A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

Saima Rashid,Abdullah M Alsharif,Zakia Hammouch,Sobia Sultana,Aasma Khalid,Rasool Shah

doi:10.3390/sym13071254

Abstract

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

Highlights

The formulation of exact and explicit partial differential equations (PDEs) solutions is essential for a good perspective on the mechanisms of diverse physical processes
We aim to develop two efficient algorithms for estimating the approximateanalytical solutions of Korteweg-de Vries (KdV) and modified coupled KdV system (MCKdV) equations that model the dynamics of the process under investigation
Using synthetic trajectories derived from the KdV and MCKdV models, we demonstrate the validity and feasibility of the suggested algorithmic approaches to deriving the approximate-analytical solutions in a simulation study

Summary

Introduction

The formulation of exact and explicit PDE solutions is essential for a good perspective on the mechanisms of diverse physical processes. Hirota and Satsuma proposed a coupled KdV framework to address the effects of two long waves with independent dispersion correlations. It was developed as an evolution equation regulating the propagation of a one-dimensional, small-amplitude, long-surface gravity wave in a shallow water channel. The non-linear coupled system of partial differential equations (PDEs) has several applications in physical systems such as fluid mechanics, aquifers, chaos, thermodynamics, plasma physics and many more. It is demonstrated that it determines the velocity [2,3] of the KdV subsystem. The fractional-order paired KdV equations are written as follows:

Objectives

Results

Conclusion