Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

Muhammad Shakeel,Nehad Shah,Jae Chung

doi:10.3390/fractalfract6010024

Abstract

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Highlights

Fractional calculus is a sort of fractional differential equation that resembles a large sweeping differential equation [1]
Reactiondiffusion-advection equations are partial differential equations that describe the evolution of a substance in a medium described by spatial coordinates, involving sheltered transport according to a physical or chemical force represented by a velocity vector, diffusion, or irregular motion of the material molecules in the medium, and reaction with other constituents present in the medium represented by a reaction vector
The explicit solutions represented various types of solitary wave solutions according to the variation of the physical parameters

Summary

Introduction

Fractional calculus is a sort of fractional differential equation that resembles a large sweeping differential equation [1]. Reactiondiffusion-advection equations are partial differential equations that describe the evolution of a substance (e.g., a drug) in a medium described by spatial coordinates, involving sheltered transport (or advection) according to a physical or chemical force represented by a velocity vector, diffusion, or irregular motion of the material molecules in the medium, and reaction (e.g., chemical) with other constituents present in the medium represented by a reaction vector They demonstrate physical properties such as the singular periodic wave solution, singular single-soliton solution, and singularly double periodic wave solution. The dynamics of waves in the fractional Schrödinger equation with harmonic potential were studied by Zhang et al [12] To explore such dynamics, they used an analytical technique with fractional Laplacian derivative and compared it to numerical simulation. A novel (G0 /G2 )-expansion method is introduced in this article, to solve the MCH equation of fractional order in the sense of Jumarie [33], and we discover a large number of new families of precise solutions.

The Methodology

Application

Results and Discussion

Conclusions