Numerical solution to nonlinear biochemical reaction model using hybrid polynomial basis differential evolution technique

Suheel Abdullah Malik,Ihsanul Haq,Muhammad Amir,Ijaz Mansoor Qureshi

doi:10.12988/asb.2014.4520

Abstract

In this paper, we present an approximate numerical solution to the well known Michaelis-Menten nonlinear biochemical reaction system using a stochastic technique based on hybrid polynomial basis evolutionary computing. The approximate solution is expanded as a linear combination of polynomial basis with unknown parameters. The system of nonlinear differential equation is transformed into an equivalent global error minimization problem. A trial solution is formulated using a fitness function with unknown parameters. Two popular evolutionary algorithms such as Genetic algorithm (GA) and Differential evolution (DE) are used to solve the minimization problem and to obtain the unknown parameters. The effectiveness of the proposed technique is demonstrated in contrast with fourth-order Runge Kutta method (RK-4) and some well known standard methods including homotopy perturbation method (HPM), variational iteration method (VIM), differential transform method (DTM), and modified Picard iteration method (Picard-Pade). The comparisons of numerical results validate the efficacy and viability of the suggested technique. The results are found to be in sharp agreement with RK-4 compared to some popular standard methods.

Highlights

A wide spectrum of scientific phenomena appearing in real life problems are governed by nonlinear ordinary differential equations (ODEs)
The optimal values of the unknown parameters corresponding to the minimum fitness achieved by genetic algorithm (GA) and Differential evolution (DE) are provided in Table 3 and Table 4 respectively
A simple yet an efficient stochastic heuristic method based on hybrid approach of polynomial basis functions and evolutionary computation has been presented for numerically solving system of nonlinear ordinary coupled differential equations

Summary

Introduction

A wide spectrum of scientific phenomena appearing in real life problems are governed by nonlinear ordinary differential equations (ODEs). Since most of nonlinear ODEs either do not have an exact solution or obtaining the same analytically is difficult, these problems must be tackled using approximate analytical and numerical methods. Very recently Malik et al [6,7] employed heuristic technique based on hybrid genetic algorithm for numerically solving the nonlinear singular boundary value problems in physiology and the Bratu problem. Arqub et al [8] used genetic algorithm (GA) based method for solving linear and nonlinear singular boundary value problems (BVPs). Caetano et al.[9] used genetic algorithm (GA) based neural network (NN) for the solution of nonlinear ODEs arising in atomic and molecular physics. Khan et al [10] used particle swarm optimization (PSO) based NN technique for solving nonlinear

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