Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function

This paper proposes a new efficient approach for obtaining approximate series solutions to fourth-order two-point boundary value problems. The proposed approach depends on constructing Green’s function and Adomian decomposition method (ADM). Unlike existing methods like ADM or modified ADM, the proposed approach avoids solving a sequence of nonlinear equations for the undetermined coefficients. In fact, the proposed method gives a direct recursive scheme for obtaining approximations of the solution with easily computable components. We also discuss the convergence and error analysis of the proposed scheme. Moreover, several numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed approach. The numerical results reveal that the proposed method is very effective and simple.     

Solving nonlinear reaction–diffusion problem in electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme systems using Taylor series method

A mathematical model of electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme is discussed. The model involves the coupled system of non-linear reaction–diffusion equations of substrate and hydrogen ion. The non-linear term in this model is related to the Michaelis–Menten reaction of the substrate and non-Michaelis–Menten kinetics of hydrogen ion. The approximate analytical expression of concentration of substrate and hydrogen ion has been derived by solving the non-linear reactions using Taylor’s series method. Reaction rate and effectiveness factor are also reported. A comparison between the analytical approximation and numerical solution is also presented. The effects of external mass transfer coefficient and the electrostatic potential on the overall reaction rate were also discussed.