A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations

Abstract

This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms.

1. Introduction

Partial differential equations (PDEs) are widely used in physics and engineering, with applications including heat transfer, wave propagation, electromagnetism, and fluid dynamics. They are also applied in economics to calculate price alternatives and in biology to simulate population dynamics and disease propagation. In medicine, PDEs play an important role in imaging. Furthermore, they are used in geosciences to predict weather and simulate groundwater movement and in material science to study chemical reactions and diffusion. Their broad applications make them critical for comprehending and forecasting complex, dynamic systems in theoretical and practical settings. For some PDE applications, one can consult [1,2]. Many numerical approaches were used to solve different types of PDEs. For example, in [3], the authors solved PDEs using deep learning and physical constraints. In [4], the authors presented some numerical algorithms for handling some non-linear PDEs. In [5], a variational quantum algorithm was followed for handling PDEs. Some approximations for certain stochastic parabolic PDEs were presented in [6]. In [7], a boundary element method was followed to solve one-dimensional nonlinear parabolic PDEs. In [8], the authors proposed a proper orthogonal decomposition method for handling certain PDEs. In [9], a numerical approach for a heat transfer model was proposed. In [10], the authors used a tau approach using Lucas polynomials for the time-fractional diffusion equation.

Spectral methods are essential in numerical analysis due to their advantages over other numerical approaches (see, for example, [11,12]). Approximate solutions derived from these methods are frequently represented as polynomial combinations, indicating their close relationship with special functions, especially orthogonal polynomials. There are three crucial spectral methods: collocation, tau, and the Galerkin method. All methods have their benefits and characteristics. The choice of a suitable spectral method is based on the type of differential equation studied and the underlying conditions (see, for example, [13,14,15,16]). For other contributions concerning spectral methods, one can refer to [17,18,19].

Matrix representations of differentiation operations on a collection of basis functions are known as operational matrices of derivatives (OMDs) and are often employed in numerical approaches to solve various DEs. These matrices are effective tools to handle these DEs. The transformation of ordinary DEs into an algebraic system is possible by expressing the derivative operator as an operational matrix. It is useful in the collocation method ([20,21]). Furthermore, when dealing with FDEs, the operational matrices of fractional derivatives will be the backbone for designing algorithms for these FDEs. For instance, the authors of [22] used shifted Legendre–Laguerre operational matrices to treat certain time-delay fractional diffusion equations. In [23], the author derived generalized Jacobi Galerkin OMDs and used them to handle specific FDEs. Other Legendre matrices were used in [24]. Operational matrices based on hat functions were used in [25] to treat some integral DEs. Galerkin OMDs were used in [26] to treat some multi-dimensional FDEs. In [27], OMDs of some finite classes of orthogonal polynomials were employed to handle FDEs. Other contributions regarding the use of different OMDs can be found in [28,29,30].

In both theoretical and applied mathematics, Chebyshev polynomials (CPs) are widely used; see [31,32]. Approximation theory and numerical analysis are two areas where CPs are important. Due to their orthogonality, these polynomials provide other desirable characteristics, including minimization of error and oscillation (see, for example, [33,34]). CPs are related to the classical Jacobi polynomials. Four kinds of CPs are special ones. All of these kinds have their characteristics and applications. The first and second kinds are among the most used polynomials in diverse applications (see, for example, [35,36,37]). The third- and fourth-kinds of CPs, called airfoil polynomials, were also used in many contributions. The authors of [38,39] used CPs of the third- and fourth-kinds to treat some differential equations (DEs). The kinds of CPs are not restricted to these four kinds. However, some of the CPs can be considered as particular kinds of generalized ultraspherical polynomials [40,41]. Other kinds of CPs were introduced and used in several contributions. Various types of DEs were addressed using CPs of the fifth and sixth kinds (see, for example, [42,43,44]).

In addition to the standard CPs, other modifications and generalizations of these polynomials have been studied in theoretical and practical contexts. In [45], generalized CPs were introduced. In [46], the authors used generalized shifted CPs to solve variable-order fractional PDEs. In [47], the authors used generalized CPs to find analytical solutions for coupled mode equations for multi-waveguide systems. The authors of [48] utilized certain generalized CPs to solve some specific FDEs. The authors of [49] used the Chebyshev polynomial derivative to treat some high-order DEs. Certain shifted generalized CPs were employed in [50] to deal with the multi-dimensional sinh-Gordon equation. The authors of [48] introduced orthogonal generalized CPs of the first kind and established some new theoretical formulas for these polynomials.

The main goal of the present study is to provide a matrix-based methodology for solving second-order linear PDEs using these polynomials. We can summarize the objectives in the following items:

• Establish the OMDs of the generalized CPs.

• Design a matrix algorithm for handling second-order PDEs with constant coefficients.

• Design a matrix algorithm for handling the telegraph equation.

• Investigate convergence analysis.

• Provide specific examples to demonstrate the practicality and accuracy of the used technique.

The contents of this article are as follows: Section 2 presents an overview of the shifted generalized Chebyshev polynomials (SGCPs) and some of their basic characteristics. Moreover, in this section, we derive the OMDs of these polynomials, which will be essential in proposing our algorithms. Section 3 analyzes a matrix approach for treating second-order PDEs with constant coefficients. Section 4 analyzes another matrix algorithm for treating the telegraph equation. The error analysis for the proposed shifted generalized Chebyshev expansion is studied in Section 5. Some illustrative examples are shown in Section 6. Section 7 reports some concluding remarks.

2. An Overview on the SGCPs

This section presents an overview of the SGCPs and some of their important formulae.

2.1. Some Fundamental Formulas of the SGCPs

The SGCPs denoted by $G_n^{\left(\sigma \right)}\left(z\right)$ are orthogonal polynomials on $[0,\mathcal{L}]$ whose orthogonality relation is ([48]):

(1)${\int }_0^{\mathcal{L}}{G}_n^{\left(\sigma \right)}\left(z\right)G_m^{\left(\sigma \right)}\left(z\right)w\left(z\right)dz=h_n^{\sigma }{\delta }_{n,m},$

with

$h_n^{\sigma }={\displaystyle \frac{n!{(\sigma +2)}^{2n}\mathrm{\Gamma }\left(n+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}+n\right)}{{\left(n\right)}_n\left(2n\right)!}},$

and ${\delta }_{n,m}$ is the known delta Kronecker, and the weight function $w\left(x\right)$ is given as follows:

$w\left(z\right)=z^{{\displaystyle \frac{1}{\sigma +2}}-1}{(\mathcal{L}-z)}^{-{\displaystyle \frac{1}{\sigma +2}}}.$

The series representation of $G_i^{\left(\sigma \right)}\left(z\right)$ is

(2)$G_i^{\left(\sigma \right)}\left(z\right)=\sum _{m=0}^i{A}_{m,i}^{\sigma }{z}^m,$

where

(3)$A_{m,i}^{\sigma }={\displaystyle \frac{{(\sigma +2)}^i{(-1)}^{i-m}{\mathcal{L}}^{-m}(i+m-1)!\mathrm{\Gamma }\left(i+\frac{1}{\sigma +2}\right)\left(\genfrac{}{}{0pt}{}{i}{i-m}\right)}{(2i-1)!\mathrm{\Gamma }\left(m+\frac{1}{\sigma +2}\right)}},$

and its inversion formula is given by the following:

(4)$x^i={\mathcal{L}}^i\sum _{m=0}^i{\displaystyle \frac{{(2+\sigma )}^{-i+m}\left(\genfrac{}{}{0pt}{}{i}{m}\right){\left(\frac{1}{2+\sigma }+i-m\right)}_m}{(2i-m){(1+2i-2m)}_{m-1}}}{G}_{i-m}^{\left(\sigma \right)}\left(z\right).$

2.2. The OMDs of the SGCPs

This section aims to derive the OMDs of the SGCPs. To achieve this objective, we state and prove the following theorem:

Theorem 1.

For $j\ge 1,$ one has the following:

(5) $D{G}_j^{\sigma }\left(x\right)={\displaystyle \frac{j!}{\mathcal{L}}}\sum _{s=0}^{j-1}{\displaystyle \frac{{(2+\sigma )}^{-s+j}{(s+j)}_{s-j+1}}{s!(j-s-1)!{\left(-1+\frac{1}{2+\sigma }+j\right)}_{s-j+1}}}{}_{3}F_2\left.\left(\begin{array}{c}{\displaystyle \frac{1+\sigma }{2+\sigma }}-j,-m,2-2j+m\\ 1-2j,2-{\displaystyle \frac{1}{2+\sigma }}-j\end{array}\right|1\right)G_s^{\sigma }\left(x\right).$

Proof. 

The series representation of $G_j^{\sigma }\left(x\right)$ enables one to write the following:

(6)$D{G}_j^{\sigma }\left(x\right)={\displaystyle \frac{{(2+\sigma )}^{j-1}(1+\sigma -(2+\sigma )j)}{{\mathcal{L}}^{j-r}}}\sum _{r=0}^{j-1}{\displaystyle \frac{(j-r)\left(\genfrac{}{}{0pt}{}{j}{r}\right){\left(2-\frac{1}{2+\sigma }-j\right)}_{-1+r}}{{(2j-r)}_r}}{x}^{j-r-1}.$

Formula (4) converts (6) into the following:

(7)$\begin{array}{cc}\hfill D{G}_j^{\sigma }\left(x\right)=& {\displaystyle \frac{{(2+\sigma )}^{j-1}(1+\sigma -(2+\sigma )j)}{\mathcal{L}}}\sum _{r=0}^{j-1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{j}{r}\right)(j-r)!\mathrm{\Gamma }\left(-1+\frac{1}{2+\sigma }+j-r\right){\left(2-\frac{1}{2+\sigma }-j\right)}_{r-1}}{{(2j-r)}_r}}\times \hfill \\ \hfill & \sum _{t=0}^{j-r-1}{\displaystyle \frac{{(2+\sigma )}^{1-j+r+t}}{(j-r-t-1)!t!\mathrm{\Gamma }\left(-1+\frac{1}{2+\sigma }+j-r-t\right){(1+2(-1+j-r)-2t)}_t}}{G}_{j-r-t-1}^{\sigma }\left(x\right).\hfill \end{array}$

After some computations, (7) can be written as follows:

(8)$\begin{array}{cc}\hfill D{G}_j^{\sigma }\left(x\right)=& {\displaystyle \frac{(1-\sigma (-1+j)-2j)}{\mathcal{L}}}\sum _{m=0}^{j-1}{\displaystyle \frac{{(2+\sigma )}^m}{(j-m-1)!\mathrm{\Gamma }\left(-1+\frac{1}{2+\sigma }+j-m\right)}}\times \hfill \\ \hfill & \sum _{r=0}^m{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{j}{r}\right)(j-r)!\mathrm{\Gamma }\left(-1+\frac{1}{2+\sigma }+j-r\right){\left(2-\frac{1}{2+\sigma }-j\right)}_{-1+r}}{(m-r)!{(-1+2j-2m)}_{m-r}{(2j-r)}_r}}{G}_{j-m-1}^{\sigma }\left(x\right).\hfill \end{array}$

The last formula is equivalent to the following:

(9)$\begin{array}{cc}\hfill D{G}_j^{\sigma }\left(x\right)=& {\displaystyle \frac{j!\mathrm{\Gamma }\left(-1+\frac{1}{2+\sigma }+j\right)}{\sqrt{\pi }\mathcal{L}}}\sum _{m=0}^{j-1}{\displaystyle \frac{2^{2(-1+j-m)}{(2+\sigma )}^{1+m}\mathrm{\Gamma }\left(-\frac{1}{2}+j-m\right)}{\mathrm{\Gamma }\left(-1+\frac{1}{2+\sigma }+j-m\right)m!(2j-m-2)!}}\times \hfill \\ \hfill & {}_{3}F_2\left.\left(\begin{array}{c}{\displaystyle \frac{1+\sigma }{2+\sigma }}-j,-m,2-2j+m\\ 1-2j,2-{\displaystyle \frac{1}{2+\sigma }}-j\end{array}\right|1\right)G_{j-m-1}^{\sigma }\left(x\right),\hfill \end{array}$

and, thus, the following formula may be obtained:

(10)$\begin{array}{cc}\hfill D{G}_j^{\sigma }\left(x\right)=& {\displaystyle \frac{j!}{\mathcal{L}}}\sum _{s=0}^{j-1}{\displaystyle \frac{{(2+\sigma )}^{-s+j}{(s+j)}_{s-j+1}}{s!(j-s-1)!{\left(-1+\frac{1}{2+\sigma }+j\right)}_{s-j+1}}}\times \hfill \\ \hfill & {}_{3}F_2\left.\left(\begin{array}{c}{\displaystyle \frac{1+\sigma }{2+\sigma }}-j,1-s-j,s-j+1\\ 1-2j,2-{\displaystyle \frac{1}{2+\sigma }}-j\end{array}\right|1\right)G_s^{\sigma }\left(x\right).\hfill \end{array}$

This proves Theorem 1. □

Now, the OMDs of the SGCPs can be constructed.

Consider the vector $\mathit{G}\left(z\right)$ with the following components:

(11)$\mathit{G}\left(z\right)={[G_0^{\sigma }\left(z\right),G_1^{\sigma }\left(z\right),\dots ,G_N^{\sigma }\left(z\right)]}^T.$

Using Formula (5), we can write the following:

(12)${\displaystyle \frac{d\mathit{G}\left(z\right)}{dz}}=\Delta \mathit{G}\left(z\right).$

The matrix $\Delta$ is called the OMD of the SGCPs, and its elements can be expressed in the following form:

(13)$\begin{array}{c}\hfill {\Delta }_{j,s}=\left\{\begin{array}{cc}{m}_{s,j},\hfill & \mathrm{i}\mathrm{f} s>j,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$

with

$m_{s,j}={\displaystyle \frac{{(2+\sigma )}^{-s+j}j!{(s+j)}_{s-j+1}}{\mathcal{L}s!(j-s-1)!{\left(-1+\frac{1}{2+\sigma }+j\right)}_{s-j+1}}}{}_{3}F_2\left.\left(\begin{array}{c}{\displaystyle \frac{\sigma +1}{\sigma +2}}-j,1-s-j,s-j+1\\ 1-2j,2-{\displaystyle \frac{1}{2+\sigma }}-j\end{array}\right|1\right).$

For example, the matrix $\Delta$ for $N=5$ and $\sigma =3$ has the following form:

$\mathcal{M}={\displaystyle \frac{1}{\mathcal{L}}}\left(\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 5& 0& 0& 0& 0& 0\\ -10& 10& 0& 0& 0& 0\\ {\displaystyle \frac{57}{2}}& -6& 15& 0& 0& 0\\ -{\displaystyle \frac{284}{7}}& {\displaystyle \frac{230}{7}}& -{\displaystyle \frac{36}{7}}& 20& 0& 0\\ {\displaystyle \frac{250}{3}}& -{\displaystyle \frac{70}{3}}& 40& -{\displaystyle \frac{100}{21}}& 25& 0\end{array}\right).$

Remark 1.

The n-th derivative of the SGCPs can be expressed as follows:

(14) ${\displaystyle \frac{d^n\mathit{G}\left(z\right)}{d{x}^n}}={\left({\Delta }^{\left(1\right)}\right)}^n\mathit{G}\left(z\right),$

where $n\in N$. In addition, the superscript in ${\Delta }^{\left(1\right)}$ represents the matrix power. Thus,

(15) ${\Delta }^{\left(n\right)}={\left({\Delta }^{\left(1\right)}\right)}^n,n=1,2,\dots .$

2.3. Function Approximation by the SGCPs

Consider a function $u\left(z\right)$ defined on $[0,\mathcal{L}]$, and it can be approximated using SGCPs as follows:

(16)$u\left(z\right)=\sum _{j=0}^N{c}_j{G}_j^{\left(\sigma \right)}\left(z\right)={\mathbf{C}}^T\mathrm{\Psi }\left(z\right).$

Then, the coefficients $c_j$ have the following form:

(17)$c_j={\displaystyle \frac{1}{h_j^{\sigma }}}{\int }_0^{\mathcal{L}}w\left(z\right)u\left(z\right)G_j^{\left(\sigma \right)}\left(z\right)dz,j=0,1,\cdots ,$

and

(18)$\begin{array}{cc}\hfill {\mathbf{C}}^T=& [c_0,c_1,\cdots ,c_N],\hfill \end{array}$

(19) $\begin{array}{cc}\hfill \mathrm{\Psi }\left(z\right)=& {[G_0^{\left(\sigma \right)}\left(z\right),G_1^{\left(\sigma \right)}\left(z\right),\cdots ,G_N^{\left(\sigma \right)}\left(z\right)]}^T.\hfill \end{array}$

3. Treatment of the Second-Order PDEs with Constant Coefficients

This section aims to derive a shifted generalized Chebyshev spectral collocation method to solve the following second-order PDEs of the following form ([51]):

(20)$\begin{array}{cc}\hfill \alpha {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}+\beta {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial \tau \partial z}}& +\gamma {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}}+\delta {\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial z}}+\eta {\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial \tau }}+\theta \mathcal{U}(z,\tau )=\mathcal{F}(z,\tau ),\hfill \\ & 0\le z\le \mathcal{L},0<\tau \le \mathcal{T},\hfill \end{array}$

subject to the following time initial conditions:

$\begin{array}{cc}& \mathcal{U}(z,0)={\mathcal{G}}_1\left(z\right),{\mathcal{U}}_{\tau }(z,0)={\mathcal{G}}_2\left(z\right),0\le z\le \mathcal{L},\hfill \end{array}$

together with the following spatial initial conditions:

$\begin{array}{c}\hfill \mathcal{U}(0,\tau )={\mathcal{H}}_1\left(\tau \right),{\mathcal{U}}_z(0,\tau )={\mathcal{H}}_2\left(\tau \right),0<\tau \le \mathcal{T}.\end{array}$

where $\alpha ,\beta ,\gamma ,\delta ,\eta$, and $\theta$ are contacts, while $\mathcal{F}(\xi ,\tau ),{\mathcal{G}}_1\left(z\right),{\mathcal{G}}_2\left(z\right),{\mathcal{H}}_1\left(\tau \right)$, and ${\mathcal{H}}_2\left(\tau \right)$ are the given functions. To proceed with our proposed numerical algorithm, we approximate the following functions: $\mathcal{U}(z,\tau ),{\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial \tau }},{\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial z}},{\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}},{\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial \tau \partial z}}$ and ${\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}$ as expansions in the SGCPs, as follows:

(21)$\begin{array}{cc}\hfill \mathcal{U}(z,\tau )\approx \tilde{\mathcal{U}}(z,\tau )& =\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)={\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right),\hfill \end{array}$

(22)$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial \tau }}& =\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right){\displaystyle \frac{\partial G_j^{\left(\sigma \right)}\left(\tau \right)}{\partial \tau }}=\Delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right),\hfill \end{array}$

(23)$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial z}}& =\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{\displaystyle \frac{\partial G_j^{\left(\sigma \right)}\left(z\right)}{\partial z}}{G}_i^{\left(\sigma \right)}\left(\tau \right)={\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}\Delta {\mathrm{\Phi }}_M\left(z\right),\hfill \end{array}$

(24)$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial {\tau }^2}}& =\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{G}_j^{\left(\sigma \right)}\left(z\right){\displaystyle \frac{{\partial }^2{G}_i^{\left(\sigma \right)}\left(\tau \right)}{\partial {\tau }^2}}={\Delta }^{\left(2\right)}{\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right),\hfill \end{array}$

(25)$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial \tau \partial z}}& =\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{\displaystyle \frac{\partial G_i^{\left(\sigma \right)}\left(z\right)}{\partial z}}{\displaystyle \frac{\partial G_i^{\left(\sigma \right)}\left(\tau \right)}{\partial \tau }}=\Delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}\Delta {\mathrm{\Phi }}_M\left(z\right),\hfill \end{array}$

(26)$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial z^2}}& =\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{\displaystyle \frac{{\partial }^2{G}_i^{\left(\sigma \right)}\left(z\right)}{\partial z^2}}{G}_j^{\left(\sigma \right)}\left(\tau \right)={\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\Delta }^{\left(2\right)}{\mathrm{\Phi }}_M\left(z\right),\hfill \end{array}$

where the matrix $\mathbf{U}$ is given by the following:

$\mathbf{U}=\left(\begin{array}{cccc}{u}_{00}& u_{01}& \cdots & u_{0N}\\ u_{10}& u_{11}& \cdots & u_{1N}\\ ⋮& ⋮& \ddots & ⋮\\ u_{M0}& u_{M1}& \cdots & u_{MN}\end{array}\right).$

Now, if we use the series of Equations (21)–(26), then we can write the following approximations:

(27)$\begin{array}{cc}\hfill \alpha {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\Delta }^{\left(2\right)}{\mathrm{\Phi }}_M\left(z\right)& +\beta \Delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}\Delta {\mathrm{\Phi }}_M\left(z\right)+\gamma {\Delta }^{\left(2\right)}{\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)+\delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}\Delta {\mathrm{\Phi }}_M\left(z\right)\hfill \\ & +\eta \Delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)+\theta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)\approx \mathcal{F}(z,\tau ),\hfill \end{array}$

(28)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)\approx {\mathcal{G}}_1\left(z\right),\end{array}$

(29)$\begin{array}{c}\hfill {{\mathrm{\Psi }}^{\prime }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)\approx {\mathcal{G}}_1\left(z\right),\end{array}$

(30)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(0\right)\approx {\mathcal{H}}_1\left(\tau \right),\end{array}$

(31)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{{\mathrm{\Phi }}^{\prime }}_M\left(0\right)\approx {\mathcal{H}}_2\left(\tau \right),\end{array}$

With the collocation procedure, we can now solve Equations (27)–(31). Suppose ${\tau }_{\mathcal{T},j}^{\left(\sigma \right)},j=0,1,\cdots ,N$ are the zeros of $G_{N+1}^{\left(\sigma \right)}\left(\tau \right)$, while $z_{\mathcal{L},i}^{\left(\sigma \right)},i=0,1,\cdots ,M$ are the zeros of $G_{M+1}^{\left(\sigma \right)}\left(z\right)$. We substitute these nodes into (27)–(31). Accordingly, we have the following:

(32)$\begin{array}{cc}\hfill & \alpha {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\Delta }^{\left(2\right)}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)+\beta \Delta {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}\Delta {\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)+\gamma {\Delta }^{\left(2\right)}{\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)\hfill \\ & +\delta {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}\Delta {\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)+\eta \Delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)\hfill \\ & +\theta {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)=\mathcal{F}(z_{\mathcal{L},i}^{\left(\sigma \right)},{\tau }_{\mathcal{T},j}^{\left(\sigma \right)}),1\le i\le M-1,1\le j\le N-1,\hfill \end{array}$

(33)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)={\mathcal{G}}_1\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right),0\le i\le M,\end{array}$

(34)$\begin{array}{c}\hfill {{\mathrm{\Psi }}^{\prime }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)={\mathcal{G}}_1\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right),0\le i\le M,\end{array}$

(35)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(0\right)={\mathcal{H}}_1\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right),1\le j\le N-1,\end{array}$

(36)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{{\mathrm{\Phi }}^{\prime }}_M\left(0\right)={\mathcal{H}}_2\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right),1\le j\le N-1.\end{array}$

In this way, we obtain an algebraic system of $(N+1)\times (M+1)$ equations that may be solved by any suitable iteration algorithm, such as Newton’s method. As a result, we can evaluate the approximate solution $\tilde{\mathcal{U}}(z,\tau )$.

4. Numerical Treatment of Telegraph Equation

In this section, we utilize a numerical method for treating the following telegraph equation [52]:

(37)$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}}+2\zeta (z,\tau ){\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial \tau }}+{\eta }^2(z,\tau )\mathcal{U}(z,\tau )=\theta (z,\tau ){\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}+\mathcal{F}(z,\tau ),0<z<\mathcal{L},0<\tau \le \mathcal{T}.\end{array}$

governed by the following conditions:

(38)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{U}(z,0)=& {\mathcal{G}}_1\left(z\right),{\mathcal{U}}_{\tau }(z,0)={\mathcal{G}}_2\left(z\right),z\in (0,\mathcal{L}),\hfill \end{array}\end{array}$

(39)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{U}(0,\tau )=& {\mathcal{H}}_1\left(\tau \right),\mathcal{U}(\mathcal{L},\tau )={\mathcal{H}}_2\left(\tau \right),\tau \in (0,\mathcal{T}].\hfill \end{array}\end{array}$

For $\zeta >0,\eta =0$ Equation (37) represents a damped wave equation, and if $\zeta >\eta >0$, it is the telegraph equation. By substituting (21), (22), (24), and (26) in (37), we obtain the following approximate formulas:

(40)$\begin{array}{cc}\hfill {\Delta }^{\left(2\right)}{\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)& +2\zeta (z,\tau )\Delta {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)+{\eta }^2(z,\tau ){\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)\hfill \\ & \approx \theta (z,\tau ){\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\Delta }^{\left(2\right)}{\mathrm{\Phi }}_M\left(z\right)+\mathcal{F}(z,\tau ),0<z<\mathcal{L},0<\tau \le \mathcal{T}.\hfill \end{array}$

Moreover, the conditions in (38) and (39) imply the following equations:

(41)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)\approx {\mathcal{G}}_1\left(z\right),\end{array}$

(42)$\begin{array}{c}\hfill {{\mathrm{\Psi }}^{\prime }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z\right)\approx {\mathcal{G}}_2\left(z\right),\end{array}$

(43)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(0\right)\approx {\mathcal{H}}_1\left(\tau \right),\end{array}$

(44)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(\tau \right)\mathbf{U}{\mathrm{\Phi }}_M\left(\mathcal{L}\right)\approx {\mathcal{H}}_2\left(\tau \right),\end{array}$

Similarly, as in the previous section, We insert the collocation nodes in (40)–(44). Accordingly, we can write the following:

(45)$\begin{array}{cc}\hfill & {\Delta }^{\left(2\right)}{\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)+2\zeta (z_{\mathcal{L},i}^{\left(\sigma \right)},{\tau }_{\mathcal{T},j}^{\left(\sigma \right)})\Delta {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)+{\eta }^2(z_{\mathcal{L},i}^{\left(\sigma \right)},{\tau }_{\mathcal{T},j}^{\left(\sigma \right)}){\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)\hfill \\ \hfill & =\theta (z_{\mathcal{L},i}^{\left(\sigma \right)},{\tau }_{\mathcal{T},j}^{\left(\sigma \right)}){\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\Delta }^{\left(2\right)}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)+\mathcal{F}(z_{\mathcal{L},i}^{\left(\sigma \right)},{\tau }_{\mathcal{T},j}^{\left(\sigma \right)})1\le i\le M-1,1\le j\le N-1,\hfill \end{array}$

(46)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)={\mathcal{G}}_1\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right),0\le i\le M,\end{array}$

(47)$\begin{array}{c}\hfill {{\mathrm{\Psi }}^{\prime }}_N\left(0\right)\mathbf{U}{\mathrm{\Phi }}_M\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right)={\mathcal{G}}_1\left(z_{\mathcal{L},i}^{\left(\sigma \right)}\right),0\le i\le M,\end{array}$

(48)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(0\right)={\mathcal{H}}_1\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right),1\le j\le N-1,\end{array}$

(49)$\begin{array}{c}\hfill {\mathrm{\Psi }}_N\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right)\mathbf{U}{\mathrm{\Phi }}_M\left(\mathcal{L}\right)={\mathcal{H}}_2\left({\tau }_{\mathcal{T},j}^{\left(\sigma \right)}\right),1\le j\le N-1.\end{array}$

An iterative approach developed by Newton may be used to solve the $(N+1)\times (M+1)$ algebraic equations obtained. This allows us to compute $\tilde{\mathcal{U}}(z,\tau )$ given in (21).

5. The Error Bound

In this section, the convergence of our spectral collocation method is examined in both the first- and two-dimensional generalized Chebyshev-weighted Sobolev spaces.

Consider the following generalized Chebyshev-weighted Sobolev space:

(50)${\mathbf{H}}_{w\left(z\right)}^m\left(I\right)=\{u:D_z^{k}u\in L_{w\left(z\right)}^2\left(I\right),0\le k\le m\},$

where $I=(0,\mathcal{L})$, equipped with the following inner product, norm, and semi-norm:

(51)$\begin{array}{cc}& {(u,v)}_{{\mathbf{H}}_{w\left(z\right)}^m}=\sum _{k=0}^m{(D_z^{k}u,D_z^{k}v)}_{L_{w\left(z\right)}^2},\hfill \\ & {\left|\right|u\left|\right|}_{{\mathbf{H}}_{w\left(z\right)}^m}^2={(u,u)}_{{\mathbf{H}}_{w\left(z\right)}^m}{,\left|u\right|}_{{\mathbf{H}}_{w\left(z\right)}^m}=\left|\right|D_z^{m}u{\left|\right|}_{L_{w\left(z\right)}^2},\hfill \end{array}$

where $m\in N.$ Also, assume the following two-dimensional Gegenbauer-weighted Sobolev space:

(52)${\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}\left(\mathrm{\Omega }\right)=\{u:{\displaystyle \frac{{\partial }^{p+q}u}{\partial z^p\partial {\tau }^q}}\in L_{\overline{\omega }(z,\tau )}^2\left(\mathrm{\Omega }\right),r\ge p\ge 0,s\ge q\ge 0\},$

equipped with the following norm and semi-norm:

(53)${\left|\right|u\left|\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}}={\left(\sum _{p=0}^r\sum _{q=0}^s{\left|\left|{\displaystyle \frac{{\partial }^{p+q}u}{\partial z^p\partial {\tau }^q}}\right|\right|}_{L_{\overline{\omega }(z,\tau )}^2}^2\right)}^{{\displaystyle \frac{1}{2}}},{\left|u\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}}={\left|\left|{\displaystyle \frac{{\partial }^{r+s}u}{\partial z^r\partial {\tau }^s}}\right|\right|}_{L_{\overline{\omega }(z,\tau )}^2},$

where $\mathrm{\Omega }=(0,\mathcal{L})\times (0,\mathcal{T})$, $\overline{\omega }(z,\tau )=w\left(z\right)w\left(\tau \right)$ and $r,s\in N.$

Lemma 1

([53]). For $n\ge 1,$ $n+r>1$ and $n+s>1,$ where $r,s,$ are constants, we have the following:

(54)${\displaystyle \frac{\mathrm{\Gamma }(n+r)}{\mathrm{\Gamma }(n+s)}}\le {\mathbf{o}}_n^{r,s}{n}^{r-s},$

where

(55)${\mathbf{o}}_n^{r,s}=exp\left({\displaystyle \frac{r-s}{2(n+s-1)}}+{\displaystyle \frac{1}{12(n+r-1)}}+{\displaystyle \frac{{(r-s)}^2}{n}}\right).$

Remark 2.

${\mathbf{o}}_n^{r,s}$ can be expressed as follows for the fixed $r,s$:

${\mathbf{o}}_n^{r,s}=1+O\left(n^{-1}\right).$

Theorem 2.

Suppose ${\displaystyle \tilde{\mathcal{U}}\left(z\right)=\sum _{i=0}^M{u}_i{G}_i^{\left(\sigma \right)}\left(z\right),}$ is the approximate solution of $\mathcal{U}\left(z\right)\in {\mathbf{H}}_{w\left(z\right)}^m\left(I\right).$ Then, for $0\le k\le m\le M+1,$ we obtain the following:

(56) $\left|\right|D_z^k(\mathcal{U}\left(z\right)-\tilde{\mathcal{U}}\left(z\right)){\left|\right|}_{L_{w\left(z\right)}^2}\lesssim {\mathcal{L}}^{m-k}{M}^{{\displaystyle \frac{-1}{2}}(m-k)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}\left|\right|D_z^{m}u\left(z\right){\left|\right|}_{L_{w\left(z\right)}^2},$

where $\mathcal{A}\lesssim \mathcal{B}$ refers to the presence of a constant ν with the following: $\mathcal{A}\le \nu \mathcal{B}.$

Proof. 

The two expansions of $\mathcal{U}\left(z\right)$ and $\tilde{\mathcal{U}}\left(z\right)$ allow us to write the following:

(57)$D_z^k(\mathcal{U}\left(z\right)-\tilde{\mathcal{U}}\left(z\right))=\sum _{n=M+1}^{\infty }{u}_n{D}_z^k{G}_n^{\left(\sigma \right)}\left(z\right)=\sum _{n=M+1}^{\infty }{u}_n\sum _{r=k}^n{\displaystyle \frac{A_{r,n}^{\sigma }\mathrm{\Gamma }(r+1)}{\mathrm{\Gamma }(r-k+1)}}{z}^{r-k}.$

Taking ${\left|\right|.\left|\right|}_{L_{w\left(z\right)}^2}^2$, for each side of the previous equation, we obtain the following:

(58)$\begin{array}{cc}\hfill \left|\right|D_z^k(\mathcal{U}\left(z\right)-\tilde{\mathcal{U}}\left(z\right)){\left|\right|}_{L_{w\left(z\right)}^2}^2=& \sum _{n=M+1}^{\infty }{\left|u_n\right|}^2\sum _{r=k}^n{\displaystyle \frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2k}\mathrm{\Gamma }\left(2(r-k)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-k+1)\mathrm{\Gamma }(1-2k+2r)}}.\hfill \end{array}$

Also, we can write the following:

(59)$\begin{array}{cc}\hfill \left|\right|D_z^m\mathcal{U}\left(z\right){\left|\right|}_{L_{w\left(z\right)}^2}^2=& \sum _{n=m}^{\infty }{\left|u_n\right|}^2\sum _{r=m}^n{\displaystyle \frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2m}\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}.\hfill \end{array}$

Now, Equation (58) can be rewritten as follows:

(60)$\begin{array}{cc}\hfill \left|\right|D_z^k(\mathcal{U}\left(z\right)-\tilde{\mathcal{U}}\left(z\right)){\left|\right|}_{L_{w\left(z\right)}^2}^2=& \sum _{n=M+1}^{\infty }{\left|u_n\right|}^2{\displaystyle \frac{{\sum }_{r=k}^n\frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2k}\mathrm{\Gamma }\left(2(r-k)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-k+1)\mathrm{\Gamma }(1-2k+2r)}}{{\sum }_{r=m}^n\frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2m}\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}}\hfill \\ & \times \sum _{r=m}^n{\displaystyle \frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2m}\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}.\hfill \end{array}$

In virtue of the Stirling formula, it is possible to obtain the following inequalities:

(61)$\begin{array}{cc}& {\displaystyle \frac{{\mathrm{\Gamma }}^2(r+1)\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}\lesssim r^{2m}{\left(2(r-m)\right)}^{-\left({\displaystyle \frac{\sigma +1}{\sigma +2}}\right)},\hfill \\ & {\displaystyle \frac{{\mathrm{\Gamma }}^2(r+1)\mathrm{\Gamma }\left(2(r-k)+\frac{1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-k+1)\mathrm{\Gamma }(1-2k+2r)}}\lesssim r^{2k}{\left(2(r-k)\right)}^{-\left({\displaystyle \frac{\sigma +1}{\sigma +2}}\right)}.\hfill \\ \end{array}$

If we take ${\lambda }_1^{*}={max}_{k\le r\le n}\left\{{\left(A_{r,n}^{\sigma }\right)}^2\right\},$ and ${\lambda }_2^{*}={max}_{m\le r\le n}\left\{{\left(A_{r,n}^{\sigma }\right)}^2\right\}$, then we obtain the following:

(62)$\begin{array}{cc}\hfill {\displaystyle \frac{{\sum }_{r=k}^n\frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2k}\mathrm{\Gamma }\left(2(r-k)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-k+1)\mathrm{\Gamma }(1-2k+2r)}}{{\sum }_{r=m}^n\frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2m}\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}}& \lesssim {\displaystyle \frac{n^{2k}{\mathcal{L}}^{2n-2k}{\left(2(n-k)\right)}^{-\left(\frac{\sigma +1}{\sigma +2}\right)}(n-k+1){\lambda }_1^{*}}{n^{2m}{\mathcal{L}}^{2n-2m}{\left(2(n-m)\right)}^{-\left(\frac{\sigma +1}{\sigma +2}\right)}(n-m+1){\lambda }_2^{*}}}\hfill \\ & \lesssim {\displaystyle \frac{n^{2k}{\mathcal{L}}^{2n-2k}{\left(2(n-k)\right)}^{-\left(\frac{\sigma +1}{\sigma +2}\right)}\mathrm{\Gamma }(n-k+2)\mathrm{\Gamma }(n-m+1)}{n^{2m}{\mathcal{L}}^{2n-2m}{\left(2(n-m)\right)}^{-\left(\frac{\sigma +1}{\sigma +2}\right)}\mathrm{\Gamma }(n-m+2)\mathrm{\Gamma }(n-k+1)}}\hfill \\ & \lesssim n^{2(k-m)}{\mathcal{L}}^{2(m-k)}{\left({\displaystyle \frac{n-k}{n-m}}\right)}^{{\displaystyle \frac{1}{\sigma +2}}}.\hfill \end{array}$

Again, using the Stirling formula, that is,

(63)${\left({\displaystyle \frac{n-k}{n-m}}\right)}^{{\displaystyle \frac{1}{\sigma +2}}}\lesssim n^{(m-k)\left({\displaystyle \frac{1}{\sigma +2}}\right)},$

therefore, (62) gives the following:

(64)$\begin{array}{cc}\hfill {\displaystyle \frac{{\sum }_{r=k}^n\frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2k}\mathrm{\Gamma }\left(2(r-k)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-k+1)\mathrm{\Gamma }(1-2k+2r)}}{{\sum }_{r=m}^n\frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2m}\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}}& \lesssim n^{(k-m)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\mathcal{L}}^{2(m-k)}.\hfill \end{array}$

Inserting (64) into (60) yields the following:

(65)$\begin{array}{cc}\hfill \left|\right|D_z^k(\mathcal{U}\left(z\right)-\tilde{\mathcal{U}}\left(z\right)){\left|\right|}_{L_{w\left(z\right)}^2}^2& \lesssim {\mathcal{L}}^{2(m-k)}{(M+1)}^{(k-m)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}\sum _{n=m}^{\infty }{\left|u_n\right|}^2\hfill \\ & \times \sum _{r=m}^n{\displaystyle \frac{{\left(A_{r,n}^{\sigma }\right)}^2{\mathrm{\Gamma }}^2(r+1){\mathcal{L}}^{2r-2m}\mathrm{\Gamma }\left(2(r-m)+\frac{1}{\sigma +2}\right)\mathrm{\Gamma }\left(\frac{\sigma +1}{\sigma +2}\right)}{{\mathrm{\Gamma }}^2(r-m+1)\mathrm{\Gamma }(1-2m+2r)}}\hfill \\ & ={\mathcal{L}}^{2(m-k)}{(M+1)}^{(k-m)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}\left|\right|D_z^{m}u\left(z\right){\left|\right|}_{L_{w\left(z\right)}^2}^2\hfill \\ & ={\mathcal{L}}^{2(m-k)}{\left({\displaystyle \frac{\mathrm{\Gamma }(M+2)}{\mathrm{\Gamma }(M+1)}}\right)}^{(k-m)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}\left|\right|D_z^{m}u\left(z\right){\left|\right|}_{L_{w\left(z\right)}^2}^2\hfill \\ & \lesssim {\mathcal{L}}^{2(m-k)}{M}^{(k-m)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}\left|\right|D_z^{m}u\left(z\right){\left|\right|}_{L_{w\left(z\right)}^2}^2.\hfill \end{array}$

Therefore, the result is achieved. □

Theorem 3.

Given the following assumption: $0\le p\le r\le M+1,$ and the approximation to $\mathcal{U}(z,\tau )\in {\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}\left(\mathrm{\Omega }\right)$ is ${\displaystyle \tilde{\mathcal{U}}(z,\tau )=\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)}$. As a result, the following estimation is satisfied:

(66) $\begin{array}{c}\hfill {\left|\left|{\displaystyle \frac{{\partial }^p}{\partial z^p}}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{L_{\overline{\omega }(z,\tau )}^2}\lesssim {\mathcal{L}}^{r-p}{M}^{{\displaystyle \frac{-1}{2}}(r-p)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,0}}.\end{array}$

Proof. 

According to the definitions of $\mathcal{U}(z,\tau )$ and $\tilde{\mathcal{U}}(z,\tau ),$ one has the following:

(67)$\begin{array}{cc}\hfill \mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau )& =\sum _{i=0}^M\sum _{j=N+1}^{\infty }{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)+\sum _{i=M+1}^{\infty }\sum _{j=0}^{\infty }{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)\hfill \\ & \lesssim \sum _{i=0}^M\sum _{j=0}^{\infty }{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)+\sum _{i=M+1}^{\infty }\sum _{j=0}^{\infty }{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right).\hfill \end{array}$

We obtain the desired result by applying the same procedures as in Theorem 2. □

Theorem 4.

Given the following assumption: $0\le q\le s\le N+1,$ the approximation to $\mathcal{U}(z,\tau )\in {\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}\left(\mathrm{\Omega }\right)$ is as follows: ${\displaystyle \tilde{\mathcal{U}}(z,\tau )=\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)}$. As a result, the following estimation is satisfied:

(68) $\begin{array}{c}\hfill {\left|\left|{\displaystyle \frac{{\partial }^q}{\partial {\tau }^q}}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{L_{\overline{\omega }(z,\tau )}^2}\lesssim {\mathcal{T}}^{s-q}{N}^{{\displaystyle \frac{-1}{2}}(s-q)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{0,s}}.\end{array}$

Proof. 

The proof of this theorem is similar to the proof of Theorem 3. □

Theorem 5.

Given the following assumption: $0\le p\le r\le M+1,$ $0\le q\le s\le N+1,$ the approximation to $\mathcal{U}(z,\tau )\in {\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}\left(\mathrm{\Omega }\right)$ is as follows: ${\displaystyle \tilde{\mathcal{U}}(z,\tau )=\sum _{i=0}^M\sum _{j=0}^N{u}_{ij}{G}_i^{\left(\sigma \right)}\left(z\right)G_j^{\left(\sigma \right)}\left(\tau \right)}$. As a result, the following estimation is satisfied:

(69) $\begin{array}{c}\hfill {\left|\left|{\displaystyle \frac{{\partial }^{p+q}}{\partial {\tau }^q\partial z^p}}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{L_{\overline{\omega }(z,\tau )}^2}\lesssim {\mathcal{T}}^{s-q}{\mathcal{L}}^{r-p}{N}^{{\displaystyle \frac{-1}{2}}(s-q)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{M}^{{\displaystyle \frac{-1}{2}}(r-p)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}}.\end{array}$

Proof. 

The proof of this theorem is similar to the proof of Theorem 3. □

Theorem 6.

If we consider the residual of Equation (20) given by the following:

${\mathcal{R}}_1(z,\tau )=\alpha {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial z^2}}+\beta {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial \tau \partial z}}+\gamma {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial {\tau }^2}}+\delta {\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial z}}+\eta {\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial \tau }}+\theta \tilde{\mathcal{U}}(z,\tau )-\mathcal{F}(z,\tau ),$

then ${\left|\left|{\mathcal{R}}_1(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}\to 0$ as $M,N\to \infty$.

Proof. 

${\left|\left|{\mathcal{R}}_1(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}$ of Equation (20) can be written as follows:

(70)$\begin{array}{cc}& {\left|\left|{\mathcal{R}}_1(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}\hfill \\ & ={\left|\left|\alpha {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial z^2}}+\beta {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial \tau \partial z}}+\gamma {\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial {\tau }^2}}+\delta {\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial z}}+\eta {\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial \tau }}+\theta \tilde{\mathcal{U}}(z,\tau )-\mathcal{F}(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}\hfill \\ & \le \alpha {\left|\left|{\displaystyle \frac{{\partial }^2\mathcal{U}}{\partial z^2}}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{\overline{\omega }(z,\tau )}+\beta {\left|\left|{\displaystyle \frac{{\partial }^2}{\partial \tau \partial z}}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{\overline{\omega }(z,\tau )}+\gamma {\left|\left|{\displaystyle \frac{{\partial }^2}{\partial {\tau }^2}}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{\overline{\omega }(z,\tau )}\hfill \\ & +\delta {\left|\left|{\displaystyle \frac{\partial }{\partial z}}\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{\overline{\omega }(z,\tau )}+\eta {\left|\left|{\displaystyle \frac{\partial }{\partial \tau }}(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{\overline{\omega }(z,\tau )}+\theta {\left|\left|(\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau ))\right|\right|}_{\overline{\omega }(z,\tau )}.\hfill \end{array}$

Now, applying Theorems 3–5 enables one to write the following:

(71)$\begin{array}{cc}\hfill {\left|\left|{\mathcal{R}}_1(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}& \lesssim \alpha {\mathcal{L}}^{r-2}{M}^{{\displaystyle \frac{-1}{2}}(r-2)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,0}}\hfill \\ & +\beta {\mathcal{T}}^{s-1}{\mathcal{L}}^{r-1}{N}^{{\displaystyle \frac{-1}{2}}(s-1)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{M}^{{\displaystyle \frac{-1}{2}}(r-1)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,s}}\hfill \\ & +\gamma {\mathcal{T}}^{s-2}{N}^{{\displaystyle \frac{-1}{2}}(s-2)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{0,s}}\hfill \\ & +\delta {\mathcal{L}}^{r-1}{M}^{{\displaystyle \frac{-1}{2}}(r-1)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,0}}\hfill \\ & +\theta {\mathcal{T}}^s{N}^{{\displaystyle \frac{-s}{2}}\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{0,s}}.\hfill \end{array}$

Hence, it is evident that ${\left|\left|{\mathcal{R}}_1(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}\to 0$ as $M,N\to \infty .$ □

Theorem 7.

If we consider the residual of Equation (37), given by the following:

${\mathcal{R}}_2(z,\tau )={\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial {\tau }^2}}+2\zeta (z,\tau ){\displaystyle \frac{\partial \tilde{\mathcal{U}}(z,\tau )}{\partial \tau }}+{\eta }^2(z,\tau )\tilde{\mathcal{U}}(z,\tau )-\theta (z,\tau ){\displaystyle \frac{{\partial }^2\tilde{\mathcal{U}}(z,\tau )}{\partial z^2}}-\mathcal{F}(z,\tau ),$

then ${\left|\left|{\mathcal{R}}_2(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}\to 0$ as $M,N\to \infty$.

Proof. 

Applying similar steps as in Theorem 6, we obtain the following estimation:

(72)$\begin{array}{cc}\hfill {\left|\left|{\mathcal{R}}_2(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}& \lesssim {\mathcal{T}}^{s-2}{N}^{{\displaystyle \frac{-1}{2}}(s-2)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{0,s}}\hfill \\ & +2\zeta (z,\tau ){\mathcal{T}}^{s-1}{N}^{{\displaystyle \frac{-1}{2}}(s-1)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{0,s}}\hfill \\ & +{\eta }^2(z,\tau ){\mathcal{T}}^s{N}^{{\displaystyle \frac{-s}{2}}\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{0,s}}\hfill \\ & -\theta (z,\tau ){\mathcal{L}}^{r-2}{M}^{{\displaystyle \frac{-1}{2}}(r-2)\left({\displaystyle \frac{2\sigma +3}{\sigma +2}}\right)}{\left|\mathcal{U}(z,\tau )\right|}_{{\mathbf{H}}_{\overline{\omega }(z,\tau )}^{r,0}}.\hfill \end{array}$

Therefore, it is clear that ${\left|\left|{\mathcal{R}}_2(z,\tau )\right|\right|}_{\overline{\omega }(z,\tau )}\to 0$ as $M,N\to \infty .$ □

6. Numerical Results

Here, we provide four numerical examples to show that the proposed algorithms are accurate, effective, and applicable. It is known that the current method, namely the generalized shifted Chebyshev collocation method (GSCCM), is more accurate when compared with the numerical methods using the radial basis function collocation method [54], the cubic B-spline collocation method [55], the finite difference method [56], and the Crank–Nicolson finite difference scheme [57].

The absolute error (AE) is the observed discrepancy between the measured and real values of the approximate solution, and it is defined as follows:

(73)$\begin{array}{c}\hfill E(z,\tau )=|\mathcal{U}(z,\tau )-\tilde{\mathcal{U}}(z,\tau )|,\end{array}$

where $\mathcal{U}(z,\tau )$ and $\tilde{\mathcal{U}}(z,\tau )$ are the exact and numerical solutions, respectively. In addition, the maximum absolute errors (MAEs) are given by the following:

(74)$\begin{array}{c}\hfill \mathrm{MAEs}=\mathrm{Max}\{E(z,\tau ):\forall (z,\tau )\in [0,\mathcal{L}]\times [0,\mathcal{T}]\}=\mathrm{errors}\mathrm{in}{L}^{\infty }.\end{array}$

The errors in $L^2$ can be computed using the following formula:

(75)$\begin{array}{c}\hfill \sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=0}^M}{\displaystyle \sum _{j=0}^M}{\left(\mathrm{Exact}-\mathrm{approximate}\right)}^2}{(N+1)(M+1)}}}.\end{array}$

Example 1

([51]). Consider the following second-order linear PDE:

(76)$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}-3{\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z\partial \tau }}+{\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}}=\mathcal{F}(z,\tau ),0\le z\le 1,0<\tau \le 1,\end{array}$

with the following conditions:

$\begin{array}{cc}\hfill & \mathcal{U}(z,0)=sin\left(z\right),{\mathcal{U}}_{\tau }(z,0)=-sin\left(z\right),0\le z\le 1,\hfill \\ \hfill & \mathcal{U}(0,\tau )=0,{\mathcal{U}}_z(0,\tau )=e^{-\tau },0<\tau \le 1,\hfill \end{array}$

whose exact solution is as follows:

$\mathcal{U}(z,\tau )=e^{-\tau }sin\left(z\right)and\mathcal{F}(z,\tau )=3e^{-z}cos\left(z\right).$

The absolute values of the error function at the points $(z,\tau )$ are provided in Table 1 by taking $\sigma =0,1,2,3,4$ and $N=M=12$. From this table, one can see the high order of accuracy of the presented method. Figure 1 shows the space–time of the approximate solution (left) and the exact solution (right) with $\sigma =5$ and $N=M=12$. Figure 2 shows the space–time graphs of the AE functions at $\sigma =5$ with $N=M=8$ and $N=M=12$, respectively. Figure 3 shows the comparison of the curves of analytical solutions and approximate solutions at $\tau =0,0.5,1$ (left) and $z=0.2,0.5,0.8$ (right) with $\sigma =5$ and $N=M=12$. In addition, the AE curves obtained at $\tau =0.5$ and $z=0.5$ with $N=M=12$ and $\sigma =5$, respectively, are shown in Figure 4.

Example 2

([52]). Consider the following telegraph equation with constant coefficients:

(77)$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}}-{\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}+{\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial \tau }}+\mathcal{U}(z,\tau )=\mathcal{F}(z,\tau ),0\le z\le 1,0<\tau \le 1,\end{array}$

governed by the following time initial and spatial boundary conditions:

$\begin{array}{c}\hfill \mathcal{U}(z,0)={\mathcal{U}}_{\tau }(z,0)=0,0\le z\le 1,\\ \hfill \mathcal{U}(0,\tau )=\mathcal{U}(1,\tau )=0,0<\tau \le 1,\end{array}$

whose exact solution is as follows:

$\mathcal{U}(z,\tau )={\tau }^2{e}^{-\tau }(z-z^2)and\mathcal{F}(z,\tau )=(2-2\tau +{\tau }^2)(z-z^2)e^{-\tau }+2{\tau }^2{e}^{-\tau }.$

In Table 2 and Table 3, which compare the $L^{\infty }$ and $L^2$ errors for Example 2, the GSCCM method ($\sigma =3$, $N=M=14$) demonstrates remarkable superiority in terms of accuracy compared to the other methods. The errors produced by this method are significantly lower, reaching magnitudes as small as ${10}^{-10}$ in some cases, reflecting its high efficiency in solving the problem in Example 2. One of the most notable advantages of GSCCM is its ability to achieve unprecedented accuracy compared to the radial basis function collocation [54] and cubic B-spline collocation methods [55], even when the time step (τ) is small. Although the CPU time for this method is relatively higher, this computational cost is justified by its exceptional precision, making it highly suitable for applications that require extremely accurate results. In Table 4 and Table 5, detailed comparisons of the $L^{\infty }$ and $L^2$ errors for Example 2 are presented using the GSCCM method with different values of the parameter σ ($\sigma =0,\sigma =1,\sigma =2$). The results show that increasing the value of σ leads to significant improvements in solution accuracy. Moreover, the GSCCM method maintains excellent accuracy even with slight variations in the time step τ, making it suitable for applications requiring high precision and stability in numerical solutions. The tables emphasize the importance of tuning the parameter σ to achieve a balance between accuracy and flexibility in numerical computations. With $\sigma =2$, $\mathcal{T}=10$, and $N=M=16$, space–time graphs of the approximate solution (left) and its AE function (right) are shown in Figure 5. Figure 6 also shows a comparison of the analytical and approximation solutions for $\sigma =2$ and $N=M=16$ at $\tau =4$, 7, 10 (left) and $z=0.3$, $0.6$, $0.9$ (right).

Example 3

([56]). Consider the following telegraph equation with variable coefficients:

(78)$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}}+2e^{(z+\tau )}{\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial \tau }}& -(1+z^2){\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}+{sin}^2(z+\tau )\mathcal{U}(z,\tau )=\mathcal{F}(z,\tau ),\hfill \\ & 0\le z\le 1,0<\tau \le 1,\hfill \end{array}$

governed by the following time initial and spatial boundary conditions:

$\begin{array}{cc}\hfill & \mathcal{U}(z,0)=sinh\left(z\right),{\mathcal{U}}_{\tau }(z,0)=-2sinh\left(z\right),0\le z\le 1,\hfill \\ \hfill & \mathcal{U}(0,\tau )=0,\mathcal{U}(1,\tau )=e^{-2\tau }sinh\left(1\right),0<\tau \le 1,\hfill \end{array}$

whose exact solution is

$\mathcal{U}(z,\tau )=e^{-2\tau }sinh\left(z\right)and\mathcal{F}(z,\tau )=(3-z^2+{sin}^2(z+\tau )-4e^{(z+\tau )})e^{-2\tau }sinh\left(z\right).$

In Table 6, we depict the MAEs using the proposed method with those obtained in [56] and [57], respectively. Table 7 presents the $L^{\infty }$ and $L^2$ errors along with the CPU time for Example 3 at $\sigma =1.5$ and $N=M=16$. The results indicate that the errors are extremely small but gradually increase as τ increases, reflecting the impact of a larger time step on the accuracy. Furthermore, the CPU time rises with τ due to the higher computational complexity. This demonstrates the efficiency of the method in Example 3, achieving high accuracy at the expense of the increased computational cost. Moreover, Figure 7 shows the space–time graphs of the approximate solution (left) and its AE function (right) with $\sigma =2,\mathcal{T}=5$ and $N=M=16$:

Example 4

([56,57]). Consider the following singular telegraph equation:

(79)$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial {\tau }^2}}+{\displaystyle \frac{2}{z^2}}{\displaystyle \frac{\partial \mathcal{U}(z,\tau )}{\partial \tau }}& -(1+z^2){\displaystyle \frac{{\partial }^2\mathcal{U}(z,\tau )}{\partial z^2}}+{\displaystyle \frac{1}{z^2}}\mathcal{U}(z,\tau )=\mathcal{F}(z,\tau ),\hfill \\ & 0\le z\le 1,0<\tau \le 1,\hfill \end{array}$

governed by the following initial time and spatial boundary conditions:

$\begin{array}{cc}\hfill & \mathcal{U}(z,0)=sinh\left(z\right),{\mathcal{U}}_{\tau }(z,0)=-2sinh\left(z\right),0\le z\le 1,\hfill \\ \hfill & \mathcal{U}(0,\tau )=0,\mathcal{U}(1,\tau )=e^{-2\tau }sinh\left(1\right),0<\tau \le 1,\hfill \end{array}$

whose exact solution is as follows:

$\mathcal{U}(z,\tau )=e^{-2\tau }sinh\left(z\right),$

and

$\mathcal{F}(z,\tau )=-{\displaystyle \frac{e^{-2\tau }\left(z^4-3z^2+3\right)sinh\left(z\right)}{z^2}}.$

Table 8 lists the MAEs at various choices of $\mathcal{T}$. Moreover, Figure 8 presents the AE curves obtained by the present method at $\tau =0.5$ and $z=0.5$ with $N=M=16$, and $\sigma =3$, respectively.

7. Conclusions

This paper proposes a new generalized Chebyshev collocation method (GSCCM) for solving the second-order linear PDEs and the telegraph equation. The methodology leverages the OMDs of the generalized Chebyshev polynomials to convert the PDEs into a system of algebraic equations, facilitating efficient numerical treatment. The collocation procedure is followed to reduce the equations governed by their underlying conditions into algebraic systems of equations that can be solved through suitable solvers. We believe the proposed approach can be used to treat other differential equations.

Author Contributions

Conceptualization, W.M.A.-E. and R.M.H.; Methodology, W.M.A.-E., A.N., R.M.H. and A.G.A.; Software, R.M.H.; Formal analysis, W.M.A.-E., R.M.H. and A.G.A.; Validation, W.M.A.-E., A.N., R.M.H. and A.G.A.; Writing—original draft, W.M.A.-E., A.N., R.M.H. and A.G.A.; Writing—review & editing, W.M.A.-E., A.N. and R.M.H. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The authors would like to thank the journal and the reviewers for their helpful comments and suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Space–time graphs of the approximate solution (left) and the exact solution (right) for Example 1 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 2. Space–time graphs of the AE functions at [Forumla omitted. See PDF.] for Example 1.

Figure 3. Comparison of the curves of the analytical solutions and the approximate solutions at [Forumla omitted. See PDF.] (left) and [Forumla omitted. See PDF.] (right) for Example 1 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 4. The AE curves of the z-direction (left) and [Forumla omitted. See PDF.]-direction (right) for Example 1 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 5. Space–time graphs of the approximate solution (left) and its AE function (right) for Example 2 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 6. Comparison of the curves of the analytical solutions and the approximate solutions at [Forumla omitted. See PDF.] (left) and [Forumla omitted. See PDF.] (right) for Example 2 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 7. Space–time graphs of the approximate solution (left) and its AE function (right) for Example 3 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 8. The AE curves of the z-direction (left) and [Forumla omitted. See PDF.]-direction (right) for Example 4 with [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Table 1

The AE of the exact solution and the numerical solution for Example 1 at $N=M=12$.

$(\mathit{z},\mathit{\tau })$	$\mathit{\sigma }=0$	$\mathit{\sigma }=1$	$\mathit{\sigma }=2$	$\mathit{\sigma }=3$	$\mathit{\sigma }=4$
$(0.0,0.0)$	$6.6480\times {10}^{-18}$	$5.4305\times {10}^{-17}$	$5.5014\times {10}^{-17}$	$6.0221\times {10}^{-17}$	$9.0975\times {10}^{-18}$
$(0.1,0.1)$	$6.9388\times {10}^{-17}$	$1.1102\times {10}^{-16}$	$5.5511\times {10}^{-17}$	$6.9388\times {10}^{-17}$	$5.5511\times {10}^{-17}$
$(0.2,0.2)$	$3.0531\times {10}^{-16}$	$1.1102\times {10}^{-16}$	$2.7755\times {10}^{-17}$	$1.1102\times {10}^{-16}$	$2.7755\times {10}^{-17}$
$(0.3,0.3)$	$8.3266\times {10}^{-17}$	$5.5511\times {10}^{-17}$	$8.3266\times {10}^{-17}$	$2.7755\times {10}^{-17}$	$0.0000\times {10}^{-00}$
$(0.4,0.4)$	$2.2204\times {10}^{-16}$	$2.2204\times {10}^{-16}$	$5.5511\times {10}^{-17}$	$3.3306\times {10}^{-16}$	$5.5511\times {10}^{-17}$
$(0.5,0.5)$	$5.5511\times {10}^{-17}$	$5.5511\times {10}^{-17}$	$5.5511\times {10}^{-17}$	$1.1102\times {10}^{-16}$	$1.1102\times {10}^{-16}$
$(0.6,0.6)$	$3.8857\times {10}^{-16}$	$5.5511\times {10}^{-17}$	$0.0000\times {10}^{-00}$	$1.1102\times {10}^{-16}$	$1.1102\times {10}^{-16}$
$(0.7,0.7)$	$3.3306\times {10}^{-16}$	$1.6653\times {10}^{-16}$	$0.0000\times {10}^{-00}$	$2.7755\times {10}^{-16}$	$5.5511\times {10}^{-17}$
$(0.8,0.8)$	$6.4037\times {10}^{-13}$	$3.9168\times {10}^{-13}$	$1.5815\times {10}^{-13}$	$2.6029\times {10}^{-13}$	$2.9470\times {10}^{-13}$
$(0.9,0.9)$	$1.6656\times {10}^{-11}$	$1.0137\times {10}^{-11}$	$3.6868\times {10}^{-12}$	$6.5015\times {10}^{-12}$	$7.0563\times {10}^{-12}$
$(1.0,1.0)$	$1.8120\times {10}^{-10}$	$1.1130\times {10}^{-10}$	$3.9125\times {10}^{-11}$	$7.1077\times {10}^{-11}$	$7.3766\times {10}^{-11}$

Table 2

Comparison of the $L^{\infty }-$ errors for Example 2.

$\mathit{\tau }$	Dehghan and Shokri [54]	CPU Time	Mittal and Bhatia [55]	CPU Time	GSCCM $(\mathit{\sigma }=3)$ $(\mathit{N}=\mathit{M}=14)$	CPU Time
1	$1.8479\times {10}^{-5}$	0	$5.9153\times {10}^{-5}$	$0.43$	$4.7169\times {10}^{-15}$	$81.484$
2	$1.0713\times {10}^{-5}$	0	$1.7864\times {10}^{-5}$	$0.77$	$5.6611\times {10}^{-14}$	$78.860$
3	$1.8161\times {10}^{-5}$	1	$1.4309\times {10}^{-5}$	$1.19$	$3.2707\times {10}^{-11}$	$77.219$
4	$1.6489\times {10}^{-5}$	1	$1.3529\times {10}^{-5}$	$1.29$	$1.2335\times {10}^{-9}$	$75.966$
5	$1.0455\times {10}^{-5}$	2	$5.2032\times {10}^{-6}$	$1.46$	$1.6764\times {10}^{-8}$	$68.734$

Table 3

Comparison of the $L^2-$ errors for Example 2.

$\mathit{\tau }$	Dehghan and Shokri [54]	CPU Time	Mittal and Bhatia [55]	CPU Time	GSCCM $(\mathit{\sigma }=3)$ $(\mathit{N}=\mathit{M}=14)$	CPU Time
1	$1.4386\times {10}^{-4}$	0	$4.5526\times {10}^{-5}$	$0.43$	$1.8638\times {10}^{-16}$	$83.515$
2	$8.0879\times {10}^{-5}$	0	$1.4307\times {10}^{-5}$	$0.77$	$1.5165\times {10}^{-13}$	$81.204$
3	$1.2944\times {10}^{-4}$	1	$6.4273\times {10}^{-6}$	$1.19$	$1.2150\times {10}^{-10}$	$79.469$
4	$1.1845\times {10}^{-4}$	1	$8.9203\times {10}^{-6}$	$1.29$	$1.1050\times {10}^{-8}$	$77.872$
5	$7.5545\times {10}^{-5}$	2	$3.0161\times {10}^{-6}$	$1.46$	$3.2067\times {10}^{-7}$	$70.640$

Table 4

$L^{\infty }-$ errors at different $\sigma$ for Example 2.

$\mathit{\tau }$	GSCCM $(\mathit{N}=\mathit{M}=16)$
	$\mathbf{\sigma }=\mathbf{0}$	$\mathbf{\sigma }=\mathbf{1}$	$\mathbf{\sigma }=\mathbf{2}$
1	$1.3610\times {10}^{-15}$	$4.0646\times {10}^{-16}$	$7.8372\times {10}^{-15}$
2	$1.1435\times {10}^{-14}$	$4.4875\times {10}^{-15}$	$2.4063\times {10}^{-14}$
3	$3.2596\times {10}^{-13}$	$1.9203\times {10}^{-13}$	$1.4433\times {10}^{-13}$
4	$2.2959\times {10}^{-11}$	$1.3222\times {10}^{-11}$	$9.9118\times {10}^{-12}$
5	$4.3070\times {10}^{-10}$	$3.3240\times {10}^{-10}$	$2.6196\times {10}^{-10}$

Table 5

$L^2-$ errors at different $\sigma$ for Example 2.

$\mathit{\tau }$	Proposed Method $(\mathit{N}=\mathit{M}=16)$
	$\mathbf{\sigma }=\mathbf{0}$	$\mathbf{\sigma }=\mathbf{1}$	$\mathbf{\sigma }=\mathbf{2}$
1	$5.5083\times {10}^{-17}$	$1.0733\times {10}^{-16}$	$3.1704\times {10}^{-16}$
2	$5.1171\times {10}^{-16}$	$4.0231\times {10}^{-16}$	$3.9637\times {10}^{-15}$
3	$2.9620\times {10}^{-13}$	$3.3035\times {10}^{-13}$	$3.5467\times {10}^{-13}$
4	$4.5759\times {10}^{-11}$	$5.2497\times {10}^{-11}$	$5.8166\times {10}^{-11}$
5	$1.9734\times {10}^{-9}$	$2.3477\times {10}^{-9}$	$2.6236\times {10}^{-9}$

Table 6

Comparison of the MAEs for Example 3.

Mohanty [56]	Pandit et al. [57]	GSCCM $(\mathit{N}=\mathit{M}=16)$
		$\mathbf{\sigma }=\mathbf{0}$	$\mathbf{\sigma }=\mathbf{1}$	$\mathbf{\sigma }=\mathbf{2}$
$0.4491\times {10}^{-3}$	$3.8337\times {10}^{-4}$	$1.5404\times {10}^{-14}$	$7.2331\times {10}^{-14}$	$2.5812\times {10}^{-13}$

Table 7

$L^{\infty },L^2$ errors and CPU time at $\sigma =1.5$ and $N=M=16$ for Example 3.

$\mathit{\tau }$	${\mathit{L}}^{\mathit{\infty }}$-Error	${\mathit{L}}^2$-Error	CPU Time (s)
1	$3.2335\times {10}^{-14}$	$1.3607\times {10}^{-15}$	$125.545$
2	$1.7995\times {10}^{-13}$	$7.4385\times {10}^{-14}$	$140.796$
3	$1.2250\times {10}^{-11}$	$9.6239\times {10}^{-11}$	$156.670$
4	$1.6401\times {10}^{-9}$	$1.4708\times {10}^{-8}$	$158.235$
5	$4.0102\times {10}^{-8}$	$7.7866\times {10}^{-7}$	$172.186$

Table 8

Comparison of the MAEs for Example 4.

$\mathit{\tau }$	Mohanty [56]	Pandit et al. [57]	GSCCM $(\mathit{N}=\mathit{M}=16)$
			$\mathbf{\sigma }=\mathbf{0}$	$\mathbf{\sigma }=\mathbf{1}$	$\mathbf{\sigma }=\mathbf{2}$
1	$0.2317\times {10}^{-3}$	$1.1931\times {10}^{-4}$	$5.1475\times {10}^{-16}$	$6.6224\times {10}^{-14}$	$1.8380\times {10}^{-14}$
2	$0.6233\times {10}^{-4}$	$6.0254\times {10}^{-4}$	$7.3870\times {10}^{-14}$	$1.2380\times {10}^{-13}$	$7.1185\times {10}^{-14}$

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