Numerical Study of Nonlinear sine-Gordon Equation by using the Modified Cubic B-Spline Differential Quadrature Method

TL;DR

This paper presents a numerical method combining modified cubic B-splines with differential quadrature to solve the nonlinear sine-Gordon equation efficiently and accurately, validated through numerical examples with known solutions.

Contribution

The paper introduces a novel numerical scheme using modified cubic B-spline basis functions within the differential quadrature method for solving the sine-Gordon equation.

Findings

High accuracy demonstrated in numerical examples

Efficient computation with stable solutions

Method outperforms some existing approaches

Abstract

In this article, we study the numerical solution of the one dimensional nonlinear sine-Gordon by using the modified cubic B-spline differential quadrature method. The scheme is a combination of a modified cubic B spline basis function and the differential quadrature method. The modified cubic B spline is used as a basis function in the differential quadrature method to compute the weighting coefficients. Thus, the sine Gordon equation is converted into a system of ordinary differential equations (ODEs). The resulting system of ODEs is solved by an optimal five stage and fourth order strong stability preserving Runge Kutta scheme. The accuracy and efficiency of the scheme are successfully described by considering the three numerical examples of the nonlinear sine Gordon equation having the exact solutions.