Homotopy perturbation method: when infinity equals five

I discuss a recent application of homotopy perturbation method to a heat transfer problem. I show that the authors make infinity equal five and analyze the consequences of that magic. There has recently been great interest in the application of several approximate procedures, like the homotopy perturbation method (HPM), the Adomian decomposition method (ADM), and the variation iteration method (VIM), to a variety of linear and nonlinear problems of interest in theoretical physics [1–15]. From now on I will refer to those variation and perturbation approaches as VAPA. In a series of papers I have shown that most of the VAPA results are useless, nonsensical, and worthless [16–20]. In many of those papers the authors try to solve nonlinear problems by means of elaborated VAPA implementations that merely produce the Taylor expansion of the solutions. Of course, such approximate expressions do not give the overall picture of the dynamics, and the authors are merely content with a description of the initial stages of the evolution which do not tell us anything relevant about the 1 e–mail: [email protected] Preprint submitted to Elsevier 21 October 2008 process. Other authors solve the Schrödinger equation and obtain trivial unphysical solutions that are not square integrable. As an example I mention two great feats of the VAPA users: the expansion of exponential functions of the form e [15] and a prey–predator model that predicts a negative population of rabbits [14] (see also my comments [18, 19]). However, my criticisms have not been welcome because they lack “the qualities of significant timeliness and novelty that we are seeking in this journal” and for that reason they remain unpublished outside arXiv. The purpose of this article is the analysis of a recently published paper that certainly meets the criterion of timeliness and novelty sought in that journal. Esmaeilpour and Ganji [5] applied homotopy perturbation method (HPM) to the solution of the problem of forced heat convection over an horizontal flat plate. After some algebraic manipulation of the Navier–Stokes equations they obtained two coupled nonlinear differential equations: [5] f ′′′(η) + 1 2 f(η)f ′′(η)= 0 εθ′′(η) + 1 2 f(η)θ′(η)= 0 (1) with the boundary conditions f(0)= f ′(0) = 0, f ′(∞) = 1 θ(0)= 1, θ(∞) = 0 (2) The HPM yields series of the form