Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay

Brajesh Kumar Singh,Pramod Kumar

doi:10.1155/2017/5206380

Abstract

This paper deals with an alternative approximate analytic solution to time fractional partial differential equations (TFPDEs) with proportional delay, obtained by using fractional variational iteration method, where the fractional derivative is taken in Caputo sense. The proposed series solutions are found to converge to exact solution rapidly. To confirm the efficiency and validity of FRDTM, the computation of three test problems of TFPDEs with proportional delay was presented. The scheme seems to be very reliable, effective, and efficient powerful technique for solving various types of physical models arising in science and engineering.

Highlights

The idea of derivatives of fractional order was described first by great mathematician Newton and Leibnitz in the seventh century and has achieved a great attention due to their numerous applications in nonlinear complex systems arising in various important phenomena in the fluid mechanics, damping laws, electrical networks, signal processing, diffusion-reaction process relaxation processes, mathematical biology, and other fields of science and engineering [1,2,3,4,5,6,7]
A fractional model of differentialdifference equation model has been studied analytically by adopting homotopy analysis transform method [27], and a hybrid computational approach based on local fractional Sumudu transform with HPM has been employed for numerical study of Klein–Gordon equations on Cantor sets [28]
The main aim of this paper is to propose an alternative approximate solution of the initial valued autonomous system of time fractional partial differential equations (TFPDEs) with proportional delay [17] by employing alternative variational iteration method (AVIM)

Summary

Introduction

The idea of derivatives of fractional order was described first by great mathematician Newton and Leibnitz in the seventh century and has achieved a great attention due to their numerous applications in nonlinear complex systems arising in various important phenomena in the fluid mechanics, damping laws, electrical networks, signal processing, diffusion-reaction process relaxation processes, mathematical biology, and other fields of science and engineering [1,2,3,4,5,6,7]. Goufo [30] adopted newly developed Caputo-Fabrizio fractional derivative without singular kernel to obtain an analytical solution of Korteweg-de Vries-Burgers equation with two perturbations’ levels. Another well-known model, time fractional nonlinear Klein–Gordon equation with proportional delay, aries in quantum field theory to describe nonlinear wave interaction:. The analytical solutions of TFPDE with proportional delay have been obtained by employing homotopy perturbation method by Sakar et al [17] and Biazar ad Ghanbari [47]. Chena and Wang [48] have adopted variational iteration method (VIM) for solving a neutral functional-differential equation with proportional delays. The main aim of this paper is to propose an alternative approximate solution of the initial valued autonomous system of TFPDE with proportional delay [17] by employing alternative variational iteration method (AVIM).

Preliminaries

AVIM for FPDEs with Proportional Delay Equations

Application of VIM to TFPDEs with Proportional Delay

Conclusion