%0 Journal Article
%T A New Spectral-Homotopy Perturbation Method and Its Application to Jeffery-Hamel Nanofluid Flow with High Magnetic Field
%A Ahmed A. Khidir
%J Journal of Computational Methods in Physics
%D 2013
%R 10.1155/2013/939143
%X We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method, and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the Jeffery-Hamel flow considering the effects of magnetic field and nanoparticle. Comparisons are made between the proposed technique, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the present approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method. 1. Introduction Many problems in the fields of physics, engineering, and biology are modeled by coupled linear or nonlinear systems of partial or ordinary differential equations. Compared to nonlinear equations, linear equations can be easily solved and finding analytical solutions to nonlinear problems on finite or infinite domains is one of the most challenging problems. Such problems do not usually admit closed from analytic solutions and in most cases we look for finding approximate solutions using numerical approximation techniques. Nonnumerical approaches include the classical power-series method and its variants for systems of nonlinear differential equations with small or large embedded parameters. One of these methods is the homotopy perturbation method (HPM). This method, which is a combination of homotopy in topology and classic perturbation techniques, provides us with a convenient way to obtain analytic or approximate solutions for a wide variety of problems arising in different scientific fields by continuously deforming the difficult problem into a set of simple linear problems that are easy to solve. It was proposed first by He [1¨C4]. He has successfully used the method to solve many types of linear and nonlinear differential equations such as Lighthill equation [1], Duffing equation [2], Blasius equation [5], wave equations [4], and boundary value problems [6]. Homotopy perturbation method has been recently intensively studied by many authors and they used the method for solving nonlinear problems and some modifications of this method have been published [7¨C12] to facilitate, make accurate calculations, and accelerate the rapid convergence of the series solution and reduce the size of work. Jalaal et al. [13] used HPM to investigate the acceleration motion of a vertically falling spherical particle in
%U http://www.hindawi.com/journals/jcmp/2013/939143/