Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations

Rashid Nawaz,Muhammad Jibran,Abraiz Khattak,Laiq Zada,Adam Khan

doi:10.1155/2019/1741958

Abstract

In this paper, the Optimal Homotopy Asymptotic Method is extended to derive the approximate solutions of fractional order two-dimensional partial differential equations. The fractional order Zakharov–Kuznetsov equation is solved as a test example, while the time fractional derivatives are described in the Caputo sense. The solutions of the problem are computed in the form of rapidly convergent series with easily calculable components using Mathematica. Reliability of the proposed method is given by comparison with other methods in the literature. The obtained results showed that the method is powerful and efficient for determination of solution of higher-dimensional fractional order partial differential equations.

Highlights

Fractional calculus is an extension of integer order calculus
Due to the complexities of fractional calculus, most of the fractional order di erential equations do not have the exact solutions, and as an alternative, the approximate methods are extensively used for solution of these types of equations [10,11,12,13,14]
Marinca and Herisanu introduced the Optimal Homotopy Asymptotic Method (OHAM) for solving nonlinear di erential equations which made the perturbation methods independent of the assumption of small parameters and huge computational work [27,28,29,30,31]. e method was recently extended by Sarwar et al for solution of fractional order di erential equations [32,33,34,35]

Summary

Introduction

Fractional calculus is an extension of integer order calculus. For many years, it was assumed that fractional calculus is a pure subject of mathematics and having no such applications in real-world phenomena, but this concept is wrong because of the recent applications of fractional calculus in modeling of the sound waves propagation in rigid porous materials [1], ultrasonic wave propagation in human cancellous bone [2], viscoelastic properties of soft biological tissues [3], the path tracking problem in an autonomous electric vehicles [4], etc. Some of the recent methods for approximate solutions of fractional order di erential equations are the Adomian Decomposition Method (ADM), the Homotopy Perturbation Method (HPM), the Variational Iteration Method (VIM), Homotopy Analysis Method (HAM), etc. E method was recently extended by Sarwar et al for solution of fractional order di erential equations [32,33,34,35]. OHAM formulation is extended to twodimensional fractional order partial di erential equations.

Basic Definitions

OHAM Analysis for Fractional Order PDEs

OHAM Convergence

Application of OHAM

Results and Discussion