On the Solution of Some Dynamical Problems of Fractional Order Arising in Mathematical Physics

Abstract

This study explains soliton solutions of nonlinear fractionl partial differential equations (FPDEs) arising in mathematical physics. Solitons are analytic solutions and achieve a big part in abundant engineering and scientific phenomena. An effort has been made to find out the exact solitary wave solutions of several well-known FPDEs. Finding their new exact solutions is not only exciting mathematically, it is also a service towards a better understanding of the phenomena. These FPDEs are used to model most of the natural problems, such as physical, chemical, fluid dynamics, biological and other fields of engineering. Exact solitary wave solutions of nonlinear dynamical problems of fractional order arising in mathematical physics has been obtained in this work. We studied some well-known physical models to find the solitary wave solutions such as the fractional Newell-Whitehead-Segel (NWS) equation, fractional Foam Drainage (FD) equation, fractional Gardner equation, time-fractional Fornberg–Whitham (FW) equation, fractional modified Boussinesq equation, fractional Hirota-Satsuma (HS) coupled KdV equation, long wave equation, Drinfeld-Sokolov-Wilson (DSW) equation, fractional Simplified Modified Camassa-Holm (MCH) equation, fractional Caudrey-Dodd-Gibbon (FCDG) equation, fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation, fractional Modified Unstable Schrödinger equation (MUSE), fractional Sharma-Tasso-Olever (STO) equation, fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation, fractional perturbed Gerdjikov-Ivanov (pGI) equation. All these dynamical problems have been resolved by some analytic techniques like fractional variational iteration method (VIM), conformable variational iteration method (CVIM), conformable fractional reduced differential transform method (CFRDTM), Exp-function method, (G/G)-expansion method, (G/G,1/G) -expansion method, tanh method, and tanh-coth method. It has come into observation that these methods construct generalized solitons, traveling wave, solitary wave, hyperbolic, periodic, trignometic, and rational solution. These types of solutions are of great interest for mathematicians and engineers working in the fields of fiber optics and wave propagation etc. We have used some transformations to convert the FPDEs, to the corresponding ordinary differential equations (ODEs). To extract soliton solutions to these FPDEs, certain balancing principles are implemented. Comparison of the obtained solutions with the existing results in the literature are also given in this work. New consequences comprise of the modifications (Page 2 of 190) in the mathematical techniques in such manner that it delivers us with multiple solitary wave solutions and optical solutions of nonlinear dynamical problems, which have never been explored earlier by these techniques. We have determined some totally novel solutions to the problems under consideration as well as we have managed to generalize many already existing solutions. So the obtained new solutions are more generalized and are supposed to give better model of the real-word problems to which the FPDEs relates. It is also predictable from the current study that the computer simulator used to wave propagation can be enriched on the basis of our exact solutions. Moreover a comparative study with the existing results described for verification and validation of our obtained results. These obtained solutions will demonstrate the physical behavior of the problem in graphical form. Descriptions of the parameters and the values that have used to simulate the waves are also provided. For graphical representations mathematical softwares (Mathematica, Maple) have been used.