DIFFERENCE SCHEME METHOD FOR THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

doi:10.52587/tmcs010202

Abstract

In this study, wave equations with initial boundary conditions have been studied. The general form of the wave equation has been derived. The first order and second order difference schemes were established for the presented IBVP. The stability of the difference schemes has been guaranteed. The approximation solution of the problem was achieved by using finite difference methods. Two different examples are provided. A comparison between the exact and approximation solution has been carried out. Absolute errors of the problem have been presented by using MATLAB software. Moreover, the comparison shows that the second order difference scheme is a more accurate result than the first order. It is shown that the results of the comparison guaranty the reliability and accuracy of the presented method.

Highlights

In the last decades, Partial differential equations (PDE) have focused on many studies because of their common appearance in several applications in mathematics, physics, seismology, science, finance, engineering, and mechanics [1, 2]
The first order and second order difference schemes were established for the presented IBVP
The approximation solution of the problem was achieved by using finite difference methods

Summary

INTRODUCTION

Partial differential equations (PDE) have focused on many studies because of their common appearance in several applications in mathematics, physics, seismology, science, finance, engineering, and mechanics [1, 2]. The wave equation is an important second order linear PDE for describing waves, such as mechanical waves or light waves [1,2,3,4,5]. Difference scheme method for the numerical solution of partial differential equations process mathematical model under two-point time conditions[6]. Finite difference approximations have been used for solving different types of partial differential equations in the works [8,9,10,11,12,13,14,15]. Different numerical methods are presented for solving wave equations with initial boundary conditions [19,20,21,22]. We will present a numerical method based on difference scheme method for solving wave equations

DERIVING THE 1D WAVE EQUATION

STABILITY AND DIFFERENCE SCHEME

THE METHOD

NUMERICAL COMPUTATION

CONCLUSION