Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

Hegagi Mohamed Ali,Sameh Askar,Hijaz Ahmad,Ismail Gad Ameen

doi:10.3390/fractalfract6010032

Abstract

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions.

Highlights

Mathematical models within fractional calculus (FC) have been widely used in various fields of natural science and engineering
This confirms that the approximate solutions obtained by the modified generalized Mittag–Leffler function method (MGMLFM) are rapidly converging to the exact solutions and this is explained in the following tables
The Laplace Adomian decomposition method (LADM) and MGMLFM solutions were presented at different values of α and for classical case (i.e., α = 1), which showed a highly coincide with the exact solutions for all considered problems

Summary

Introduction

Mathematical models within fractional calculus (FC) have been widely used in various fields of natural science and engineering. The GMLFM is applied to find analytical and approximate solutions for nonlinear systems that have applications such as the smoking model [31,32], Lorenz system [33], Riccati differential equations [34], and so on. We can say that the LADM demonstrates how the Laplace transform may be combined with the ADM to obtain an analytic approximate solution of nonlinear differential equations.

Preliminaries

Idea of the Used Methods

Analysis of the MGMLFM

LADM for System of FDEs

Applications and Results

Conclusions