Approximate Solution of Non-Linear Fractional Klein-Gordon Equation Using Spectral Collocation Method

Rubayyi T Alqahtani

doi:10.4236/am.2015.613190

Abstract

In this paper, we implement the spectral collocation method with the help of the Legendre poly-nomials for solving the non-linear Fractional (Caputo sense) Klein-Gordon Equation (FKGE). We present an approximate formula of the fractional derivative. The Legendre collocation method is used to reduce FKGE to the solution of system of ODEs which is solved by using finite difference method. The results of applying the proposed method to the non-linear FKGE show the simplicity and the efficiency of the proposed method.

Highlights

The theory of fractional calculus is initiated by Leibniz, Liouville, Riemann, Grunwald and Letnikov and since has been found many applications in science and engineering
The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory [1]
The proposed method gives excellent results when it is applied to Fractional (Caputo sense) Klein-Gordon Equation (FKGE)

Summary

Introduction

The theory of fractional calculus is initiated by Leibniz, Liouville, Riemann, Grunwald and Letnikov and since has been found many applications in science and engineering. Finding the exact solution for most of these equations is not an easy task, analytical and numerical methods must be used. The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory [1]. (2015) Approximate Solution of Non-Linear Fractional Klein-Gordon Equation Using Spectral Collocation Method. The study of numerical solutions of the Klein-Gordon equation has been investigated considerably in the last few years. We apply spectral collocation method (with the help of Legendre polynomials) to obtain the numerical solution of the non-linear FKGE of the form utt ( x,t ) + aDαu ( x,t ) + bu ( x,t ) + cuγ ( x= ,t ) f ( x,t ), x ∈ (0, L),t > 0, α ∈ For more details about the fractional calculus see ([5]-[8]) and for more details about the Legendre collocation method see ([9]-[18])

An Approximate Formula of the Fractional Derivative

Conclusion and Remarks