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Solving the fractional heat-like and wave-like equations with variable coefficients utilizing the Laplace homotopy method

Muhammad Nadeem (Department of Mathematical Science, Dalian University of Technology, Dalian, China)
Shao-Wen Yao (School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 22 May 2020

Issue publication date: 12 January 2021

93

Abstract

Purpose

This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that the results are very close to the exact solution.

Design/methodology/approach

First with the help of some basic properties of fractional derivatives, a scheme that has the capability to solve fractional partial differential equations is constructed. Then, TFH-L and W-L equations with variable coefficients are solved by this scheme, which yields results very close to the exact solution. The derived results demonstrate that this scheme is very effective. Finally, the convergence of this method is discussed.

Findings

A traditional method is combined with the Laplace transform to construct this scheme. To decompose the nonlinear terms, this paper introduces the homotopy perturbation method with He’s polynomials and thus the solution is provided in the form of a series that converges to the exact solution very quickly.

Originality/value

The proposed approach is original and very effective because this approach is, to the authors’ knowledge, used for the first time very successfully to tackle the fractional partial differential equations, which are of great interest.

Keywords

Acknowledgements

The authors thank the Editor-in-Chief and unknown referees for the fruitful comments and significant remarks that helped them in improving the quality and readability of the paper, which led to a significant improvement of the paper. This work was supported by National Natural Science Foundation of China (No. 71601072) and Key Scientific Research Project of Higher Education Institution in Henan Province (No. 20B110006).

Citation

Nadeem, M. and Yao, S.-W. (2021), "Solving the fractional heat-like and wave-like equations with variable coefficients utilizing the Laplace homotopy method", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 31 No. 1, pp. 273-292. https://doi.org/10.1108/HFF-02-2020-0111

Publisher

:

Emerald Publishing Limited

Copyright © 2020, Emerald Publishing Limited

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