Fractional View Analysis of Third Order Kortewege-De Vries Equations, Using a New Analytical Technique

Rasool Shah,Muhammad Arif,Umar Farooq,Poom Kumam,Hassan Khan,Dumitru Baleanu

doi:10.3389/fphy.2019.00244

Abstract

In the present article, fractional view of third order Kortewege-De Vries equations is presented by a sophisticated analytical technique called Mohand decomposition method. The Caputo fractional-derivative operator is used to express fractional derivatives, containing in the targeted problems. Some numerical examples are presented to show the effectiveness of the method for both fractional and integer order problems. The solution graphs have revealed that if the sequence of fractional-orders is approaches to integer order, then the fractional-order solutions of the problems are converge to an integer order solution. Moreover, the proposed method is straight forward and easy to implement and therefore can be used for other non-linear fractional-order partial differential equations.

Highlights

The class of partial differential equations known as Korteweg-De Vries (KDV) equation which play a vital role in the diverse field of physics such as fluid mechanics, signal processing, hydrology, viscoelasticity and fractional kinetics [1, 2]
A partial differential Kortewege-De Vries equation of third order is applied to study the non-linear model of water waves in superficial canal certain namely canal [3], in the time when wave in water was of important concentration in applications in navigational design and for the awareness of flood and tides [4, 5]
We considered the third order time fractional KDV equation in the form [1]

Summary

INTRODUCTION

The class of partial differential equations known as Korteweg-De Vries (KDV) equation which play a vital role in the diverse field of physics such as fluid mechanics, signal processing, hydrology, viscoelasticity and fractional kinetics [1, 2]. A partial differential Kortewege-De Vries equation of third order is applied to study the non-linear model of water waves in superficial canal certain namely canal [3], in the time when wave in water was of important concentration in applications in navigational design and for the awareness of flood and tides [4, 5]. We have applied the Mohand transform with decomposition procedure for the analytical treatment of time fractional KDV equation. We present some related definitions of fractional calculus and basic concepts of Mohand transform. We present some basic necessary definitions and preliminaries concepts related to fractional calculus and Mohand transform.

IMPLEMENTATION OF MOHAND TRANSFORM

APPLICATIONS AND DISCUSSION

RESULTS AND DISCUSSION

CONCLUSION

DATA AVAILABILITY STATEMENT