Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

Rasool Shah,Dumitru Baleanu,Hassan Khan

doi:10.3390/axioms8040125

Rasool Shah, Dumitru Baleanu + Show 1 more

Open Access

https://doi.org/10.3390/axioms8040125

Copy DOI

Journal: Axioms	Publication Date: Nov 7, 2019
Citations: 37	License type: CC BY 4.0

Affiliation: Abdul Wali Khan University Mardan, Çankaya University

Abstract

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science.

Highlights

The modern, broadly considered concept of fractional calculus was developed from a question raised by L’Hospital to Gottfried Wilhelm Leibniz in 1695
Researchers have recognized that fractional-order differential equations contributed, in a natural way, to the study of different physical problems, such as diffusion processes, signal processing, viscoelastic systems, control processing, fractional stochastic systems, biology and ecology, quantum mechanics, wave theory, biophysics, and other research fields [3,4]
Non-linear Partial differential equations (PDEs) are important tools that can be used in various fields such as plasma physics, mathematical biology, solid state physics, and fluid dynamics for modeling nonlinear dynamic phenomena [5]

Summary

Introduction

The modern, broadly considered concept of fractional calculus was developed from a question raised by L’Hospital to Gottfried Wilhelm Leibniz in 1695. Μ(α, η ) and ν(α, η ) describe the straight velocity and height, which deviate from the equilibrium situation of the fluid, respectively, and p and q are constants expressed in various diffusion forces Investigating solutions to such nonlinear PDEs over the last several decades it is an important research area [10]. Several scientists have developed numerous mathematical techniques to explore the approximate solutions to nonlinear PDEs. Aminikhah and Biazar [11] used the HPM (homotopy perturbation method) to solve the coupled model of Brusselator and Burger equations. Noor and Mohyud-Din [12] utilized HPM to examine the solutions of different classical orders of PDEs. Ahmad et al [5] studied a coupled scheme result of WBKEs by the Adomian decomposition method (ADM). This article introduces an approximate analytical solution of a multi dimensional, time fractional model of the Whitham–Broer–Kaup equation by implementing NDM and q-HATM

Preliminaries Concepts

Fundamental Idea of q-Homotopy Analysis Transform Method

Numerical Examples

Conclusions