Approximate analytical solutions of Schnakenberg systems by homotopy analysis method

Abstract

In this paper, the homotopy analysis method (HAM) has been employed to obtain analytical solution of a two reaction–diffusion systems of fractional order (fractional Schnakenberg systems) which has been modeling morphogen systems in developmental biology. Different from all other analytic methods, HAM provides us with a simple way to adjust and control the convergence region of solution series by choosing proper values for auxiliary parameter h. The fractional derivative is described in the Caputo sense. The reason of using fractional order differential equations (FOD) is that FOD are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived of the fractional system are of a more general nature. Respectively, solutions of FOD spread at a faster rate than the classical differential equations, and may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.