set Ω is called an RKHS if there exists an RK $$K(x,\cdot)$$ of H.

### Theorem 2.3

(See )

Suppose that $$\boldsymbol{\mathsf{H}}$$ is an RKHS with RK $$K:\Omega \times \Omega \longrightarrow \mathbf{R}$$. Then $$K(x,\cdot)$$ is positive definite. Moreover, $$K(x,\cdot)$$ is strictly positive definite if and only if the point evaluation functionals $$\bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} I_{x}: \boldsymbol{\mathsf{H}}\longrightarrow \mathbf{R}, \\ I_{x}(u)=u(x) \end{array}}$$ are linearly independent in $$\boldsymbol{\mathsf{H}}^{*}$$, where $$\boldsymbol{\mathsf{H}}^{*}$$ is the space of bounded linear functionals on $$\boldsymbol{\mathsf{H}}$$.

### Definition 2.4

(See . One-dimensional RKHS)

The inner product space $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$ for a function u is defined as

\begin{aligned} \boldsymbol{\mathsf{H}}_{p}[0,b]=\bigl\{ u|u(x),u^{\prime }(x),u^{\prime \prime }(x)\in AC[0,b], u^{\prime \prime \prime }(x) \in L^{2}[0,b],u(0)=u(b)=0, x\in [0,b]\bigr\} . \end{aligned}

The inner product in $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$ is in the form

\begin{aligned} \langle u,v\rangle _{\boldsymbol{\mathsf{H}}_{p}}=u(0)v(0)+u(b)v(b)+u^{\prime }(0)v^{\prime }(0)+ \int _{a}^{b}u^{\prime \prime \prime }(x)v^{\prime \prime \prime }(x)\,dx, \end{aligned}
(2.5)

and the norm $$\|u\|_{\boldsymbol{\mathsf{H}}}$$ is defined by

\begin{aligned} \Vert u \Vert _{\boldsymbol{\mathsf{H}}_{p}}=\sqrt{\langle u,u\rangle _{\mathsf{H}}}, \end{aligned}
(2.6)

where $$u,v\in \boldsymbol{\mathsf{H}}_{p}[0,b]$$.

The space $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$ is an RKHS and the RK $$K_{y}(x)$$ can be denoted by 

\begin{aligned} K_{y}(x)= \textstyle\begin{cases} \frac{1}{120b^{2}}(b-x)y(120bx+x(6b^{2}x-120-4bx^{2}+x^{3})y \\ \quad {}-5bxy^{3}+(b+x)y^{4}), \quad y< x, \\ \frac{1}{120b^{2}}(b-y)x(120by+y(6b^{2}y-120-4by^{2}+y^{3})x \\ \quad {}-5byx^{3}+(b+y)x^{4}), \quad y\geq x. \end{cases}\displaystyle \end{aligned}
(2.7)

With the help of pseudo-spectral method based on RK, we will illustrate how to derive the numerical solution. Now, we will give the representation of a numerical solution to the semi-discrete problem (2.3) in the RKHS $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$. Let $$\mathcal{B}_{M}=\{x_{j}\}_{j = 1}^{M}$$ be a distinct subset of Ω̅. We consider the finite-dimensional space

\begin{aligned} \mathcal{U}_{M}=\operatorname{\textbf{span}}\bigl\{ \psi _{j}(x)=K_{y}(x)|_{y=x_{j}}, x_{j} \in \mathcal{B}_{M}\bigr\} \subset \boldsymbol{\mathsf{H}}_{p}[0,b], \end{aligned}

where $$K_{y}(x)$$ is the RK constructed in $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$.

The semi-discrete problem can be written into following equivalent form $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$ to $$C[0,b]$$:

\begin{aligned} \textrm{L} U^{n+1}(x)=F^{n+1}(x),\quad 0\leq n \leq N-1, \end{aligned}

where

$$F^{n+1}(x)=\textstyle\begin{cases} -a(\alpha ,\tau )(-2\phi _{1}(x)-2\tau \phi _{2}(x))+ f^{1}(x),\quad n=0, \\ -a(\alpha ,\tau )(-2U^{n}+U^{n-1})-a(\alpha ,\tau )\sum_{j=1}^{n}b_{j}(U^{n+1-j} \\ \quad {}-2U^{n-j}+U^{n-1-j})+ f^{n+1}(x),\quad 1\leq n \leq N-2, \end{cases}$$

and $$F^{n+1}\in C[0,b]$$ as $$u^{k}\in \boldsymbol{\mathsf{H}}_{p}[0,b]$$.

An approximant $$U_{M}^{n+1}$$ to $$U^{n+1}$$ can be obtained by calculating a truncated series based on trial functions as follows:

$\begin{array}{rcl}{U}^{n+1}\left(x\right)& \approx & {U}_{M}^{n+1}\left(x\right):=\sum _{j=1}^{M}{\alpha }_{j}^{n+1}{\psi }_{j}\left(x\right)\\ & =& \left[{\psi }_{1}\left(x\right),{\psi }_{2}\left(x\right),\dots ,{\psi }_{M}\left(x\right)\right]\left(\begin{array}{c}{\alpha }_{1}^{n+1}\\ {\alpha }_{2}^{n+1}\\ ⋮\\ {\alpha }_{M}^{n+1}\end{array}\right).\end{array}$
(2.8)

To determine the interpolation coefficients $$\{\alpha ^{n+1}_{j}\}_{j=1}^{M}$$, the set of collocation conditions is used by applying (2.3) to $$\mathcal{B}_{M}$$. Thus

$$\lambda _{i}\bigl[U_{M}^{n+1} \bigr]:=\textrm{L} U_{M}^{n+1}(x_{i})=\sum _{j=1}^{M} \alpha ^{n+1}_{j} \textrm{L}\psi _{j}(x_{i})=F^{n+1}(x_{i}),\quad i=1,2, \ldots ,M,$$
(2.9)

where the functional $$\lambda _{i}$$, ($$1\leq i \leq M$$) is defined by applying the differential operator followed by a point evaluation at $$x_{i} \in \mathcal{B}_{n}$$. In general, a single set $$\Lambda _{M}:=\{\lambda _{i}\}_{i=1}^{M}$$ of functionals contains several types of differential operators.

The arising collocation matrix K is unsymmetric and has the ij-entries:

$$\textbf{K}_{ij}:=\lambda _{i}[\psi _{j}]=\lambda ^{x}_{i} K_{y}(x)|_{y=x_{j}},\quad 1 \leq i,j \leq M,$$
(2.10)

where the subscript x in $$\lambda ^{x}_{j}$$ indicates that $$\lambda ^{x}_{j}$$ applies to the function of x.

Therefore the unknown coefficients $$\alpha ^{n+1}_{j}$$, $$j=1,2,\ldots ,M$$, can be obtained by solving the following system:

\begin{aligned}& \textbf{K}[\alpha ]^{n+1}=\textbf{F}^{n+1}, \end{aligned}

where

\begin{aligned}& [\alpha ]^{n+1}=\bigl[\alpha _{1}^{n+1},\alpha _{2}^{n+1},\ldots ,\alpha _{M}^{n+1} \bigr]^{T},\\& \textbf{F}^{n+1}=\bigl[F^{n+1}(x_{1}),F^{n+1}(x_{2}), \ldots ,F^{n+1}(x_{M})\bigr]^{T}, \end{aligned}

and

$\mathbf{\text{K}}=\left(\begin{array}{cccc}{\lambda }_{1}^{x}{K}_{y}\left(x\right){|}_{y={x}_{1}}& {\lambda }_{2}^{x}{K}_{y}\left(x\right){|}_{y={x}_{1}}& \cdots & {\lambda }_{M}^{x}{K}_{y}\left(x\right){|}_{y={x}_{1}}\\ {\lambda }_{1}^{x}{K}_{y}\left(x\right){|}_{y={x}_{2}}& {\lambda }_{2}^{x}{K}_{y}\left(x\right){|}_{y={x}_{2}}& \cdots & {\lambda }_{M}^{x}{K}_{y}\left(x\right){|}_{y={x}_{2}}\\ ⋮& ⋮& \ddots & ⋮\\ {\lambda }_{1}^{x}{K}_{y}\left(x\right){|}_{y={x}_{M}}& {\lambda }_{2}^{x}{K}_{y}\left(x\right){|}_{y={x}_{M}}& \cdots & {\lambda }_{M}^{x}{K}_{y}\left(x\right){|}_{y={x}_{M}}\end{array}\right).$
(2.11)

We know that

\begin{aligned} \textbf{U}^{n+1}=\textbf{A}[\alpha ]^{n+1}, \end{aligned}
(2.12)

where

\begin{aligned} \textbf{A}=[A_{ij}]_{M \times M},\qquad A_{ij}=\psi _{j}(x_{i}) \end{aligned}

and

\begin{aligned} \textbf{U}^{n+1}=\bigl[U^{n+1}(x_{1}),U^{n+1}(x_{2}), \ldots ,U^{n+1}(x_{M})\bigr]^{T}. \end{aligned}

The following matrix vector form is achieved by differentiating (2.12) with respect to x and evaluating it at the point girds $$x_{i}\in \mathcal{B}_{M}$$:

\begin{aligned} \Delta \textbf{U}^{n+1}=\textbf{A}_{xx}[\alpha ]^{n+1}, \end{aligned}

where

\begin{aligned} \Delta \textbf{U}^{n+1}=\bigl[\Delta U^{n+1}(x_{1}), \Delta U^{n+1}(x_{2}), \ldots ,\Delta U^{n+1}(x_{M}) \bigr]^{T} \end{aligned}

and

\begin{aligned} \textbf{A}_{xx}=[A_{xx,ij}]_{M \times M},\qquad A_{xx,ij}= \frac{\partial ^{2}\psi _{j}}{\partial x^{2}}|_{x=x_{i}}. \end{aligned}

Now, from $$\textbf{U}^{n+1}=\textbf{A}[\alpha ]^{n+1}$$ we have

\begin{aligned}{} [\alpha ]^{n+1}=\textbf{A}^{-1}\textbf{U}^{n+1}, \end{aligned}

and therefore

\begin{aligned} \Delta \textbf{U}^{n+1}=\textbf{A}_{xx} \textbf{A}^{-1}\textbf{U}^{n+1}. \end{aligned}
(2.13)

Now, by using (2.13), we can write

\begin{aligned} \textbf{K} \textbf{U}^{n+1}=\textbf{F}^{n+1},\quad 0 \leq n \leq N-2, \end{aligned}

where

\begin{aligned} \textbf{K}=\textstyle\begin{cases} ((2a(\alpha ,\tau )+1)\textbf{I}-\textbf{A}_{xx}\textbf{A}^{-1} ) ,& n=0, \\ ((a(\alpha ,\tau )+1)\textbf{I}-\textbf{A}_{xx}\textbf{A}^{-1} ),& 1\leq n \leq N-2, \end{cases}\displaystyle \end{aligned}
(2.14)

and

$$\textbf{F}^{n+1}=\textstyle\begin{cases} a(\alpha ,\tau )(2\Phi _{1}+2\tau \Phi _{2})+ \textbf{f}^{1},\quad n=0, \\ -a(\alpha ,\tau )(-2\textbf{U}^{n}+\textbf{U}^{n-1})-a(\alpha ,\tau ) \sum_{j=1}^{n}b_{j}(\textbf{U}^{n+1-j} \\ \quad {}-2\textbf{U}^{n-j}+\textbf{U}^{n-1-j})+ \textbf{f}^{n+1}, \quad 1\leq n \leq N-2, \end{cases}$$

in which

\begin{aligned} \textbf{f}^{n+1}=\bigl[ f^{n+1}(x_{1}), f^{n+1}(x_{2}),\ldots ,f^{n+1}(x_{M}) \bigr]^{T} \end{aligned}

and

\begin{aligned} \Phi _{j}=\bigl[ \phi _{j}(x_{1}), \phi _{j}(x_{2}),\ldots ,\phi _{j}(x_{M}) \bigr]^{T},\quad j=1,2. \end{aligned}

### Lemma 2.5

(See )

Let $$K_{y}(x)$$ be the reproducing kernel of the space $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$, then

$$\frac{\partial ^{i+j}K_{y}(x)}{\partial x^{i} \partial y^{j}}, \quad 0 \leq i+j \leq 2,$$
(2.15)

is absolutely continuous with respect to x and y.

### Lemma 2.6

Let $$\{x_{j}\}_{j=1}^{\infty}$$ be dense in the domain $$[0,b]$$ and the set of functions $$\{\lambda ^{x}_{j} K(x,\cdot)\}_{j=1}^{M}$$ be linearly independent on the reproducing kernel space $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$. Then the set of vectors $$\{( \lambda ^{x}_{j} K_{y}(x)|_{y=x_{1}},\lambda ^{x}_{j} K_{y}(x)|_{y=x_{2}}, \ldots )^{T}\}_{j=1}^{M}$$ is linearly independent.

### Proof

If $$\{c_{j}\}_{j=1}^{M}$$ satisfies $$\sum_{j=1}^{M}c_{j}( \lambda ^{x}_{j} K_{y}(x)|_{y=x_{1}},\lambda ^{x}_{j} K_{y}(x)|_{y=x_{2}},\ldots )^{T}=0$$, one can deduce that

\begin{aligned} \sum_{j=1}^{M}c_{j} \lambda ^{x}_{j} K_{y}(x)|_{y=x_{i}}=0, \quad i\geq 1. \end{aligned}
(2.16)

From Lemma 2.5 it is clear that the functions $$\lambda ^{x}_{j} K(x,\cdot)$$ for $$\lambda _{j} \in \Lambda _{M}$$ are continuous. Furthermore, note that $$\{x_{i}\}_{i \geq 1}$$ is a dense set. Therefore $$\sum_{j=1}^{M}c_{j} \lambda ^{x}_{j} K(x,\cdot)=0$$, which implies that $$c_{j}=0$$ ($$j=1,2,\ldots ,M$$). This completes the proof. □

From Lemma 2.6 we can extract the following theorem.

### Theorem 2.7

Let the set of functions $$\{\lambda ^{x}_{j} K(x,\cdot)\}_{j=1}^{M}$$ be linearly independent on $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$. Then there exist m points $$\mathcal{B}_{n}=\{x_{j}\}_{j=1}^{M}$$ such that the collocation matrix K is nonsingular.

### Lemma 2.8

Let the set of functionals $$\{\lambda _{j}\}_{j=1}^{M}$$ be linearly independent on $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$. Then the set of functions $$\{\lambda ^{x}_{j} K(x,\cdot)\}_{j=1}^{M}$$ is linearly independent.

### Proof

If $$\{c_{j}\}_{j=1}^{M}$$ satisfies $$\sum_{j=1}^{M}c_{j}\lambda ^{x}_{j} K(x,\cdot)=0$$, then we get

\begin{aligned} 0 =&\Biggl\langle U(\cdot),\sum_{j=1}^{M}c_{j} \lambda ^{x}_{j} K(x,\cdot)\Biggr\rangle _{ \boldsymbol{\mathsf{H}}_{p}} \\ =&\sum _{j=1}^{M}c_{j}\lambda ^{x}_{j} \bigl\langle U(\cdot), K(x,\cdot)\bigr\rangle _{\boldsymbol{\mathsf{H}}_{p}} \\ =&\sum_{j=1}^{M}c_{j}\lambda _{j}[U], \quad \forall U \in \boldsymbol{\mathsf{H}}_{p}[0,b], \end{aligned}
(2.17)

which implies that $$c_{j}=0$$ ($$j=1,2,\ldots ,M$$), and this completes the proof. □

From Lemma 2.8 and Theorem 2.7, we can derive the following theorem.

### Theorem 2.9

Let the set of functionals $$\{\lambda _{j}\}_{j=1}^{M}$$ be linearly independent on $$\boldsymbol{\mathsf{H}}_{p}[0,b]$$. Then there exist n points $$\mathcal{B}_{n}=\{x_{j}\}_{j=1}^{n}$$ such that the collocation matrix K is nonsingular.

### Error analysis

Suppose that $$\mathcal{B}_{M}=\{x_{i}\}_{i = 1}^{M}$$ and $$\mathcal{U}_{M}=\operatorname{Span}\{\psi _{1},\psi _{2},\ldots , \psi _{M}\}$$. Applying the Gram–Schmidt orthogonalization process to $$\{\psi _{1},\psi _{2},\ldots , \psi _{M}\}$$, we can obtain

\begin{aligned} \overline{\psi}_{i}(x)=\sum _{k=1}^{i}\beta _{ik}\psi _{k}(x),\quad ( \beta _{ii}>0,i=1,2,\ldots ,M). \end{aligned}
(2.18)

Therefore $$\{\overline{\psi}_{1},\overline{\psi}_{2},\ldots , \overline{\psi}_{n} \}$$ is an orthonormal basis for $$\mathcal{U}_{M}$$.

Therefore, we can write the interpolant $$U_{M}^{n+1}(x)$$ to $$U^{n+1}$$ at $$\mathcal{B}_{n}$$ in the following form:

$$U^{n+1}(x)\approx U_{M}^{n+1}(x)= \sum_{i=1}^{m}U^{n+1}(x_{i}) \overline{\psi}_{i}(x).$$
(2.19)

### Theorem 2.10

Suppose that $$U_{M}^{n+1}(x) \in \boldsymbol{\mathsf{H}}_{p}[0,b]$$ and $$U^{n+1}(x)$$ are the approximate solution and exact solution for (2.3), respectively. Then, for any $$U^{n+1}(x) \in \boldsymbol{\mathsf{H}}_{p}[0,b]$$, we have

\begin{aligned} \bigl\vert U^{n+1}(x)-U_{M}^{n+1}(x) \bigr\vert \leq \bigl\Vert U^{n+1} \bigr\Vert _{\boldsymbol{\mathsf{H}}_{p}} \Biggl\Vert K(x,\cdot)-\sum_{i=1}^{M}\overline{ \psi}_{i}(x)\psi _{i} \Biggr\Vert _{ \boldsymbol{\mathsf{H}}_{p}}. \end{aligned}
(2.20)

### Proof

Using the reproducing property, we have

\begin{aligned} U_{M}^{n+1}(x) =&\sum_{i=1}^{M}U^{n+1}(x_{i}) \overline{\psi}_{i}(x) \\ =&\sum_{i=1}^{M}\bigl\langle U^{n+1},\psi _{i}\bigr\rangle _{ \boldsymbol{\mathsf{H}}_{p}}\overline{ \psi}_{i}(x) \\ =& \Biggl\langle U^{n+1},\sum_{i=1}^{M} \overline{\psi}_{i}(x)\psi _{i} \Biggr\rangle _{\boldsymbol{\mathsf{H}}_{p}}. \end{aligned}
(2.21)

Thus

\begin{aligned} \bigl\vert U^{n+1}(x)-U_{M}^{n+1}(x) \bigr\vert =& \Biggl\vert \Biggl\langle U^{n+1},K(x,\cdot)-\sum _{i=1}^{M} \overline{\psi}_{i}(x)\psi _{i}\Biggr\rangle _{\boldsymbol{\mathsf{H}}_{p}} \Biggr\vert \\ \leq& \bigl\Vert U^{n+1} \bigr\Vert _{\boldsymbol{\mathsf{H}}_{p}} \Biggl\Vert K(x,\cdot)-\sum_{i=1}^{M} \overline{ \psi}_{i}(x)\psi _{i} \Biggr\Vert _{\boldsymbol{\mathsf{H}}_{p}}. \end{aligned}
(2.22)

Thus, the proof is completed. □

### Lemma 2.11

(See )

Suppose that $$u \in C^{3}[0,b]$$ and $$\mathcal{B}_{M}=\{x_{i}\}_{i = 1}^{M} \subset [0,b]$$ is a distinct subset of $$[0,b]$$, then

$$\Vert U \Vert _{L^{2}[0,b]} \leq d \max _{x_{j} \in \mathcal{B}_{n}} \bigl\vert U(x_{j}) \bigr\vert +ch^{3} \biggl\Vert \frac{d^{3}U}{dx^{3}} \biggr\Vert _{L^{2}[0,b]},\quad h=\sup_{x \in [0,b]}\min_{x_{j} \in \mathcal{B}_{M}} \Vert x-x_{j} \Vert ,$$
(2.23)

where c and d are real constants.

### Theorem 2.12

If $$U^{n+1}_{M}$$ is the approximate solution of (2.3) in the space $$\boldsymbol{\mathsf{H}}_{p}$$. Then the following relation holds:

\begin{aligned} \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert _{L^{2}[0,b]} \leq ch^{3} \bigl\Vert U^{n+1} \bigr\Vert _{ \boldsymbol{\mathsf{H}}_{p}}, \end{aligned}
(2.24)

where c is a real constant.

### Proof

According to Lemma 2.11, we have

\begin{aligned}& \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert _{L^{2}[0,b]} \\& \quad \leq d \max_{x_{j} \in \mathcal{B}_{n}} \bigl\vert U^{n+1}(x_{j})-U_{M}^{n+1}(x_{j}) \bigr\vert \\& \qquad {} +ch^{3} \biggl\Vert \frac{d^{3}U^{n+1}}{dx^{3}}- \frac{d^{3}U_{M}^{n+1}}{dx^{3}} \biggr\Vert _{L^{2}[0,b]} \\& \quad \leq ch^{3} \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert _{\boldsymbol{\mathsf{H}}_{p}}, \end{aligned}
(2.25)

where d and c are constants.

We know that

\begin{aligned} \bigl\Vert U^{n+1} \bigr\Vert ^{2}_{\boldsymbol{\mathsf{H}}_{p}}= \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert ^{2}_{ \boldsymbol{\mathsf{H}}_{p}}+ \bigl\Vert U_{M}^{n+1} \bigr\Vert ^{2}_{\boldsymbol{\mathsf{H}}_{p}}+2 \langle U-U_{M},U_{M} \rangle _{\boldsymbol{\mathsf{H}}_{p}}. \end{aligned}

Since

\begin{aligned} \langle U-U_{M},U_{M}\rangle _{\boldsymbol{\mathsf{H}}_{p}} =&\Biggl\langle U-U_{M}, \sum_{j=1}^{M} \alpha ^{n+1}_{j}\psi _{j}\Biggr\rangle _{ \boldsymbol{\mathsf{H}}_{p}} \\ =&\sum_{j=1}^{M}\alpha ^{n+1}_{j}\bigl\langle U-U_{M},K(\cdot ,x_{j}) \bigr\rangle _{ \boldsymbol{\mathsf{H}}_{p}} \\ =&\sum_{j=1}^{M} \alpha ^{n+1}_{j} (U-U_{M}) (x_{j})=0, \end{aligned}

then

\begin{aligned} \bigl\Vert U^{n+1} \bigr\Vert ^{2}_{\boldsymbol{\mathsf{H}}_{p}}= \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert ^{2}_{ \boldsymbol{\mathsf{H}}_{p}}+ \bigl\Vert U_{M}^{n+1} \bigr\Vert ^{2}_{\boldsymbol{\mathsf{H}}_{p}}. \end{aligned}

Therefore, we have

\begin{aligned} \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert _{\boldsymbol{\mathsf{H}}_{p}} \leq \bigl\Vert U^{n+1} \bigr\Vert _{ \boldsymbol{\mathsf{H}}_{p}}. \end{aligned}
(2.26)

Now, from (2.25) and (2.26), we can obtain

\begin{aligned}& \bigl\Vert U^{n+1}-U_{M}^{n+1} \bigr\Vert _{L^{2}[0,b]} \leq ch^{3} \bigl\Vert U^{n+1} \bigr\Vert _{ \boldsymbol{\mathsf{H}}_{p}}. \end{aligned}
(2.27)

Thus, the proof is completed. □

## Illustrative test problems

We have studied some example tests to illustrate the performance of the proposed methods. We show the stability and accuracy of the proposed method for different values of M and N.

As the exact solution is known, the root mean square error $$L_{rms}$$ and the maximum absolute error $$L_{\infty}$$ are measured with the following formulas:

\begin{aligned} L_{rms}= \sqrt{\frac{1}{M}\sum _{i=1}^{M} \bigl\vert u^{N}(x_{i})-U^{N}_{M}(x_{i}) \bigr\vert ^{2}} \end{aligned}

and

\begin{aligned} L_{\infty}= \max_{1\leq i \leq M} \bigl\vert u^{N}(x_{i})-U^{N}_{M}(x_{i}) \bigr\vert . \end{aligned}

### Example 3.1

In this example, we deal with the following problem:

\begin{aligned}& \partial ^{\alpha}_{t} u(x,t)=\Delta u(x,t)-u(x,t)+f(x,t),\quad (x,t) \in (0,1) \times (0,1], \end{aligned}
(3.1)
\begin{aligned}& u(0,t)=0,\qquad u(1,t)=0,\quad t \in [0,1], \end{aligned}
(3.2)
\begin{aligned}& u(x,0)=0, \qquad \partial _{t}u(x,0)=0,\quad x \in [0,1], \end{aligned}
(3.3)

where

\begin{aligned} f(x,t)=\sin (\pi x) \biggl(\frac{2 t^{-\alpha +2}}{\Gamma (-\alpha +3)} +t^{2} \pi ^{2} +t^{2} \biggr). \end{aligned}

The exact solution u is given by $$u(x,t)=t^{2} \sin (\pi x)$$. The proposed method in the previous section is tested on this problem with $$\{t_{n}=\frac{n}{N} \}_{n=0}^{N}$$, $$N=100$$, $$\{x_{i}=\frac{i}{M+1}\}_{i=1}^{M}$$, $$M=5,10,20,40$$.

We consider the RKHS $$\boldsymbol{\mathsf{H}}_{p}[0,1]$$ with the following RK:

\begin{aligned} K_{x}(y)= \textstyle\begin{cases} \frac{1}{120}(1-x)y(120x+x(6x-120-4x^{2}+x^{3})y-5xy^{3}+(1+x)y^{4}), & y< x, \\ \frac{1}{120}(1-y)x(120y+y(6y-120-4y^{2}+y^{3})x-5yx^{3}+(1+y)1x^{4}), & y\geq x. \end{cases}\displaystyle \end{aligned}

In Tables 1 and 2, we present, the root mean square error $$L_{rms}$$, the maximum absolute error $$L_{\infty}$$, and the convergence ratio in the computed solutions for Example 3.1 with $$\alpha =1.2,1.4,1.8,1.9$$. In a considerable number of cases, an exciting agreement between the results is observed, which confirms the excellent validity of the proposed method.

### Example 3.2

In this example, we deal with the following problem:

\begin{aligned}& \partial ^{\alpha}_{t} u(x,t)=\Delta u(x,t)-u(x,t)+f(x,t),\quad (x,t) \in (0,2) \times (0,1], \end{aligned}
(3.4)
\begin{aligned}& u(0,t)=0, \qquad u(2,t)=0, \quad t \in [0,1], \end{aligned}
(3.5)
\begin{aligned}& u(x,0)=0, \qquad \partial _{t}u(x,0)=0,\quad x \in [0,1], \end{aligned}
(3.6)

where

\begin{aligned} f(x,t)= \frac{e^{-x}(8t^{2-\alpha}\sin (\frac{\pi}{2}x)+4t^{2}\cos (\frac{\pi}{2}x)\pi \Gamma (3-\alpha )+t^{2}\sin (\frac{\pi}{2}x)\pi ^{2}\Gamma (3-\alpha ))}{4\Gamma (3-\alpha )}. \end{aligned}

The exact solution u is given by $$u(x,t)=t^{2} e^{-x}\sin (\frac{\pi}{2}x)$$. The proposed method in the previous section is tested on this problem with $$\{t_{n}=\frac{n}{N} \}_{n=0}^{N}$$, $$N=120$$, $$\{x_{i}=\frac{2i}{M+1}\}_{i=1}^{M}$$, $$M=5,10,20,40$$.

We consider the RKHS $$\boldsymbol{\mathsf{H}}_{p}[0,2]$$ with the following RK:

\begin{aligned} K_{x}(y)= \textstyle\begin{cases} \frac{1}{480}(2-x)y(240x+x(24x-120-8x^{2}+x^{3})y-10xy^{3}+(2+x)y^{4}), & y< x, \\ \frac{1}{120}(2-y)x(240y+y(24y-120-8y^{2}+y^{3})x-10yx^{3}+(2+y)x^{4}), & y\geq x. \end{cases}\displaystyle \end{aligned}

In Tables 3 and 4, we present the root mean square error $$L_{rms}$$, the maximum absolute error $$L_{\infty}$$, and the convergence ratio in the computed solutions for Example 3.2 with $$\alpha =1.3,1.5,1.7,1.95$$. In a considerable number of cases, an exciting agreement between the results is observed, which confirms the excellent validity of the proposed method.

## Conclusion

In this paper, a finite difference/pseudo-spectral method is presented to solve time-fractional diffusion-wave equation. The method is based on a finite difference method in a temporal direction to obtain a semi-discrete configuration, whereas a pseudo-spectral method based on RK introduces spatial discretization. The implementation of the proposed method is very simple and has reasonable accuracy. It can be seen from the error norms and numerical results that the proposed method for solving time-fractional diffusion-wave equation is an excellent method.

## Availability of data and materials

No data to be declared.

## References

1. Hifer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)

2. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

3. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

4. Ortigueira, M.D.: Fractional Calculus for Scientists and Engineer. Lecture Notes in Electrical. Springer, Berlin (2011)

5. Qiao, L., Wang, Z., Xu, D.: An alternating direction implicit orthogonal spline collocation method for the two dimensional multi-term time fractional integro-differential equation. Appl. Numer. Math. 151, 199–212 (2020)

6. Wang, Z., Cen, D., Mo, Y.: Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels. Appl. Numer. Math. 159, 190–203 (2021)

7. Cen, D., Wang, Z., Mo, Y.: Second order difference schemes for time-fractional KdV–Burgers’ equation with initial singularity. Appl. Math. Lett. 112, 106829 (2021)

8. Cen, D., Wang, Z.: Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations. Appl. Math. Lett. 129, 107919 (2022)

9. Fardi, M., Khan, Y.: A fast difference scheme on a graded mesh for time-fractional and space distributed-order diffusion equation with nonsmooth data. Int. J. Mod. Phys. B 36(15), 2250076 (2022)

10. Fardi, M., Ghasemi, M.: A numerical solution strategy based on error analysis for time-fractional mobile/immobile transport model. Soft Comput. 25, 11307–11331 (2021)

11. Fardi, M., Khan, Y.: A novel finite difference-spectral method for fractal mobile/immobile transport model based on Caputo–Fabrizio derivative. Chaos Solitons Fractals 143, 110573 (2021)

12. Abu-Gdairi, R., Hasan, S., Al-Omari, S., Al-Smadi, M., Momani, S.: Attractive multistep reproducing kernel approach for solving stiffness differential systems of ordinary differential equations and some error analysis. Comput. Model. Eng. Sci. 130(1), 299–313 (2022)

13. Shqair, M., Alabedalhad, M., Al-Omari, S., Al-Smadi, M.: Abundant exact travelling wave solutions for a fractional massive Thirring model using extended Jacobi elliptic function method. Fractal Fract. 6(252), 1–16 (2022)

14. Ayasrah, M., Al-Smadi, M., Al-Omari, S., Baleanu, D., Momani, S.: Structure of optical soliton solution for nonlinear resonant space-time Schrodinger equation in conformable sense with full nonlinearity term. Phys. Scr. 95(10), 105215 (2020)

15. Al-Qudah, Y., Alaroud, M., Qoqazeh, H., Jaradat, A., AlHazmi, S., Al-Omari, S.: Approximate analytic-numeric fuzzy solutions of fuzzy fractional equations using residual power series approach. Symmetry 14(804), 1–19 (2022)

16. Abdelhadi, M., Alhazmi, S., Al-Omari, S.K.: On a class of partial differential equations and their solution via local fractional integrals and derivatives. Fractal Fract. 6(210), 1–15 (2022)

17. Benson, D.A., Schumer, R., Meerschaert, M.M., Wheatcraft, S.W.: Fractional dispersion, Levy motion, and the MADE tracer tests. Transp. Porous Media 42(1), 211–240 (2001)

18. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Levy motion. Water Resour. Res. 36(6), 1413–1423 (2000)

19. Gafiychuk, V., Datsko, B., Meleshko, V.: Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1–2), 215–225 (2008)

20. Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. Phys. A, Stat. Mech. Appl. 370(1), 114–118 (2006)

21. Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Phys. A, Stat. Mech. Appl. 314(1–4), 749–755 (2002)

22. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29(1), 129–143 (2002)

23. Pachepsky, Y., Timlin, D., Rawls, W.: Generalized Richards’ equation to simulate water transport in unsaturated soils. J. Hydrol. 272(1–4), 3–13 (2003)

24. O’Shaughnessy, B., Procaccia, I.: Analytical solutions for diffusion on fractal objects. Phys. Rev. Lett. 54(5), 455 (1985)

25. Ubriaco, M.R.: A simple mathematical model for anomalous diffusion via Fisher’s information theory. Phys. Lett. A 373(44), 4017–4021 (2009)

26. Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional in space reaction-diffusion equations. BIT Numer. Math. 54, 937–954 (2014)

27. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

28. Chen, W., Ye, L., Sun, H.: Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59, 1614–1620 (2010)

29. Wang, J.G., Wei, T.: An iterative method for backward time-fractional diffusion problem. Numer. Methods Partial Differ. Equ. 30, 2029–2041 (2014)

30. Szekeres, B.J., Izsák, F.: A finite difference method for fractional diffusion equations with Neumann boundary conditions. Open Math. 13, 581–600 (2015)

31. Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. R. Soc. Lond. Ser. A 209, 415–446 (1909)

32. Aronszajn, N.: Theory of Reproducing Kernels. Harvard University Press, Cambridge (1951)

33. Cui, M., Deng, Z.H.: Numerical Functional Method In Reproducing Kernel space. The Publication of Harbin Institute of Technology (1988)

34. Cui, M., Yan, Y.: The representation of the solution of a kind of operator equation $$Au = f$$. Numer. Math. J. Chin. Univ. 1, 82–86 (1995)

35. Geng, F., Qian, S.: Piecewise reproducing kernel method for singularly perturbed delay initial value problems. Appl. Math. Lett. 37, 67–71 (2014)

36. Wang, Y., Li, Z., Cao, Y., Wan, X.: A new method for solving a class of mixed boundary value problems with singular coefficient. Appl. Math. Comput. 217(6), 2768–2772 (2010)

37. Jiang, W., Cui, M., Lin, Y.: Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method. Commun. Nonlinear Sci. Numer. Simul. 15(7), 1754–1758 (2010)

38. Zhou, Y., Cui, M., Lin, Y.: Numerical algorithm for parabolic problems with non-classical conditions. J. Comput. Appl. Math. 230(2), 770–780 (2009)

39. Xu, M., Lin, Y.: Simplified reproducing kernel method for fractional differential equations with delay. Appl. Math. Lett. 52, 156–161 (2016)

40. Cui, M., Lin, Y.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Publ. (Nova Science Publishers), New York (2009)

41. Al-Smadi, M., Djeddi, N., Momani, S., Al-Omari, S., Araci, S.: An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space. Adv. Differ. Equ. 2021, 271 (2021)

42. Momani, S., Djeddi, N., Al-Smadi, M., Al-Omari, S.: Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method. Appl. Numer. Math. 170, 418–434 (2021)

43. Fardi, M., Ghasemi, M.: Numerical solution of singularly perturbed 2D parabolic initial-boundary-value problems based on reproducing kernel theory: error and stability analysis. Numer. Methods Partial Differ. Equ. 38(4), 876–903 (2022)

44. Fardi, M., Khan, Y.: Numerical simulation of squeezing Cu–Water nanofluid flow by a kernel-based method. Int. J. Model. Simul. Sci. Comput. 13(01), 2250005 (2022)

45. Fardi, M., Khan, Y.: A kernel-based method for Volterra delay integro-differential equations. Hacet. J. Math. Stat. (2022). https://doi.org/10.15672/hujms.879507

46. Fardi, M., Pishkar, I., Alidousti, J., Khan, Y.: Numerical investigation of the MHD suction–injection model of viscous fluid using a kernel-based method. Arch. Appl. Mech. 91, 4205–4221 (2021)

47. Yaseen, M., Abbas, M., Nazir, T., Baleanu, D.: A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation. Adv. Differ. Equ. 2017, 274 (2017)

48. Fasshauer, G.E., McCourt, M.: Kernel-Based Approximation Methods with Matlab. World Scientific, New Jersey (2016)

## Acknowledgements

The authors would like to thank Springer Nature for their support and the reviewers for their valuable comments which improved the content of the article.

## Author information

Authors

### Contributions

MF wrote the original draft preparation. SA and MF performed formal analysis and methodology. SA and S Araci performed investigation. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Shrideh K. Qasem Al-Omari.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions 