Mohammad Zurigat, Shaher Momani and Ahmad Alawneh, The multistage homotopy analysis method: Application to a biochemical reaction model of fractional order, International Journal of Computer Mathematics, 91 (5) (2014), 1030-1040.

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme?substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge?Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order.