Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

A G Atta,W M Abd-Elhameed,G M Moatimid,Y H Youssri

doi:10.1007/s40096-022-00460-6

A G Atta, W M Abd-Elhameed + Show 2 more

Open Access

https://doi.org/10.1007/s40096-022-00460-6

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Journal: Mathematical Sciences	Publication Date: Mar 6, 2022
Citations: 13	License type: CC BY 4.0

Affiliation: Ain Shams University, Cairo University

Abstract

A new numerical scheme based on the tau spectral method for solving the linear hyperbolic telegraph type equation is presented and implemented. The derivation of this scheme is based on utilizing certain modified shifted Chebyshev polynomials of the sixth-kind as basis functions. For this purpose, some new formulas concerned with the modified shifted Chebyshev polynomials of the sixth-kind have been stated and proved, and after that, they serve to study the hyperbolic telegraph type equation with our proposed scheme. One advantage of using this scheme is that it reduces the problem into a system of algebraic equations that can be simplified using the Kronecker algebra analysis. The convergence and error estimate of the proposed technique are analyzed in detail. In the end, some numerical tests are presented to demonstrate the efficiency and high accuracy of the proposed scheme.

Highlights

The telegraph equation is a second-order hyperbolic partial differential equation (HPDE)
A Haar wavelet collocation approach is followed in [7] for treating one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations. Another numerical algorithm depending on the meshless approach is developed for the numerical solution of some types of hyperbolic telegraph equation in [8–10]
We have presented in this paper a numerical algorithm designed for approximating the solutions for the hyperbolic telegraph type problem based using the spectral tau approach

Summary

Introduction

The telegraph equation is a second-order hyperbolic partial differential equation (HPDE). A Haar wavelet collocation approach is followed in [7] for treating one and two-dimensional second-order linear and nonlinear hyperbolic telegraph equations. There are considerable contributions concerned with the first-, second-, third- and fourth- kinds of Chebyshev polynomials These four kinds of Chebyshev polynomials have played important roles in the numerical solutions of various types of differential equations using the different versions of spectral methods (see, [38–42]). The authors in [44] and [28] used, respectively, the fifth- and the sixth-kinds Chebyshev polynomials as basis functions to solve some types of linear and nonlinear fractional-order differential equations. We resort to the celebrated algorithm of Zeilberger ( [48]) to show that G ,i satisfies the following recurrence relation:

Proof First set

Investigation of the convergence and error analysis

Illustrative examples

Concluding remarks