Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.

Ram K Pandey,Om P Singh,Vipul K Baranwal

doi:10.1155/2014/847419

Abstract

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

Highlights

Nonlinear problems have posed a challenge to the scientific world since long and many scientists and researchers have been working hard to find methods to solve these problems
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations
The aim of the present paper is to propose an optimal variational asymptotic method (OVAM) to solve the following initial value nonlinear time fractional PDE: Ly + Ny = g (x, t), x ∈ Ω = [a, b] ⊂ R, t > 0, (1)

Summary

Introduction

Nonlinear problems have posed a challenge to the scientific world since long and many scientists and researchers have been working hard to find methods to solve these problems. There are several analytical methods, such as homotopy analysis method (HAM) [1], homotopy perturbation method (HPM) [2], Adomian decomposition method (ADM) [3], variational iteration method (VIM) [4], and a new iterative method [5], are available to solve nonlinear fractional partial differential equations. The aim of the present paper is to propose an optimal variational asymptotic method (OVAM) to solve the following initial value nonlinear time fractional PDE: Ly + Ny = g (x, t) , x ∈ Ω = [a, b] ⊂ R, t > 0, (1). The proposed method is the generalization of the method given in [5]

Objectives

Results

Conclusion