The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

Abstract

Purpose

The purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems.

Design/methodology/approach

In Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution.

Findings

The HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models.

Originality/value

The HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus.

Citation

Nadeem, M. and He, J.-H. (2022), "The homotopy perturbation method for fractional differential equations: part 2, two-scale transform", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 32 No. 2, pp. 559-567. https://doi.org/10.1108/HFF-01-2021-0030