Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators

Ai-Min Yang,Zeng-Shun Chen,Xiao-Jun Yang,H M Srivastava

doi:10.1155/2013/259125

Abstract

We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

Highlights

The Helmholtz equation is known to arise in several physical problems such as electromagnetic radiation, seismology, and acoustics
Using separation of variables in nondifferentiable functions, the three-dimensional Helmholtz equation involving local fractional derivative operators was suggested by the following expression [39]:
The two-dimensional Helmholtz equation involving local fractional derivative operators is expressed as follows:

Summary

Introduction

The Helmholtz equation is known to arise in several physical problems such as electromagnetic radiation, seismology, and acoustics It is a partial differential equation, which models the normal and nonfractal physical phenomena in both time and space [1]. The variational iteration method was used to solve the Helmholtz equation in [4]. The main objective of the present paper is to solve the Helmholtz equation involving the local fractional derivative operators by means of the local fractional series expansion method and the variational iteration method.

Helmholtz Equations within Local Fractional Derivative Operators

Analysis of the Methods Used

Local Fractional Series Expansion Method for the Helmholtz Equation

Local Fractional Variational Iteration Method for the Helmholtz Equation

Conclusions