Numerical solutions of time‐fractional Burgers equations: A comparison between generalized differential transformation technique and homotopy perturbation method

Abstract

Purpose

The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

In these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.

Findings

The paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.

Research limitations/implications

Burgers and coupled Burgers equations with time‐fractional derivative used.

Practical implications

The implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.

Originality/value

The numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

Citation

Alam Khan, N., Ara, A. and Mahmood, A. (2012), "Numerical solutions of time‐fractional Burgers equations: A comparison between generalized differential transformation technique and homotopy perturbation method", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 22 No. 2, pp. 175-193. https://doi.org/10.1108/09615531211199818