Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

Weishi Yin,Fei Xu,Weipeng Zhang,Yixian Gao

doi:10.1155/2016/4632163

Abstract

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

Highlights

The Kuramoto-Sivashinsky (KS) equation in one space dimension, Dtu (x, t) + Dx4 u (x, t) + Dx2 u (x, t) + u (x, t) Dxu (x, t) (1) = 0, has attracted a great deal of interest as a model for complex spatiotemporal dynamics in spatially extended systems and as a paradigm for finite-dimensional dynamics in a partial differential equation
The main advantage of the residual power series (RPS) method is that it can be applied directly for all types of differential equation, because it depends on the recursive differentiation of time-fractional derivative and uses given initial conditions to calculate coefficients of the multiple fractional power series solution with minimal calculations
The fractional power series solutions can be obtained by the recursive equation (35) with time-fractional derivative, while it will use the given initial conditions

Summary

Introduction

The main advantage of the RPS method is that it can be applied directly for all types of differential equation, because it depends on the recursive differentiation of time-fractional derivative and uses given initial conditions to calculate coefficients of the multiple fractional power series solution with minimal calculations. Another important advantage is that this method does not require linearization, perturbation, or discretization of the variables; it is not affected by computational round-off errors and does not require large computer memory and extensive time.

Some Concepts on the Theory of Fractional Calculus

Algorithm of RPS Method

Application of RPS Method to Time-Space Fractional KS Equation

Concluding Remarks