Solving Helmholtz Equation with Local Fractional Derivative Operators

Jassim Jassim,Baleanu Baleanu,Al Qurashi Al Qurashi

doi:10.3390/fractalfract3030043

Jassim Jassim, Baleanu Baleanu + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract3030043

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Journal: Fractal and Fractional	Publication Date: Aug 1, 2019
Citations: 28	License type: CC BY 4.0

Affiliation: Thi Qar University

Abstract

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

Highlights

The Helmholtz equation often arises in the study of physical problems involving partial differential equation (PDEs) in both space and time
It is important to note that the new modification reduces the size of calculations compared to the local fractional variational iteration method (LFVIM)
The local fractional Laplace variational iteration method was proved to be an effective approach for solving partial differential equations with local fractional derivative operators due to the excellent agreement between the obtained approximate solution and the exact solution

Summary

Introduction

The Helmholtz equation often arises in the study of physical problems involving partial differential equation (PDEs) in both space and time. The Helmholtz equation with local fractional derivative operators in two-dimensional case was suggested in [1,2] as follows:. While coupled Helmholtz equations with local fractional derivative operators in two-dimensional case was introduced in [2] as follows:. The main aim of this work is to propose the local fractional Laplace variational iteration method to solve Helmholtz and coupled Helmholtz equations with LFDOs. It is important to note that the new modification reduces the size of calculations compared to the LFVIM.

Basic Definitions of Local Fractional Calculus

Analysis of the Method

Illustrated Examples

The plot of the solutions to the coupled

Conclusions