Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method

Abstract

Purpose

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.

Findings

HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.

Originality/value

The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

Citation

Yıldırım, A. (2010), "Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 20 No. 2, pp. 186-200. https://doi.org/10.1108/09615531011016957