Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel

Abstract

In applied science, the exact solution (when available) for any physical model is of great importance. Such exact solution not only leads to the correct physical interpretation, but also very useful in validating the approximate analytical or numerical methods. However, the exact solution is not always available for the reason that many authors resort to the approximate solutions by using any of the analytical or the numerical methods. To ensure the accuracy of these approximate solutions, the convergence issue should be addressed, otherwise, such approximate solutions inevitably lead to incorrect interpretations for the considered model. Recently, several peristaltic flow problems have been solved via the homotopy perturbation method, which is an approximate analytical method. One of these problems is selected in this paper to show that the solutions obtained by the homotopy perturbation method were inaccurate, especially, when compared with the exact solutions provided currently and also when compared with a well known accurate numerical method. The comparisons reveal that great remarkable differences have been detected between the exact current results and those approximately obtained in the literatures for the temperature distribution and the nano-particle concentration. Hence, many similar problems that have been approximately solved by using the homotopy perturbation method should be re-investigated by taking the convergence issue into consideration, otherwise, the published results were really incorrect.