Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

Rajeev Kumar,Sanjeev Kumar,Shrishty Jain,Sukhneet Kaur

doi:10.48185/jfcns.v2i2.315

Abstract

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

Highlights

Non-linear partial differential equations (PDEs) have been used extensively to model many real world problems
We have resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform
The present work aims at finding the exact solution of the following time fractional generalized Korteweg-de Vries equation (TFg KdVe):

Summary

Introduction

Non-linear partial differential equations (PDEs) have been used extensively to model many real world problems. The present work aims at finding the exact solution of the following time fractional generalized Korteweg-de Vries equation (TFg KdVe):. For this we have explored the method discussed in [12, 13, 14]. In 2012, Sahadevan and Bakkyaraj [15], had derived the Lie point symmetries of TFg KdVe(1.1) and used the obtained symmetries to transform it into non-linear FODE as follows:. They concluded that equation (1.1) cannot be solved in general except for α = p = 1.

Preliminaries

Power Series Solution using Erdèyli-Kober Operator

Traveling wave solution in sense of modified Riemann-Liouville derivative

Convergence Analysis

Traveling Wave Solution in Sense of Conformable Derivative

Conclusion