Local Fractional Fourier Series Method for Solving Nonlinear Equations with Local Fractional Operators

Yong-Ju Yang,Shun-Qin Wang

doi:10.1155/2015/481905

Yong-Ju Yang, Shun-Qin Wang

Open Access

https://doi.org/10.1155/2015/481905

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Abstract

We apply the local fractional Fourier series method for solving nonlinear equation with local fractional operators. This method is the coupling of the local fractional Fourier series expansion method with other methods, such as the Yang-Laplace transformation method and the local fractional power series method, which effectively separates the variables of partial differential equation. Some testing nonlinear equations and equation systems are given to demonstrate the accuracy and applicability of the proposed approach.

Highlights

There are many definitions of fractional derivative and integral, such as Riesz, Caputo, Riemann-Liouville, Marchaud, and Sonin-Letnikov [1, 2]
Nonlinear fractional differential equation and nonlinear fractional integral-differential equation are the promising fields of research in technology and science
The local fractional differential and calculus theory is introduced in [13, 14], which is set up on fractal geometry and which is the best candidate for depicting the nondifferential function defined on Cantor sets

Summary

Introduction

There are many definitions of fractional derivative and integral, such as Riesz, Caputo, Riemann-Liouville, Marchaud, and Sonin-Letnikov [1, 2]. It is very difficult to solve these nonlinear equations with fractional differential or fractional integral operator. The local fractional Fourier series method has been proposed in [22], which is the coupling of the local fractional Fourier series expansion method with the Yang-Laplace transformation method for solving local fractional linear differential equations. By coupling of the local fractional Fourier series expansion method with a few methods, the Yang-Laplace method, we generalize and enrich the local fractional Fourier series method for solving some nonlinear equations within the local fractional differential or fractional integral operator.

Mathematical Fundamentals

Local Fractional Fourier Series Method

Illustrative Examples

Conclusion