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Solutions of time-fractional third- and fifth-order Korteweg–de-Vries equations using homotopy perturbation transform method

Perumandla Karunakar (Department of Mathematics, National Institute of Technology, Rourkela, India)
Snehashish Chakraverty (Department of Mathematics, National Institute of Technology, Rourkela, India)

Engineering Computations

ISSN: 0264-4401

Article publication date: 5 August 2019

Issue publication date: 12 September 2019

Abstract

Purpose

This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM).

Design/methodology/approach

HPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials.

Findings

The study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders.

Originality/value

Although third-order tfKdV and mKdV equations have already been solved by ADM and HPM, respectively, the fifth-order tfKdV equation has not been solved yet. Accordingly, here HPTM is applied to two tfKdV equations of order three and five which are used for modeling various wave phenomena. The results of third-order KdV and KdV equations are compared with existing results.

Keywords

Acknowledgements

The authors would like to thank the learned reviewers and the editor for their valuable comment(s) and suggestion(s).

Citation

Karunakar, P. and Chakraverty, S. (2019), "Solutions of time-fractional third- and fifth-order Korteweg–de-Vries equations using homotopy perturbation transform method", Engineering Computations, Vol. 36 No. 7, pp. 2309-2326. https://doi.org/10.1108/EC-01-2019-0012

Publisher

:

Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited