An approach based on the pseudospectral method for fractional telegraph equations

Haifa Bin Jebreen; Beatriz Hernández-Jiménez; Haifa Bin Jebreen; Beatriz Hernández-Jiménez

doi:10.3934/math.20231496

AIMS Mathematics

2023, Volume 8, Issue 12: 29221-29238. doi: 10.3934/math.20231496

Previous Article Next Article

Research article Special Issues

An approach based on the pseudospectral method for fractional telegraph equations

Haifa Bin Jebreen ^{1
,
,},
Beatriz Hernández-Jiménez ²

1.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Departamento de Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, 41013 Sevilla, Spain

Received: 16 August 2023 Revised: 08 October 2023 Accepted: 12 October 2023 Published: 26 October 2023
MSC : 54A25, 65M70, 65Bxx, 35L20

We aim to implement the pseudospectral method on fractional Telegraph equation. To implement this method, Chebyshev cardinal functions (CCFs) are considered bases. Introducing a matrix representation of the Caputo fractional derivative (CFD) via an indirect method and applying it via the pseudospectral method helps to reduce the desired problem to a system of algebraic equations. The proposed method is an effective and accurate numerical method such that its implementation is easy. Some examples are provided to confirm convergence analysis, effectiveness and accuracy.

Keywords:

Citation: Haifa Bin Jebreen, Beatriz Hernández-Jiménez. An approach based on the pseudospectral method for fractional telegraph equations[J]. AIMS Mathematics, 2023, 8(12): 29221-29238. doi: 10.3934/math.20231496

Related Papers:

[1]	Haifa Bin Jebreen, Beatriz Hernández-Jiménez . Pseudospectral method for fourth-order fractional Sturm-Liouville problems. AIMS Mathematics, 2024, 9(9): 26077-26091. doi: 10.3934/math.20241274
[2]	A. K. Mittal, L. K. Balyan . Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain. AIMS Mathematics, 2020, 5(3): 1642-1662. doi: 10.3934/math.2020111
[3]	Haifa Bin Jebreen, Hongzhou Wang . On the effective method for the space-fractional advection-diffusion equation by the Galerkin method. AIMS Mathematics, 2024, 9(9): 24143-24162. doi: 10.3934/math.20241173
[4]	M. Mossa Al-Sawalha, Azzh Saad Alshehry, Kamsing Nonlaopon, Rasool Shah, Osama Y. Ababneh . Fractional view analysis of delay differential equations via numerical method. AIMS Mathematics, 2022, 7(12): 20510-20523. doi: 10.3934/math.20221123
[5]	Dina Abdelhamid, Wedad Albalawi, Kottakkaran Sooppy Nisar, A. Abdel-Aty, Suliman Alsaeed, M. Abdelhakem . Mixed Chebyshev and Legendre polynomials differentiation matrices for solving initial-boundary value problems. AIMS Mathematics, 2023, 8(10): 24609-24631. doi: 10.3934/math.20231255
[6]	Haifa Bin Jebreen, Yurilev Chalco Cano, Ioannis Dassios . An efficient algorithm based on the multi-wavelet Galerkin method for telegraph equation. AIMS Mathematics, 2021, 6(2): 1296-1308. doi: 10.3934/math.2021080
[7]	S. Mohammadian, Y. Mahmoudi, F. D. Saei . Solution of fractional telegraph equation with Riesz space-fractional derivative. AIMS Mathematics, 2019, 4(6): 1664-1683. doi: 10.3934/math.2019.6.1664
[8]	Shazia Sadiq, Mujeeb ur Rehman . Solution of fractional boundary value problems by $\psi$ -shifted operational matrices. AIMS Mathematics, 2022, 7(4): 6669-6693. doi: 10.3934/math.2022372
[9]	Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri . Spectral tau solution of the linearized time-fractional KdV-Type equations. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830
[10]	Sunyoung Bu . A collocation methods based on the quadratic quadrature technique for fractional differential equations. AIMS Mathematics, 2022, 7(1): 804-820. doi: 10.3934/math.2022048

Abstract

1. Introduction

The objective of this project is to employ the pseudospectral approach to approximate the solution of the fractional Telegraph equation

\frac{\partial^{η} w (x, t)}{\partial t^{η}} + s_{1} \frac{\partial^{η - 1} w (x, t)}{\partial t^{η - 1}} + s_{2} w (x, t) = s_{3} \frac{\partial^{2} w (x, t)}{\partial x^{2}} + q (x, t),

$\begin{equation} \frac{\partial^{\eta}w(x, t)}{\partial t^{\eta}}+s_1\frac{\partial^{\eta-1}w(x, t)}{\partial t^{\eta-1}}+s_2w(x, t) = s_3\frac{\partial^{2}w(x, t)}{\partial x^{2}}+q(x, t), \end{equation}$

(1.1)

subjected to boundary and initial conditions

w (0, t) = f_{0} (t), w (1, t) = f_{1} (t), t \in [0, 1],

$\begin{equation} w(0, t) = f_0(t), \quad w(1, t) = f_1(t), \quad t\in [0, 1], \end{equation}$

(1.2)

w (x, 0) = p_{0} (x), w^{'} (x, 1) = p_{1} (x), x \in [0, 1] .

$\begin{equation} w(x, 0) = p_0(x), \quad w'(x, 1) = p_1(x), \quad x\in [0, 1]. \end{equation}$

(1.3)

Here, $s_1$ , $s_2$ and $s_3$ are constants, and $q(x, t)$ is a known function. The fractional derivative is of the CFD type, so we will introduce it later.

As it is widely acknowledged, partial differential equations (PDEs) play a significant role in the simulation of multitudinous physical phenomena. Among them, the Telegraph equation, due to its application in modeling several phenomena such as electrical phenomena, signal processing and wave propagation, has been the focus of researchers. For this reason, solving it can be a fascinating challenge and attract the focus of scientists. Several methods are devoted to solving this equation. Interpolating scaling functions are applied to solve the problem via the collocation method ^[1]. Yang et al. ^[2] used orthogonal spline collocation method to solve the sub-diffusion equation. In ^[3], the authors proposed the collocation method for the desired equation using Chebyshev cardinal functions. Dehghan et al. ^[4] proposed a collocation method based on splines radial basis function. In ^[5], authors applied meshfree collocation method to solve Telegraph equation. The numerical results raised of the collocation method, taking the B-spline functions, are investigated in ^[6]. We list several numerical methods proposed to solve this equation, including shifted Jacobi collocation ^[7], multi-wavelet Galerkin method ^[8], Tchebyshev-Galerkin method ^[9], tau method ^[10] and so on.

In recent years, fractional calculations in modeling various physical phenomena have attracted the attention of engineers and specialists. Moreover, many types of equations with fractional derivatives have been considered, and many numerical schemes have been introduced to solve them, including the implementation of the Galerkin method for the fractional Riccati equation where biorthogonal Hermite cubic Spline is considered ^[11]. The application of the Tau method for a solution of space-time fractional PDEs has been considered based on interpolating scaling functions ^[12]. The multiwavelet method for solving the Cauchy-type problem is considered in ^[13]. In ^[14], the authors introduced the hybrid clique functions and applied them via the collocation method for fractional Schrödinger equation. A predictor-corrector compact difference scheme for a nonlinear fractional differential equation is applied in ^[15]. A nonlinear finite volume method is used to solve multi-term fractional sub-diffusion equation on polygonal meshes ^[16]. Zhang et al. ^[17] utilized collocation method based on Spline functions to solve the nonlinear fourth-order reaction sub-diffusion equation. In ^[18], the authors used the collocation method based on cubic B-spline functions to solve the time-fractional cable model. In ^[19], the authors applied a novel numerical technique to solve the time fractional reaction-diffusion model with a non-singular kernel, etc.

Among fractional PDEs, the fractional Telegraph equation, due to its application, has been considered as a challenging problem by many scientists. Hosseini et al. ^[20] applied a hybrid method based on finite differences and radial basis functions to solve the Eq (1.1). In ^[21], the spectral Galerkin method based on Legendre polynomials is studied for solving the problem. Mollahasani et al. ^[22] utilized a hybrid function scheme based on Block-Pulse-Functions and Legendre polynomials for solving (1.1). There is another form of time-fractional Telegraph equation

\frac{\partial^{2 η} w (x, t)}{\partial t^{η}} + s_{1} \frac{\partial^{η} w (x, t)}{\partial t^{η}} = s_{2} \frac{\partial^{2} w (x, t)}{\partial x^{2}} + q (x, t),

$\begin{equation} \frac{\partial^{2\eta}w(x, t)}{\partial t^{\eta}}+s_1\frac{\partial^{\eta}w(x, t)}{\partial t^{\eta}} = s_2\frac{\partial^{2}w(x, t)}{\partial x^{2}}+q(x, t), \end{equation}$

(1.4)

along with the initial conditions

w (x, 0) = p_{0} (x), w^{'} (x, 1) = p_{1} (x), x \in [0, 1],

$\begin{equation} w(x, 0) = p_0(x), \quad w'(x, 1) = p_1(x), \quad x\in [0, 1], \end{equation}$

(1.5)

and boundary conditions

w (1, t) + a_{2} w_{x} (1, t) = f_{1} (t), w (0, t) + a_{1} w_{x} (0, t) = f_{0} (t), t \in [0, 1] .

$\begin{equation} w(1, t)+a_2w_x(1, t) = f_1(t), \quad w(0, t)+a_1w_x(0, t) = f_0(t), \quad t\in [0, 1]. \end{equation}$

(1.6)

For solving this equation, Saadatmandi et al. ^[23] used the Legendre polynomials and Tau method. In ^[24], the separable variable method is considered such as an analytical method to solve Eq (1.4). In ^[25], the authors solved (1.4) via the bireproducing kernel theorem. Akram et al. ^[19] used modified extended cubic B-spline functions to solve the Non-Linear Time-Fractional Telegraph Equation. The 2D time-fractional Telegraph equation is solved using modified fractional group iterative scheme ^[26].

The Chebyshev cardinal functions are applied as attractive bases for solving various kinds of equations. Owing to their abilities, they can be used in the pseudospectral and Galerkin methods. Recently, Shahriari et al. ^[27] studied the fractional Dirac problem using these bases. These bases are applied for solving the fractional Sturm-Liouville problem in ^[28]. Bin Jebreen et al. solved a family of time-fractional partial differential equation by these bases.

In this paper, for the first time, the Hyperbolic fractional Telegraph equation is solved using Cardinal Chebyshev functions. As you know, existing fractional derivatives and the nature of the hyperbolic partial differential equations are two factors that cause problems in the numerical solution. To overcome these issues, we apply the pseudospectral method based on Chebyshev cardinal functions. In this study, a matrix representation of the Caputo fractional derivative is introduced via an indirect method (using the relation between the Caputo fractional derivative and fractional integration) which plays a key role in our algorithm. An analysis of convergence is investigated to show the effectiveness and efficiency of the method.}

The paper is organized as follows. The CCFs along with their properties are briefly described in Section 2. In Section 3, the pseudospectral method implements for approximating the solution of the problem. In this section, also, convergence analysis is investigated for the presented method. Some numerical examples are considered to give an affirmation of the method's efficiency.

2. Chebyshev cardinal functions (CCFs)

Consider the Chebyshev nodes as a set of numbers

Y := {y_{j} : T_{n + 1} (y_{j}) = 0, j \in Ω}, Ω := {1, 2, \dots, n + 1},

$\mathcal{Y}: = \{y_j:T_{n+1}(y_j) = 0, j\in \Omega\}, \quad \Omega: = \{1, 2, \ldots, n+1\},$

in which $T_{n+1}$ is Tchebyshev polynomial of order $n+1$ , $n$ is a positive integer number, and $\{y_j\}_{j\in\Omega}$ are the roots of $T_{n+1}$ on $[-1, 1]$ . As we know, the roots of this polynomial are obtained via

y_{j} := \cos (\frac{2 j - 1}{2 n + 2} π), \forall j \in Ω .

$\begin{equation} y_j: = \cos\left(\frac{2j-1}{2n+2}\pi\right), \quad \forall j\in \Omega. \end{equation}$

(2.1)

It is so easy to verify that the Tchebyshev polynomials can be shifted on any arbitrary interval using a proper change of variables. The generated polynomials are known as the shifted Chebyshev polynomials, given by

T_{n + 1}^{*} (x) := T_{n + 1} (\frac{2 (x - a)}{b - a} - 1), x \in [a, b] .

$\begin{equation} T^*_{n+1}(x): = T_{n+1}\left(\frac{2(x-a)}{b-a}-1\right), \quad x\in [a, b]. \end{equation}$

(2.2)

Due to the change of variable $y = \left(\frac{2(x-a)}{b-a}-1\right)$ , the roots of the Chebyshev polynomials are also shifted to interval [a, b] and are given by $x_j = \frac{(y_j+1)(b-a)}{2}+a$ .

The CCFs are the momentous type of cardinal functions that use orthogonal polynomials. These polynomials are defined via

ψ_{j} (x) = \frac{T_{n + 1}^{*} (x)}{T_{n + 1, x}^{*} (x_{j}) (x - x_{j})}, j \in Ω,

$\begin{equation} \psi_j(x) = \frac{T^*_{n+1}(x)}{T^*_{n+1, x}(x_j)(x-x_j)}, \quad j\in \Omega, \end{equation}$

(2.3)

in which the subscript $x$ demonstrates differentiation with respect to $x$ ( $T^*_{n+1, x}(x_j): = \frac{d}{dx}T^*_{\omega+1}(x)|_{x = x_j}$ ). For simplicity and computational demands, these polynomials can be demonstrated as

ψ_{j} (x) = ρ \prod_{k = 1, k \neq j}^{n + 1} (x - x_{k}),

$\begin{equation} \psi_j(x) = \rho \prod\limits_{{k = 1, k\neq j}}^{n+1}(x-x_k), \end{equation}$

(2.4)

where $\rho = 2^{2n+1}/((b-a)^{n+1}T^*_{n+1, x}(x_j))$ . These types of functions have significant property so that make them powerful tools for solving differential equations. According to Eq (2.3), It is obvious that

ψ_{j} (x_{i}) = δ_{j i} = {\begin{cases} 1, & j = i, \\ 0, & j \neq i, \end{cases} i \in Ω,

$\begin{equation} \psi_j(x_i) = \delta_{ji} = \left\{ \begin{array}{ll} 1, & { j = i }, \\ 0, & { j\neq i }, \end{array} \right.\quad i\in \Omega, \end{equation}$

(2.5)

where $\delta_{ji}$ determines the Kronecker $\delta$ -function. This property is sufficient to demonstrate that any function $p(x)$ can be easily represented as an expansion based on Chebyshev cardinal functions, i.e.,

p (x) \approx p_{n} (x) = \sum_{j = 1}^{n + 1} p (x_{j}) ψ_{j} (x) .

$\begin{equation} p(x)\approx p_{n}(x) = \sum\limits_{j = 1}^{n+1}p(x_j)\psi_j(x). \end{equation}$

(2.6)

Let $\mathcal{D}$ be the derivative operator. Given $\omega\in \mathbb{N}$ , the Sobolev space $H^{\omega}([0, 1])$ is specified by

H^{ω} ([0, 1]) = {p \in L_{2} ([0, 1]) : \forall n^{'} \leq ω, D^{n^{'}} p \in L_{2} ([0, 1])} .

$H^{\omega}([0, 1]) = \{p\in L_2([0, 1]):\forall n'\leq \omega, \mathcal{D}^{n'}p\in L_2([0, 1])\}.$

This space is equipped with norm

‖ p ‖_{H^{ω} ([0, 1])}^{2} = \sum_{k = 0}^{ω} ‖ p^{(k)} ‖_{L_{2} ([0, 1])}^{2},

$\begin{equation} \|p\|_{H^{\omega}([0, 1])}^2 = \sum\limits_{k = 0}^{\omega}\|p^{(k)}\|_{L_2([0, 1])}^2, \end{equation}$

(2.7)

and semi-norm

| p |_{H^{ω, n} ([0, 1])}^{2} = \sum_{k = min {ω, n}}^{n} ‖ p^{(k)} ‖_{L_{2} ([0, 1])}^{2} .

$\begin{equation} |p|_{H^{\omega, n}([0, 1])}^2 = \sum\limits_{k = \min\{\omega, n\}}^n\|p^{(k)}\|^2_{L_2([0, 1])}. \end{equation}$

(2.8)

Lemma 1. ^[29] Assuming the set of points $\{x_j\}_{j\in\Omega}$ as the shifted Gauss-Chebyshev points, we say that the bound of error for (2.6) can be approximated by

‖ p - p_{n} ‖_{L_{2} ([0, 1])} \leq C_{0} n^{- ω} | p |_{H^{ω, n} ([0, 1])},

$\begin{equation} \|p-p_{n}\|_{L_2([0, 1])}\leq C_0n^{-\omega}|p|_{H^{\omega, n}([0, 1])}, \end{equation}$

(2.9)

where $\omega\in \mathbb{N}$ and $C_0$ is a constant and independent of $\omega$ .

Putting the Chebyshev cardinal functions $\psi_j(x)$ into an $(n+1)$ -dimensional vector, we introduce the vector function $\Psi(x)$ whose $j$ -th entry is $\psi_j(x)$ .

Lemma 2. The derivative operator $\mathcal{D}$ can be represented by a square matrix $D$ as

D (Ψ) (x) \approx D Ψ (x),

$\begin{equation} \mathcal{D}(\Psi)(x)\approx D\Psi(x), \end{equation}$

(2.10)

whose entries are given by

D_{j, i} = D (ψ_{j}) (x_{i}) = {\begin{cases} \sum_{\begin{matrix} l = 1 \\ l \neq i \end{matrix}}^{n + 1} \frac{1}{(x_{i} - x_{l})}, & j = i, \\ ρ \prod_{\begin{matrix} l = 1 \\ l \neq i, j \end{matrix}}^{n + 1} (x_{i} - x_{l}), & j \neq i . \end{cases}

$\begin{equation} {D}_{j, i} = \mathcal{D}(\psi_j)(x_i) = \left\lbrace\begin{array}{ll} \sum\limits_{\substack{l = 1 \\ l\neq i}}^{n+1}\frac{1}{(x_i-x_l)}, & j = i, \\ \rho\prod\limits_{\substack{l = 1 \\ l\neq i, j}}^{n+1}(x_i-x_l), & j\neq i. \end{array} \right. \end{equation}$

(2.11)

Proof. Using (2.6) and (2.10), we can easily verify that the elements of matrix $D$ are computed by

D_{j, i} = D (ψ_{j}) (x_{i}) .

$\begin{equation} {D}_{j, i} = \mathcal{D}(\psi_j)(x_i). \end{equation}$

(2.12)

Motivated by (2.4), to derive the entries of matrix $D$ , the following results can be obtained by taking the derivative with respect to the variable $x$ from both sides of (2.4), viz,

D (ψ_{j}) (x) = ρ D \prod_{\begin{matrix} k = 1 \\ k \neq j \end{matrix}}^{n + 1} (x - x_{k}) = ρ \sum_{\begin{matrix} l = 1 \\ l \neq j \end{matrix}}^{n + 1} \prod_{\begin{matrix} k = 1 \\ k \neq j, l \end{matrix}}^{n + 1} (x - x_{k}) = \sum_{\begin{matrix} l = 1 \\ l \neq j \end{matrix}}^{n + 1} \frac{T_{n + 1}^{*} (x)}{(x - x_{j}) (x - x_{l}) T_{n + 1, x}^{*} (x_{j})} = \sum_{\begin{matrix} l = 1 \\ l \neq j \end{matrix}}^{n + 1} \frac{1}{(x - x_{l})} ψ_{j} (x) .

$\begin{equation} \mathcal{D}(\psi_j)(x) = \rho \mathcal{D} \prod\limits_{\substack{k = 1 \\ k\neq j}}^{n+1}(x-x_k) = \rho\sum\limits_{\substack{l = 1 \\ l\neq j}}^{n+1}\prod\limits_{\substack{k = 1 \\ k\neq j, l}}^{n+1}(x-x_k)\\ = \sum\limits_{\substack{l = 1 \\ l\neq j}}^{n+1}\frac{T^*_{n+1}(x)}{(x-x_j)(x-x_l)T^*_{n+1, x}(x_j)}\\ = \sum\limits_{\substack{l = 1 \\ l\neq j}}^{n+1}\frac{1}{(x-x_l)}\psi_j(x). \end{equation}$

(2.13)

Thus, this gives rise to (2.11) and we have

\begin{aligned} D (ψ_{j}) (x_{i}) & = \sum_{\begin{matrix} l = 1 \\ l \neq i \end{matrix}}^{n + 1} \frac{1}{(x_{i} - x_{l})}, i = j, \\ D (ψ_{j}) (x_{i}) & = ρ \prod_{\begin{matrix} l = 1 \\ l \neq i, j \end{matrix}}^{n + 1} (x_{i} - x_{l}), i \neq j . \end{aligned}

$\begin{align*} \mathcal{D}(\psi_j)(x_i)& = \sum\limits_{\substack{l = 1 \\ l\neq i}}^{n+1}\frac{1}{(x_i-x_l)}, \quad i = j, \\ \mathcal{D}(\psi_j)(x_i)& = \rho\prod\limits_{\substack{l = 1 \\ l\neq i, j}}^{n+1}(x_i-x_l), \quad i\neq j. \end{align*}$

□

The matrix $D$ can be considered instead of the derivative from bases in the numerical method. This matrix is used to reduce and simplify the calculations. It is worthwhile to mention that when we utilize the spectral methods, it is no longer a need to find the derivative of the bases. Instead, we can use the matrix $D$ .

Before introducing a matrix such as the operational matrix $I_{\eta}$ for the fractional integral operator $\mathcal{I}_0^{\eta}$ of order $\eta > 0$ , let us make some preliminaries about the fractional integral.

Definition 1. Given $\eta\in \mathbb{R}^+$ . Assuming the local integrable function $p:[0, 1]\to \mathbb{R}$ , the operator of the fractional integral $\mathcal{I}_0^{\eta}$ of order $\eta > 0$ is specified by

I_{0}^{η} (p) (x) := \frac{1}{Γ (η)} \int_{0}^{x} (x - z)^{η - 1} p (z) d z, x \in [0, 1], p \in L_{1} [0, 1] .

$\begin{equation} \mathcal{I}_0^{\eta}(p)(x): = \frac{1}{\Gamma(\eta)}\int_0^x(x-z)^{\eta-1}p(z)dz, \quad x\in[0, 1], \quad p\in L_1[0, 1]. \end{equation}$

(2.14)

With simple calculations, it can be shown that acting this operator on a power function is also a power function

I_{0}^{η} (x^{β}) = \frac{Γ (β + 1)}{Γ (β + η + 1)} x^{β + η} .

$\begin{equation} \mathcal{I}_0^{\eta}(x^{\beta}) = \frac{\Gamma(\beta+1)}{\Gamma(\beta+\eta+1)}x^{\beta+\eta}. \end{equation}$

(2.15)

Motivated by ^[30], the fractional integral can be bounded. To specify this bound, one can refer to the following Lemma.

Lemma 3. The operator $\mathcal{I}^{\eta}_0$ is bounded in $L_p([0, 1])$ , i.e.,

‖ I_{0}^{η} (p) ‖_{q} \leq \frac{1}{Γ (η + 1)} ‖ p ‖_{q}, 1 \leq q \leq \infty .

$\begin{equation} \|\mathcal{I}^{\eta}_0 (p) \|_{q}\leq \frac{1}{\Gamma(\eta+1)}\|p\|_{q}, \quad 1\leq q\leq \infty. \end{equation}$

(2.16)

Lemma 4. Given $\eta\in \mathbb{R}^+$ , one can approximate acting the fractional integral operator on $\Psi$ as follows

I_{0}^{η} (Ψ) (x) \approx I_{η} Ψ (x),

$\begin{equation} \mathcal{I}^{\eta}_0(\Psi)(x)\approx I_{\eta}\Psi(x), \end{equation}$

(2.17)

in which $I_{\eta}$ is a square matrix of order $(n+1)$ whose entries are obtained by

[I_{η}]_{j, i} = ρ \sum_{k = 0}^{n} z_{j, k} \frac{Γ (n - k + 1)}{Γ (n - k + η + 1)} x_{i}^{n - k + η} .

$\begin{equation} [I_{\eta}]_{j, i} = \rho\sum\limits_{k = 0}^{n}z_{j, k}\frac{\Gamma(n-k+1)}{\Gamma(n-k+\eta+1)}x^{n-k+\eta}_i. \end{equation}$

(2.18)

Proof. It is convenient to verify that

\prod_{\begin{matrix} k = 1 \\ k \neq i \end{matrix}}^{n + 1} (x - x_{k}) = \sum_{k = 0}^{n} z_{i, k} x^{n - k},

$\begin{equation} \prod\limits_{\substack{k = 1 \\ k\neq i}}^{n+1}(x-x_k) = \sum\limits_{k = 0}^{n}z_{i, k}x^{n-k}, \end{equation}$

(2.19)

where

z_{i, 0} = 1, z_{i, k} = \frac{1}{k} \sum_{l = 0}^{k} c_{i, l} z_{i, k - l}, k = 1, 2, \dots, n, i = 1, 2, \dots, n + 1,

$z_{i, 0} = 1, z_{i, k} = \frac{1}{k}\sum\limits_{l = 0}^kc_{i, l}z_{i, k-l}, \ k = 1, 2, \ldots, n, \ i = 1, 2, \ldots, n+1,$

and

c_{i, k} = \sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n + 1} x_{j}^{k}, k = 1, 2, \dots, n, i = 1, 2, \dots, n + 1.

$c_{i, k} = \sum\limits_{\substack{j = 1 \\ j\neq i}}^{n+1}x_j^k, \ k = 1, 2, \ldots, n, \ i = 1, 2, \ldots, n+1.$

Using the aforementioned changes, Chebyshev cardinal functions can be rewritten as

ψ_{j} (x) = ρ \sum_{k = 0}^{n} z_{j, k} x^{n - k} .

$\begin{equation} \psi_j(x) = \rho\sum\limits_{k = 0}^{n}z_{j, k}x^{n-k}. \end{equation}$

(2.20)

Putting (2.20) back into (2.14) and using Eq (2.15), one can write

\begin{aligned} I_{0}^{η} ψ_{j} (x) & = ρ I_{0}^{η} (\sum_{k = 0}^{n} z_{j, k} x^{n - k}) \\ = ρ \sum_{k = 0}^{n} z_{j, k} I_{0}^{η} (x^{η - k}) \\ = ρ \sum_{k = 0}^{n} z_{j, k} \frac{Γ (n - k + 1)}{Γ (n - k + η + 1)} x^{n - k + η} . \end{aligned}

$\begin{align*} \mathcal{I}_0^{\eta}\psi_j(x)& = \rho\mathcal{I}_0^{\eta}(\sum\limits_{k = 0}^{n}z_{j, k}x^{n-k})\\ & = \rho\sum\limits_{k = 0}^{n}z_{j, k}\mathcal{I}_0^{\eta}(x^{\eta-k})\\ & = \rho\sum\limits_{k = 0}^{n}z_{j, k}\frac{\Gamma(n-k+1)}{\Gamma(n-k+\eta+1)}x^{n-k+\eta}. \end{align*}$

This gives rise to reaching the desired result. □

2.1. Matrix representation of Caputo fractional derivative operator

Before introducing the operational matrix $D_{\eta}$ for the fractional derivative operator $^{c}\mathcal{D}_{0}^{\eta}$ , it is necessary to state some preliminaries about the CFD. Let $AC^{\eta}([0, 1])$ is a space of functions such that

A C^{η} [0, 1] = {p : [0, 1] \to C, & D^{(η - 1)} (p) \in A C [0, 1]} .

$AC^{\eta}[0, 1] = \{p:[0, 1]\rightarrow \mathbb{C}, \quad \& \quad \mathcal{D}^{(\eta-1)}(p)\in AC[0, 1]\}.$

Assuming $p(x)\in AC^{\eta}[0, 1]$ , the Caputo fractional derivative, characterized by

(^{c} D_{0}^{η} p) (x) = \frac{1}{Γ (κ - η)} \int_{0}^{x} \frac{p^{(κ)} (t) d t}{(x - t)^{η - κ + 1}} =: I_{0}^{κ - η} D^{κ} (p) (x),

$\begin{equation} (^{c}\mathcal{D}_{0}^{\eta}p)(x) = \dfrac{1}{\Gamma(\kappa-\eta)}\int_{0}^{x}\dfrac{p^{(\kappa)}(t)dt}{(x-t)^{\eta-\kappa+1}} = :\mathcal{I}_{0}^{\kappa-\eta}\mathcal{D}^{\kappa}(p)(x), \end{equation}$

(2.21)

exists for almost every $x\in [0, 1]$ . As a consequence of this definition, it is convenient to verify that ^[30]

(^{c} D_{0}^{η} (x)^{α - 1}) (x) = \frac{Γ (α)}{Γ (α - η)} x^{α - η}, (α > κ) .

$\begin{equation} (^{c}\mathcal{D}_{0}^{\eta}(x)^{\alpha-1})(x) = \dfrac{\Gamma(\alpha)}{\Gamma(\alpha-\eta)}x^{\alpha-\eta}, \quad (\alpha > \kappa). \end{equation}$

(2.22)

Here, our objective is to introduce a square matrix $D_{\eta}$ so that it satisfies

^{c} D_{0}^{η} (Ψ (x)) \approx D_{η} Ψ (x) .

$\begin{equation} {}^c\mathcal{D}_{0}^{\eta}(\Psi(x))\approx D_{\eta}\Psi(x). \end{equation}$

(2.23)

However, to gain the entries of the matrix $D_{\eta}$ , we avoid finding them directly in a manner that is proposed for fractional integral. Instead, motivated by (2.21), matrix $I_{\eta}$ can be used to obtain $D_{\eta}$ , viz

\begin{aligned} ^{c} D_{0}^{η} (Ψ (x)) = I_{0}^{κ - η} D^{κ} (Ψ (x)) & \approx I_{0}^{κ - η} (D^{κ} Ψ (x)) \\ = D^{κ} I_{0}^{κ - η} (Ψ (x)) \approx D^{κ} I_{κ - η} (Ψ (x)) . \end{aligned}

$\begin{align*} {}^c\mathcal{D}_{0}^{\eta}(\Psi(x)) = \mathcal{I}_{0}^{\kappa-\eta}\mathcal{D}^{\kappa}(\Psi(x))&\approx \mathcal{I}_{0}^{\kappa-\eta} (D^{\kappa}\Psi(x))\\ & = D^{\kappa}\mathcal{I}_{0}^{\kappa-\eta}(\Psi(x))\approx D^{\kappa}I_{\kappa-\eta}(\Psi(x)). \end{align*}$

Consequently, operational matrix $D_{\eta}$ obtain by

D_{η} := D^{κ} I_{κ - η} .

$\begin{equation} D_{\eta}: = D^{\kappa}I_{\kappa-\eta}. \end{equation}$

(2.24)

Therefore, to obtain the operational matrix for the fractional derivative $D_{\eta}$ , it is enough to obtain the operational matrix of the fractional integral of order $\kappa-\eta$ and multiply it by the $\kappa$ power of the operational matrix of the derivative.

3. Method description

We emphasize that our objective is to approximate the solution of the Telegraph equation (1.1) by implementing the pseudospectral method. To develop the pseudospectral method for solving (1.1), the process begins by considering the unknown solution as an expansion based on CCFs

w (x, t) \approx w_{n} (x, t) = \sum_{i = 1}^{n + 1} \sum_{j = 1}^{n + 1} W_{i, j} Ψ_{i} (x) Ψ_{j} (t) = Ψ^{T} (x) W Ψ (t),

$\begin{equation} w(x, t)\approx w_n(x, t) = \sum\limits_{i = 1}^{n+1}\sum\limits_{j = 1}^{n+1} W_{i, j}\Psi_i(x)\Psi_j(t) = \Psi^T(x)W\Psi(t), \end{equation}$

(3.1)

in which $W$ is a square matrix of order $n+1$ whose elements should be found. Here and throughout the text, the superscript "T" denotes transpose.

Putting (1.3) back into (1.1) gives rise to

\frac{\partial^{η} w_{n} (x, t)}{\partial t^{η}} + s_{1} \frac{\partial^{η - 1} w_{n} (x, t)}{\partial t^{η - 1}} + s_{2} w_{n} (x, t) = s_{3} \frac{\partial^{2} w_{n} (x, t)}{\partial x^{2}} + q (x, t) .

$\begin{equation} \frac{\partial^{\eta}w_n(x, t)}{\partial t^{\eta}}+s_1\frac{\partial^{\eta-1}w_n(x, t)}{\partial t^{\eta-1}}+s_2w_n(x, t) = s_3\frac{\partial^{2}w_n(x, t)}{\partial x^{2}}+q(x, t). \end{equation}$

(3.2)

Using operational matrices $D$ and $D_{\eta}$ , one can introduce the residual in the approximation as

r (x, t) = Ψ^{T} (x) (W D_{η} + s_{1} W D_{η - 1} + s_{2} W - s_{3} {D^{2}}^{T} W - Q) Ψ (t),

$\begin{equation} r(x, t) = \Psi^T(x)\left(WD_{\eta}+s_1W D_{\eta-1}+s_2W-s_3{D^2}^TW-Q\right)\Psi(t), \end{equation}$

(3.3)

where $Q$ is an $n+1$ -dimensional matrix and obtain by

Q_{i, j} = q (x_{i}, x_{j}) .

$Q_{i, j} = q(x_i, x_j).$

Let $WD_{\eta}+s_1W D_{\eta-1}+s_2W-s_3{D^2}^TW = UW$ (this is always applicable), then Eq (3.3) may be rewritten as

r (x, t) = Ψ^{T} (x) (U W - Q) Ψ (t) .

$\begin{equation} r(x, t) = \Psi^T(x)\left(UW-Q\right)\Psi(t). \end{equation}$

(3.4)

Pick distinct collocation points $\{x_j:j\in \Omega\}$ , the pseudospectral method requires

r (x_{i}, x_{j}) = 0, i, j \in Ω .

$\begin{equation} r(x_i, x_j) = 0, \quad i, j\in \Omega. \end{equation}$

(3.5)

This leads to specifying $U$ as the solution of the linear system

U W = Q .

$\begin{equation} UW = Q. \end{equation}$

(3.6)

To solve this system, we convert $U$ and $Q$ to $\tilde{U}$ and $\tilde{Q}$ , respectively. So we have a new system

A \tilde{U} = \tilde{Q} .

$\begin{equation} A\tilde{U} = \tilde{Q}. \end{equation}$

(3.7)

3.1. Convergence analysis

Considering $p$ as a polynomial that interpolates the given sufficiently smooth function $q$ at the points

q_{i, j} = q (x_{i}, t_{j}), i, j = 1, 2, \dots, n + 1,

$q_{i, j} = q(x_i, t_j), \quad i, j = 1, 2, \ldots, n+1,$

the reminder formula is obtained by ^[31]

| q (x, t) - p (x, t) | = \frac{\partial^{n}}{\partial x^{n}} q (ξ, t) \frac{Π_{i = 1}^{n} (x - x_{i})}{n!} + \frac{\partial^{n}}{\partial t^{n}} q (x, τ) \frac{Π_{j = 1}^{n} (t - t_{j})}{n!} - \frac{\partial^{2 n}}{\partial x^{n} t^{n}} q (ξ^{'}, τ^{'}) \frac{Π_{i = 1}^{n} (x - x_{i}) Π_{j = 1}^{n} (t - t_{j})}{n! n!}, τ, ξ, τ^{'}, ξ^{'} \in [0, 1] .

$\begin{equation} |q(x, t)-p(x, t)| = \frac{\partial^n}{\partial x^n}q(\xi, t)\frac{\Pi_{i = 1}^n(x-x_i)}{n!}+\frac{\partial^n}{\partial t^n}q(x, \tau)\frac{\Pi_{j = 1}^n(t-t_j)}{n!}\\ -\frac{\partial^{2n}}{\partial x^nt^n}q(\xi', \tau')\frac{\Pi_{i = 1}^n(x-x_i)\Pi_{j = 1}^n(t-t_j)}{n!n!}, \quad \tau, \xi, \tau', \xi' \in [0, 1]. \end{equation}$

(3.8)

Selecting the Chebyshev polynomials zeros as the interpolation nodes, (3.7) can be written as follows

| q (x, t) - p (x, t) | \leq {(\frac{1}{2})}^{n} \frac{1}{2^{n - 1} n!} sup_{ξ \in [0, 1)} | \frac{\partial^{n}}{\partial x^{n}} q (ξ, t) | + {(\frac{1}{2})}^{n} \frac{1}{2^{n - 1} n!} sup_{τ \in [0, 1)} | \frac{\partial^{n}}{\partial t^{n}} q (x, τ) | + {(\frac{1}{2})}^{2 n} \frac{1}{4^{n - 1} (n!)^{2}} sup_{ξ^{'}, τ^{'} \in [0, 1)} | \frac{\partial^{2 n}}{\partial x^{n} \partial t^{n}} q (ξ^{'}, τ^{'}) | \leq M_{q} {(\frac{1}{2})}^{m} \frac{1}{2^{m - 1} m!} (2 + {(\frac{1}{2})}^{m} \frac{1}{2^{m - 1} m!}),

$\begin{equation} |q(x, t)-p(x, t)| \leq \left(\frac{1}{2}\right)^n\frac{1}{2^{n-1}n!}\sup\limits_{\xi\in [0, 1)}|\frac{\partial^n}{\partial x^n}q(\xi, t)|+\left(\frac{1}{2}\right)^n\frac{1}{2^{n-1}n!}\sup\limits_{\tau\in [0, 1)}|\frac{\partial^n}{\partial t^n}q(x, \tau)|\\ +\left(\frac{1}{2}\right)^{2n}\frac{1}{4^{n-1}(n!)^2}\sup\limits_{\xi', \tau'\in [0, 1)}|\frac{\partial^{2n}}{\partial x^n\partial t^n}q(\xi', \tau')|\\ \leq M_{q}\left(\frac{1}{2}\right)^m\frac{1}{2^{m-1}m!}\left(2+\left(\frac{1}{2}\right)^m\frac{1}{2^{m-1}m!}\right), \end{equation}$

(3.9)

in which

M_{q} = max {sup_{ξ \in [0, 1)} | \frac{\partial^{n}}{\partial x^{n}} q (ξ, t) |, sup_{τ \in [0, 1)} | \frac{\partial^{n}}{\partial t^{n}} q (x, τ) |, sup_{ξ^{'}, τ^{'} \in [0, 1)} | \frac{\partial^{2 n}}{\partial x^{n} \partial t^{n}} q (ξ^{'}, τ^{'}) |} .

$M_{q} = \max\left\{\sup\limits_{\xi\in [0, 1)}|\frac{\partial^n}{\partial x^n}q(\xi, t)|, \sup\limits_{\tau\in [0, 1)}|\frac{\partial^n}{\partial t^n}q(x, \tau)|, \sup\limits_{\xi', \tau'\in [0, 1)}|\frac{\partial^{2n}}{\partial x^n\partial t^n}q(\xi', \tau')|\right\}.$

Given $e = w-w_n$ , subtracting (3.2) from

\frac{\partial^{η} w_{n} (x, t)}{\partial t^{η}} + s_{1} \frac{\partial^{η - 1} w_{n} (x, t)}{\partial t^{η - 1}} + s_{2} w_{n} (x, t) = s_{3} \frac{\partial^{2} w_{n} (x, t)}{\partial x^{2}} + q_{n} (x, t),

$\frac{\partial^{\eta}w_n(x, t)}{\partial t^{\eta}}+s_1\frac{\partial^{\eta-1}w_n(x, t)}{\partial t^{\eta-1}}+s_2w_n(x, t) = s_3\frac{\partial^{2}w_n(x, t)}{\partial x^{2}}+q_n(x, t),$

the global error satisfies

\frac{\partial^{η} e (x, t)}{\partial t^{η}} + s_{1} \frac{\partial^{η - 1} e (x, t)}{\partial t^{η - 1}} + s_{2} e (x, t) = s_{3} \frac{\partial^{2} e (x, t)}{\partial x^{2}} + q (x, t) - q_{n} (x, t) .

$\begin{equation} \frac{\partial^{\eta}e(x, t)}{\partial t^{\eta}}+s_1\frac{\partial^{\eta-1}e(x, t)}{\partial t^{\eta-1}}+s_2e(x, t) = s_3\frac{\partial^{2}e(x, t)}{\partial x^{2}}+q(x, t)-q_n(x, t). \end{equation}$

(3.10)

Let the residual corresponding to (3.10) is

R_{n} (x, t) = \frac{\partial^{η} e (x, t)}{\partial t^{η}} + s_{1} \frac{\partial^{η - 1} e (x, t)}{\partial t^{η - 1}} + s_{2} e (x, t) - s_{3} \frac{\partial^{2} e (x, t)}{\partial x^{2}} - q (x, t) + q_{n} (x, t) .

$\begin{equation} R_n(x, t) = \frac{\partial^{\eta}e(x, t)}{\partial t^{\eta}}+s_1\frac{\partial^{\eta-1}e(x, t)}{\partial t^{\eta-1}}+s_2e(x, t)-s_3\frac{\partial^{2}e(x, t)}{\partial x^{2}}-q(x, t)+q_n(x, t). \end{equation}$

(3.11)

Motivated by Theorem 2.2 ^[30], we have

\begin{aligned} | \frac{\partial^{η}}{\partial t^{η}} e (x, t) | & = | I_{0}^{κ - η} \frac{\partial^{κ}}{\partial t^{κ}} e (x, t) | \leq \frac{1}{Γ (κ - η) (κ - η + 1)} ‖ \frac{\partial^{κ}}{\partial t^{κ}} e (x, t) ‖_{C}, \\ | \frac{\partial^{η - 1}}{\partial t^{η - 1}} e (x, t) | & = | I_{0}^{κ - η} \frac{\partial^{κ - 1}}{\partial t^{κ - 1}} e (x, t) | \leq \frac{1}{Γ (κ - η) (κ - η + 1)} ‖ \frac{\partial^{κ - 1}}{\partial t^{κ - 1}} e (x, t) ‖_{C} . \end{aligned}

$\begin{align*} |\frac{\partial^{\eta}}{\partial t^{\eta}}e(x, t)|& = |\mathcal{I}_0^{\kappa-\eta}\frac{\partial^{\kappa}}{\partial t^{\kappa}}e(x, t)|\leq \frac{1}{\Gamma(\kappa-\eta)(\kappa-\eta+1)}\|\frac{\partial^{\kappa}}{\partial t^{\kappa}}e(x, t)\|_C, \\ |\frac{\partial^{\eta-1}}{\partial t^{\eta-1}}e(x, t)|& = |\mathcal{I}_0^{\kappa-\eta}\frac{\partial^{\kappa-1}}{\partial t^{\kappa-1}}e(x, t)|\leq \frac{1}{\Gamma(\kappa-\eta)(\kappa-\eta+1)}\|\frac{\partial^{\kappa-1}}{\partial t^{\kappa-1}}e(x, t)\|_C. \end{align*}$

Substituting these equations into (3.11) and using triangle inequality, we get

\begin{aligned} | R_{n} (x, t) | & \leq | \frac{\partial^{η} e (x, t)}{\partial t^{η}} | + | s_{1} \frac{\partial^{η - 1} e (x, t)}{\partial t^{η - 1}} | + | s_{2} e (x, t) | + | s_{3} \frac{\partial^{2} e (x, t)}{\partial x^{2}} | + | q (x, t) - q_{n} (x, t) | \\ \leq \frac{1}{Γ (κ - η) (κ - η + 1)} (‖ \frac{\partial^{κ}}{\partial t^{κ}} e (x, t) ‖_{C} + | s_{1} | ‖ \frac{\partial^{κ - 1}}{\partial t^{κ - 1}} e (x, t) ‖_{C}) \\ + | s_{2} | ‖ e (x, t) ‖_{C} + | s_{3} | ‖ \frac{\partial^{2} e (x, t)}{\partial x^{2}} ‖_{C} + | q (x, t) - q_{n} (x, t) | . \end{aligned}

$\begin{align*} |R_n(x, t)|&\leq |\frac{\partial^{\eta}e(x, t)}{\partial t^{\eta}}|+|s_1\frac{\partial^{\eta-1}e(x, t)}{\partial t^{\eta-1}}|+|s_2e(x, t)|+|s_3\frac{\partial^{2}e(x, t)}{\partial x^{2}}|+|q(x, t)-q_n(x, t)|\\ &\leq \frac{1}{\Gamma(\kappa-\eta)(\kappa-\eta+1)}\left(\|\frac{\partial^{\kappa}}{\partial t^{\kappa}}e(x, t)\|_C+|s_1|\|\frac{\partial^{\kappa-1}}{\partial t^{\kappa-1}}e(x, t)\|_C\right)\\ &+|s_2|\|e(x, t)\|_C+|s_3|\|\frac{\partial^{2}e(x, t)}{\partial x^{2}}\|_C+|q(x, t)-q_n(x, t)|. \end{align*}$

To proceed, using (3.9), it can be found

| R_{n} (x, t) | \leq {(\frac{1}{2})}^{m} \frac{1}{2^{m - 1} m!} (2 + {(\frac{1}{2})}^{m} \frac{1}{2^{m - 1} m!}) (\frac{M_{\frac{\partial^{κ} w}{\partial t^{κ}}} + | s_{1} | M_{\frac{\partial^{κ - 1} w}{\partial t^{κ - 1}}}}{Γ (κ - η) (κ - η + 1)} + | s_{2} | M_{w} + | s_{3} | M_{\frac{\partial^{2} w}{\partial x^{2}}} + M_{q}) .

$\begin{equation} |R_n(x, t)| \leq \left(\frac{1}{2}\right)^m\frac{1}{2^{m-1}m!}\left(2+\left(\frac{1}{2}\right)^m\frac{1}{2^{m-1}m!}\right) \left(\frac{M_{\frac{\partial^{\kappa}w}{\partial t^{\kappa}}}+|s_1|M_{\frac{\partial^{\kappa-1}w}{\partial t^{\kappa-1}}}}{\Gamma(\kappa-\eta)(\kappa-\eta+1)}\right.\\ \left.+|s_2|M_{w}+|s_3|M_{\frac{\partial^2w}{\partial x^2}}+M_q \right). \end{equation}$

(3.12)

Putting $C_y = \left(\frac{M_{\frac{\partial^{\kappa}w}{\partial t^{\kappa}}}+|s_1|M_{\frac{\partial^{\kappa-1}w}{\partial t^{\kappa-1}}}}{\Gamma(\kappa-\eta)(\kappa-\eta+1)}+|s_2|M_{w}+|s_3|M_{\frac{\partial^2w}{\partial x^2}}+M_q \right)$ back into (3.12), we have

| R_{n} (x, t) | \leq C_{y} {(\frac{1}{2})}^{m} \frac{1}{2^{m - 1} m!} (2 + {(\frac{1}{2})}^{m} \frac{1}{2^{m - 1} m!}) .

$\begin{equation} |R_n(x, t)|\leq C_y \left(\frac{1}{2}\right)^m\frac{1}{2^{m-1}m!}\left(2+\left(\frac{1}{2}\right)^m\frac{1}{2^{m-1}m!}\right). \end{equation}$

(3.13)

Thus $|R_n(x, t)|\rightarrow 0$ as $m\to \infty$ .

4. Illustrative examples

To demonstrate the performance of the proposed method, some examples are provided in this section. To illustrate the results and make a global view of the present method and its efficiency, sometimes, the absolute errors

e = | w (x_{i}, t_{j}) - w_{n} (x_{i}, t_{j}) |, i . j = 1, \dots, n,

$e = |w(x_i, t_j)-w_n(x_i, t_j)|, \quad i.j = 1, \ldots, n,$

and $L_2$ error

L_{2} - e r r o r = {(\int_{0}^{1} \int_{0}^{1} | w (x, t) - w_{n} (x, t) |^{2} d x d t)}^{1 / 2},

$L_2-error = \left(\int_0^1\int_0^1|w(x, t)-w_n(x, t)|^2dxdt\right)^{1/2},$

are reported in tables or plotted in figures.

Example 1. We dedicate the first example to the fractional Telegraph equation as

\frac{\partial^{η} w (x, t)}{\partial t^{η}} + \frac{\partial^{η - 1} w (x, t)}{\partial t^{η - 1}} + w (x, t) = π \frac{\partial^{2} w (x, t)}{\partial x^{2}} + q (x, t), 1 < η \leq 2,

$\frac{\partial^{\eta}w(x, t)}{\partial t^{\eta}}+\frac{\partial^{\eta-1}w(x, t)}{\partial t^{\eta-1}}+w(x, t) = \pi\frac{\partial^{2}w(x, t)}{\partial x^{2}}+q(x, t), \ 1 < \eta\leq 2,$

with initial and boundary conditions

w (0, t) = 0, w (1, t) = t^{3} \sin^{2} (1), w (x, 0) = 0, w^{'} (x, 0) = 0,

$w(0, t) = 0, \ w(1, t) = t^3\sin^2(1), \ w(x, 0) = 0, \ w'(x, 0) = 0,$

in which

q (x, t) = \frac{6 t^{3 - η} (\sin^{2} (x))}{Γ (4 - η)} + \frac{6 t^{4 - η} (\sin^{2} (x))}{Γ (5 - η)} + t^{3} (\sin^{2} (x)) - π (2 t^{3} (\cos^{2} (x)) - 2 t^{3} (\sin^{2} (x))) .

$q(x, t) = \frac{6 t^{3-\eta} \left(\sin^{2}\left(x \right)\right)}{\Gamma \! \left(4-\eta \right)}+\frac{6 t^{4-\eta} \left(\sin^{2}\left(x \right)\right)}{\Gamma \! \left(5-\eta \right)}+t^{3} \left(\sin^{2}\left(x \right)\right)-\pi \left(2 t^{3} \left(\cos^{2}\left(x \right)\right)-2 t^{3} \left(\sin^{2}\left(x \right)\right)\right).$

The exact solution for this equation is given by $w(x, t) = t^3\sin^2(x)$ ^[20].

To show the algorithm of the proposed method to solve this example, we describe it step by step here.

(1) Chose n;

(2) construct the Chebyshev cardinal functions of order $n$ (refer to (2.3));

(3) compute the Matrices $D$ , $I_{\eta}$ and $D_{\eta}$ (refer to Lemma 2, Lemma 4 and (2.24), respectively);

(4) approximate $w(x, t)$ using $w_n(x, t)$ (refer to (1.3));

(5) put $w_n(x, t)$ back into (1.1) (refer to (3.2));

(6) compute the residual $r(x, t)$ (refer to (3.4));

(7) obtain the linear system (3.6) using the shifted Chebyshev nodes $x_j = \frac{(y_j+1)}{2}$ ( $j = 1, \ldots, n$ );

(8) solve the linear system (3.7).

For instance, the coefficients' matrix in the obtained linear system for this example, taking $\eta = 1.75$ and $n = 3$ , is equal to

[\begin{array}{ccccccccc} 0.08931 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 1.2440 & 0.0 & 0.0 \\ 0.0 & 0.08931 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 1.2440 & 0.0 \\ 0.0 & 0.0 & 0.08931 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 1.2440 \\ 0.0 & 0.0 & 0.0 & 0.08931 & - 0.33333 & 1.2440 & 0.0 & 0.0 & 0.0 \\ 0.0 & - 5.3324 & 0.0 & 5.5237 & 2.7602 & 3.3810 & 0.0 & - 5.3324 & 0.0 \\ 0.0 & 0.0 & 0.0 & - 1.5121 & 5.3333 & - 3.8215 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.08931 & - 0.33333 & 1.2440 \\ 0.0 & 1.2440 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 0.089316 & 0.0 \\ 0.0 & 0.0 & 1.2440 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 0.089316 \end{array}] .

$\left[\begin{array}{ccccccccc} 0.08931 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 1.2440 & 0.0 & 0.0 \\ 0.0 & 0.08931 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 1.2440 & 0.0 \\ 0.0 & 0.0 & 0.08931 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 1.2440 \\ 0.0 & 0.0 & 0.0 & 0.08931 & - 0.33333 & 1.2440 & 0.0 & 0.0 & 0.0 \\ 0.0 & - 5.3324 & 0.0 & 5.5237 & 2.7602 & 3.3810 & 0.0 & - 5.3324 & 0.0 \\ 0.0 & 0.0 & 0.0 & - 1.5121 & 5.3333 & - 3.8215 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.08931 & - 0.33333 & 1.2440 \\ 0.0 & 1.2440 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 0.089316 & 0.0 \\ 0.0 & 0.0 & 1.2440 & 0.0 & 0.0 & - 0.33333 & 0.0 & 0.0 & 0.089316 \end{array}\right].$

Recall that the CFD of a function $w$ tends to integer derivative as $\eta \rightarrow \kappa$ . To demonstrate this effect, our results illustrated in , obviously, demonstrate it. illustrates the effect of parameter $n$ on $L_2$ error and confirm the convergence analysis. As you see, when $n$ increases, the error decreases exponentially. demonstrates the approximate solution and corresponding absolute errors. Absolute error is reported for different values of $x$ and $t$ , taking $n = 9$ and $\eta = 1.75$ , respectively, . It is worthwhile to mention that the process took a total of $77.390$ seconds of CPU time.

Figure 1. Approximate solution, taking different

η

$\eta$ for Example 1.

DownLoad: Full-Size Img PowerPoint

Figure 2. The

L_{2}

$L_2$ -error obtained by different number of

n

$n$ for Example 1.

DownLoad: Full-Size Img PowerPoint

Figure 3. The plots of approximate and corresponding absolute error, taking

n = 9

$n = 9$ and

η = 1.75

$\eta = 1.75$ , for Example 1.

DownLoad: Full-Size Img PowerPoint

Table 1. Absolute errors, taking

n = 9

$n = 9$ and

η = 1.75

$\eta = 1.75$ for Example 1.

x	t=0.2	t=0.4	t=0.6	t=0.8	t=1
$0.1$	$2.409e-09$	$1.711e-09$	$1.647e-09$	$1.903e-09$	$6.011e-10$
$0.3$	$7.426e-09$	$3.795e-09$	$2.722e-09$	$4.259e-09$	$8.402e-09$
$0.5$	$1.107e-08$	$4.569e-09$	$1.723e-09$	$1.055e-09$	$6.911e-10$
$0.7$	$1.033e-08$	$4.387e-09$	$1.637e-10$	$3.377e-09$	$1.876e-08$
$0.9$	$4.454e-09$	$1.497e-09$	$1.348e-09$	$4.402e-10$	$1.384e-08$

| Show Table

DownLoad: CSV

Example 2. We devote this example to the fractional Telegraph equation

\frac{\partial^{η} w (x, t)}{\partial t^{η}} + \frac{\partial^{η - 1} w (x, t)}{\partial t^{η - 1}} + w (x, t) = \frac{\partial^{2} w (x, t)}{\partial x^{2}} + (t^{2} - 2 t + 2) (x - x^{2}) e^{- t} + 2 t^{2} e^{- t}

$\frac{\partial^{\eta}w(x, t)}{\partial t^{\eta}}+\frac{\partial^{\eta-1}w(x, t)}{\partial t^{\eta-1}}+w(x, t) = \frac{\partial^{2}w(x, t)}{\partial x^{2}}+\left(t^{2}-2 t +2\right) \left(x-x^{2} \right) {\mathrm e}^{-t}+2 t^{2} {\mathrm e}^{-t}$

with boundary and initial conditions

w (x, 0) = 0, w^{'} (x, 0) = 0, w (0, t) = 0, w (1, t) = 0.

$w(x, 0) = 0, \ w'(x, 0) = 0, \ w(0, t) = 0, \ w(1, t) = 0.$

For this example, the exact solution is given by $w(x, t) = (x-x^2)e^{-t}t^2$ ^[4,21,22].

The approximate solution at $t = 0.1$ , taking different values of $\eta$ , is plotted in Figure 4. In Table 2, we observe a comparison between the present method and the method proposed in ^[22]. To demonstrate the accuracy and ability of the present method, the maximum absolute error at different times is tabulated in Table 3. The approximate solution and corresponding absolute errors are illustrated in . For more evidence of accuracy, is reported to show the $L_2$ errors with different choices of $n$ for this example.

Figure 4. Approximate solution at

t = 0.1

$t = 0.1$ , taking different

η

$\eta$ for Example 2.

DownLoad: Full-Size Img PowerPoint

Table 2. The comparison between hybrid functions method and the proposed method for Example 2.

	Proposed method		^[22]
$t$	$n=6$	$n=10$	$n=6$	$n=10$
$0.6$	$5.21\times 10^{-5}$	$3.67\times 10^{-10}$	$4.43\times 10^{-3}$	$2.19\times 10^{-3}$
$0.7$	$4.39\times 10^{-5}$	$2.88\times 10^{-10}$	$2.67\times 10^{-3}$	$1.45\times 10^{-3}$
$0.8$	$2.92\times 10^{-5}$	$1.76\times 10^{-10}$	$2.14\times 10^{-3}$	$1.62\times 10^{-3}$
CPU time	$0.156$	$0.563$	$-$	$-$

| Show Table

DownLoad: CSV

Table 3. The

L_{2}

$L_2$ errors at different times with different choices of

n

$n$ , taking

η = 2

$\eta = 2$ for Example 2.

t	n=3	n=5	n=7	n=9	n=11
$0.1$	$9.67e-04$	$9.10e-05$	$9.10e-07$	$2.70e-09$	$5.18e-13$
$0.3$	$6.00e-03$	$3.77e-04$	$2.48e-06$	$6.47e-09$	$8.23e-13$
$0.5$	$1.06e-02$	$4.20e-04$	$2.84e-06$	$7.29e-09$	$3.27e-12$
$0.7$	$1.08e-02$	$2.97e-04$	$2.04e-06$	$5.23e-09$	$8.10e-12$
$0.9$	$4.60e-03$	$1.24e-04$	$6.41e-07$	$1.32e-09$	$1.74e-11$
CPU time	$0.062$	$0.141$	$0.187$	$0.344$	$0.672$

| Show Table

DownLoad: CSV

Figure 5. The plots of approximate and corresponding error, taking

n = 8

$n = 8$ and

η = 2

$\eta = 2$ , for Example 2.

DownLoad: Full-Size Img PowerPoint

Figure 6. Effect of

n

$n$ on

L_{2}

$L_2$ error, taking

n = 8

$n = 8$ and

η = 2

$\eta = 2$ , for Example 2.

DownLoad: Full-Size Img PowerPoint

Example 3. We devote this example to the fractional Telegraph equation

\frac{\partial^{η} w (x, t)}{\partial t^{η}} + \frac{\partial^{η - 1} w (x, t)}{\partial t^{η - 1}} + w (x, t) = \frac{\partial^{2} w (x, t)}{\partial x^{2}} + Γ (α + 1) \sin (x) + \frac{Γ (α + 1) t^{α - β} \sin (x)}{Γ (α + 1 - β)} + 2 t^{α} \sin (x)

$\frac{\partial^{\eta}w(x, t)}{\partial t^{\eta}}+\frac{\partial^{\eta-1}w(x, t)}{\partial t^{\eta-1}}+w(x, t) = \frac{\partial^{2}w(x, t)}{\partial x^{2}}+\Gamma \! \left(\alpha +1\right) \sin \! \left(x \right)+\frac{\Gamma \! \left(\alpha +1\right) t^{\alpha -\beta} \sin \! \left(x \right)}{\Gamma \! \left(\alpha +1-\beta \right)}+2 t^{\alpha} \sin \! \left(x \right)$

with boundary and initial conditions

w (x, 0) = 0, w^{'} (x, 0) = 0, w (0, t) = 0, w (1, t) = \sin (1) t^{η} .

$w(x, 0) = 0, \ w'(x, 0) = 0, \ w(0, t) = 0, \ w(1, t) = \sin(1)t^{\eta}.$

For this example, the exact solution is given by $w(x, t) = \sin(x)t^{\eta}$ .

shows the $L_2$ -error at different times and different choices of $\eta$ . In this table, the CPU time is also reported. It is clear that when the parameter n increases, the error decreases. The approximate solution and corresponding absolute errors are illustrated in Figure 7.

Table 4.

L_{2}

$L_2$ -error at different times for Example 3.

	$\eta=1.75$		$\eta=1.90$
$t$	$n=7$	$n=10$	$n=7$	$n=10$
$0.1$	$1.32\times 10^{-3}$	$7.14\times 10^{-4}$	$4.60\times 10^{-4}$	$2.94\times 10^{-4}$
$0.3$	$2.02\times 10^{-3}$	$9.89\times 10^{-4}$	$9.38\times 10^{-4}$	$2.53\times 10^{-4}$
$0.5$	$1.44\times 10^{-3}$	$6.75\times 10^{-4}$	$8.78\times 10^{-4}$	$4.35\times 10^{-4}$
$0.7$	$4.04\times 10^{-4}$	$1.77\times 10^{-4}$	$5.17\times 10^{-4}$	$2.08\times 10^{-4}$
$0.9$	$4.21\times 10^{-4}$	$2.27\times 10^{-4}$	$5.10\times 10^{-4}$	$3.21\times 10^{-4}$
CPU time	$2.624$	$5.215$	$2.765$	$6.002$

| Show Table

DownLoad: CSV

Figure 7. The plots of approximate and corresponding error, taking

n = 10

$n = 10$ and

η = 1.75

$\eta = 1.75$ , for Example 3.

DownLoad: Full-Size Img PowerPoint

5. Conclusions

The pseudospectral method based on CCFs can be solved by the fractional Telegraph equation accurately. The presented method is easy to implement, and it solves problems of this type effectively and with appropriate accuracy. The convergence analysis also proves the method is convergent, and numerical examples confirm this investigation. Due to the cardinality property of the bases used, there is no need for integration to find the coefficients in the expansions, and this reduces the computational time and computational cost. For future work, we can use the pseudospectral method directly or apply the finite difference method and collocation method to solve the sophisticated models and the generalization of the method to two and three dimensions ^[32].

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This project was supported by the Researchers Supporting Project number (RSP2023R210), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	M. Lakestani, B. N. Saray, Numerical solution of Telegraph equation using interpolating scaling functions, Comput. Math. Appl., 60 (2010) 1964–1972. https://doi.org/10.1016/j.camwa.2010.07.030
[2]	X. Yang, H. Zhang, J. Tang, The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions, Comput. Math. Appl., 82 (2021), 1–12.
[3]	M. Dehghan, M. Lakestani, The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation, Numer. Meth. Part. D. E., 25 (2009), 931–938. https://doi.org/10.1002/num.20382 doi: 10.1002/num.20382
[4]	M. Dehghan, A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Math. Part. D. E., 24 (2008), 1080–1093. https://doi.org/10.1002/num.20306 doi: 10.1002/num.20306
[5]	O. Nikan, Z. Avazzadeh, J. A. T. Machado, M. N. Rasoulizadeh, An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals, Eng. Comput., 39 (2023), 2327–2344. https://doi.org/10.1007/s00366-022-01630-9 doi: 10.1007/s00366-022-01630-9
[6]	S. Sharifi, J. Rashidinia, Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method, Appl. Math. Comput., 281 (2016), 28–38. https://doi.org/10.1016/j.amc.2016.01.049 doi: 10.1016/j.amc.2016.01.049
[7]	R. M. Hafez, Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method, Comput. Appl. Math., 37 (2018), 5253–5273. https://doi.org/10.1007/s40314-018-0635-1 doi: 10.1007/s40314-018-0635-1
[8]	H. B. Jebreen, Y. C. Cano, I. Dassios, An efficient algorithm based on the multi-wavelet Galerkin method for telegraph equation, AIMS Math., 6 (2020), 1296–1308. https://doi.org/10.3934/math.2021080 doi: 10.3934/math.2021080
[9]	W. M. Abd-Elhameed, E. H. Doha, Y. H. Youssri, M. A. Bassuony, New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations, Numer. Meth. Part. D. E., 32 (2016), 1553–1571. https://doi.org/10.1002/num.22074 doi: 10.1002/num.22074
[10]	A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method, Numer. Meth. Part. D. E., 26 (2010), 239–252. https://doi.org/10.1002/num.20442 doi: 10.1002/num.20442
[11]	H. B. Jebreen, I. Dassios, A biorthogonal hermite cubic spline Galerkin method for solving fractional riccati equation, Mathematics, 10 (2022), 1461. https://doi.org/10.3390/math10091461 doi: 10.3390/math10091461
[12]	H. B. Jebreen, C. Cattani, Interpolating scaling functions tau method for solving space-time fractional partial differential equations, Symmetry, 14 (2022), 2463. https://doi.org/10.3390/sym14112463 doi: 10.3390/sym14112463
[13]	M. Asadzadeh, B. N. Saray, On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem, BIT, 62 (2022), 383–1416. https://doi.org/10.1007/s10543-022-00915-1 doi: 10.1007/s10543-022-00915-1
[14]	M. H. Heydari, M. Razzaghi, Highly accurate solutions for space-time fractional Schrödinger equations with non-smooth continuous solution using the hybrid clique functions, Math. Sci., 17 (2023), 31–42. https://doi.org/10.1007/s40096-021-00437-x doi: 10.1007/s40096-021-00437-x
[15]	X. Jiang, J. Wang, W. Wang, H. Zhang, A predictor-corrector compact difference scheme for a nonlinear fractional differential equation, Fractal Fract., 7 (2023), 521. https://doi.org/10.3390/fractalfract7070521 doi: 10.3390/fractalfract7070521
[16]	X. Yang, Q. Zhang, G. Yuan, Z. sheng, On positivity preservation in nonlinear finite volume method for multi-term fractional subdiffusion equation on polygonal meshes, Nonlinear Dynam., 92 (2018), 595–612.
[17]	H. Zhang, X. Yang, D. Xu, An efficient spline collocation method for a nonlinear fourth-order reaction subdiffusion equation, J. Sci. Comput., 85 (2020). https://doi.org/10.1007/s10915-020-01308-8
[18]	A. Iqbal, T. Akram, A numerical study of anomalous electro-diffusion cells in cable sense with a non-singular kernel, Demonstr. Math., 55 (2022), 574–586. https://doi.org/10.1515/dema-2022-0155 doi: 10.1515/dema-2022-0155
[19]	T. Akram, M. Abbas, A. Ali, A. Iqbal, D. Baleanu, A numerical approach of a time fractional reaction-diffusion model with a non-singular kernel, Symmetry, 12 (2020), 1653. https://doi.org/10.3390/sym12101653 doi: 10.3390/sym12101653
[20]	V. R. Hosseini, W. Chen, Z. Avazzadeh, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. Bound. Elem., 38 (2014), 31–39. https://doi.org/10.1016/j.enganabound.2013.10.009 doi: 10.1016/j.enganabound.2013.10.009
[21]	Y. H. Youssri, W. M. Abd-Elhameed, Numerical spectral Legendre-Galerkin algorithm for solving time fractional telegraph equation, Rom. J. Phys., 63 (2018), 1–16.
[22]	N. Mollahasani, M. M. Mohseni, K. Afrooz, A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Appl. Math. Model., 40 (2016), 2804–2814. https://doi.org/10.1016/j.apm.2015.08.020 doi: 10.1016/j.apm.2015.08.020
[23]	A. Saadatmandi, M. Mohabbati, Numerical solution of fractional telegraph equation via the tau method, Math. Rep., 17 (2015), 155–166.
[24]	J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364–1377. https://doi.org/10.1016/j.jmaa.2007.06.023 doi: 10.1016/j.jmaa.2007.06.023
[25]	W. Jiang, Y. Lin, Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci., 16 (2011), 3639–3645. https://doi.org/10.1016/j.cnsns.2010.12.019 doi: 10.1016/j.cnsns.2010.12.019
[26]	A. Ali, T. Abdeljawad, A. Iqbal, T. Akram, M. Abbas, On unconditionally stable new modified fractional group iterative scheme for the solution of 2D time-fractional telegraph model, Symmetry, 13 (2021), 2078. https://doi.org/10.3390/sym13112078 doi: 10.3390/sym13112078
[27]	M. Shahriari, B. N. Saray, B. Mohammadalipour, S. Saeidian, Pseudospectral method for solving the fractional one-dimensional Dirac operator using Chebyshev cardinal functions, Phys. Scripta., 98 (2023), 055205. https://doi.org/10.1088/1402-4896/acc7d3 doi: 10.1088/1402-4896/acc7d3
[28]	A. Afarideh, F. D. Saei, M. Lakestani, B. N. Saray, Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions, Phys. Scripta, 96 (2021), 125267. https://doi.org/10.1088/1402-4896/ac3c59 doi: 10.1088/1402-4896/ac3c59
[29]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods fundamentals in single domains, Berlin: Springer-Verlag, 2006.
[30]	A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[31]	G. Dahlquist, A. Björck, Numerical methods, Englewood Cliffs: Prentice Hall, 1974.
[32]	H. Zhang, Y. Liu, X. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651–674.

This article has been cited by:

Jiakun Zhang, Fengying Zhou, Naiwei Mao, Numerical optimization algorithm for solving time-fractional telegraph equations, 2025, 100, 0031-8949, 045237, 10.1088/1402-4896/adbf6f

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1253) PDF downloads(51) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(7) / Tables(4)

AIMS Mathematics

An approach based on the pseudospectral method for fractional telegraph equations

Related Papers:

Abstract

1. Introduction

2. Chebyshev cardinal functions (CCFs)

2.1. Matrix representation of Caputo fractional derivative operator

3. Method description

3.1. Convergence analysis

4. Illustrative examples

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Chebyshev cardinal functions (CCFs)

2.1. Matrix representation of Caputo fractional derivative operator

3. Method description

3.1. Convergence analysis

4. Illustrative examples

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

AIMS Mathematics

An approach based on the pseudospectral method for fractional telegraph equations

Related Papers:

Abstract

1. Introduction

2. Chebyshev cardinal functions (CCFs)

2.1. Matrix representation of Caputo fractional derivative operator

3. Method description

3.1. Convergence analysis

4. Illustrative examples

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Chebyshev cardinal functions (CCFs)

2.1. Matrix representation of Caputo fractional derivative operator

3. Method description

3.1. Convergence analysis

4. Illustrative examples

5. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References