A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model

In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM). Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT) is applied to enlarge the convergence region and rate of solution series given by the HAM

The relationship of degenerate kernel and projection methods on Fredholm integral equations of the second kind

In this paper, we show that the degenerate kernel method for some cases, on the condition that the source function is approximated by the same way of producing degenerate kernel, becomes as a projection method. We consider two ways, including Lagrange interpolation and best approximation methods, of producing degenerate kernel approximations of more general Fredholm integral equation of the second kind. For these two ways, we show that the degenerate kernel method becomes as a Lagrange-collocation method and Galerkin method respectively

A Note on the Homotopy Analysis Method

The present work is devoted to using an analytic approach, namely the homotopy analysis method, to obtain convergent series solutions of strongly nonlinear problems. On the basis of the homotopy derivative concept described in Liao (2009) [3], a theorem is proved here which generalizes some lemmas and theorems provided in Liao (2009) [3] and Molabahrami and Khani (2007) [4]. Significant applicability of the theorem obtained here in some practical situations is demonstrated. (C) 2010 Elsevier Ltd. All rights reserved.WoSScopu