Numerical Solutions of Time Fractional Zakharov-Kuznetsov Equation via Natural Transform Decomposition Method with Nonsingular Kernel Derivatives

Nehad Ali Shah,K Aruna,Hadi Rezazadeh,K Raghavendar,Mei-Xiu Zhou,Ayman A Aly,Mustafa Inc,A S V Ravi Kanth

doi:10.1155/2021/9884027

Abstract

In this paper, we have studied the time-fractional Zakharov-Kuznetsov equation (TFZKE) via natural transform decomposition method (NTDM) with nonsingular kernel derivatives. The fractional derivative considered in Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in Caputo sense (ABC). We employed natural transform (NT) on TFZKE followed by inverse natural transform, to obtain the solution of the equation. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results.

Highlights

Fractional calculus is an emerging field in various branches of engineering science
Fractional differential equations attracted researchers as they used to model a variety of diverse applications such as visco elasticity, heat conduction, biology, and dynamical systems [1,2,3,4,5,6,7]
Two nonsingular kernel fractional derivative definitions are developed by Atangana-Baleanu and CaputoFabrizio

Summary

Introduction

Fractional differential equations attracted researchers as they used to model a variety of diverse applications such as visco elasticity, heat conduction, biology, and dynamical systems [1,2,3,4,5,6,7]. Due to its importance in diverse fields, considerable methods developed to study the exact and computational solutions of fractional differential equations. Several fractional derivative definitions developed in the last few decades. Some of the popular definitions in the literature are Riemann-Liouville (R-L), Caputo, CF, ABC, Grunwald-Letnikov, and Riesz fractional derivatives. R-L and Caputo fractional derivatives have a singular kernel. Two nonsingular kernel fractional derivative definitions are developed by Atangana-Baleanu and CaputoFabrizio. Several methods are being investigated for the analysis of fractional differential equations for accuracy and reliable solutions. Some of the popular semi analytical and numerical methods are variational iteration method (VIM) [10], fractional differential transform method [11,12,13,14], homotopy perturbation transform method (HPTM) [15], homotopy analysis transform method [16, 17], residual power series method (RPS) [18], q-homotopy analysis transform method (q-HATM) [19,20,21], operational matrix method [22], tension spline method [23], parametric cubic

Objectives

Methods

Results