The multistage homotopy analysis method: application to a biochemical reaction model of fractional order

Abstract

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.

Keywords:

• fractional differential equations
• enzyme kinetics
• homotopy analysis method
• numerical solution
• mathematical modelling
• multistage homotopy method
• Runge–Kutta method

2010 AMS Subject Classifications:

• 34A30
• 92Bxx
• 33F05
• 34K28
• 93A30
• 49Mxx