Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids

Haiyong Qin,Raghda A M Attia,Mostafa M A Khater

doi:10.1155/2020/3498796

Abstract

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.

Highlights

Fractional nonlinear evolution equation is one of the noticeable branches of science, in recent years
Fractional calculus has a great profound physical background where it is able to formulate many various phenomena in distinct fields such as physics, mechanical engineering, economics, chemistry, signal processing, food supplement, applied mathematics, quasichaotic dynamical systems, hydrodynamics, system identification, statistics, finance, fluid mechanics, solid-state biology, dynamical systems with chaotic dynamical behavior, optical fibers, electric control theory, and economics and diffusion problems. e mathematical modeling of these phenomena will contain a fractional derivative which provides a great explanation of the nonlocal property of these models since it depends on both historical and current states of the problem in contrast with the classical calculus which depends on the current state only
Based on the importance of this kind of calculus, many definitions have been being derived such as conformable fractional derivative, fractional Riemann–Liouville derivatives, Caputo, and Caputo–Fabrizio definition [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. ese definitions have been being employed to convert the fractional nonlinear partial differential equations to nonlinear integer-order ordinary differential equation and the computational and numerical schemes can be applied to get various types of solutions for these models and the examples of these schemes [18,19,20,21,22,23,24,25,26,27,28,29,30]

Summary

Introduction

Fractional nonlinear evolution equation is one of the noticeable branches of science, in recent years. E auxiliary equation of the mK method is given by It leads to the equalling of the mK auxiliary equation with many other analytical methods, but the mK method can obtain more solutions than most of them.

Results

Conclusion