On Some Approximate Solutions of Nonlinear Physical and Biological Problems

Abstract

Most of the perturbation methods are developed to find the periodic solutions of nonlinear systems with small nonlinearities, transients are not considered. In 1947, first Russian scientists Krylov and Bogo Liubov introduced a perturbation method to discuss the transient' s response in the second order autonomous differential systems with small nonlinearities and this method is well known as "an asymptotic averaging method" in the theory of nonlinear oscillations. Later, this method has been amplified and justified by Bogo Liubov and Mitropolskii in 1961 and this extended method is known as the KBM method in literature. In this dissertation, we have presented an analytical technique based on He's homotopy perturbation technique and the extended form of the KBM method to investigate the solutions of second order strongly nonlinear physical and oscillating processes in biological systems with significant damping effects. Also, we have extended the KBM method to investigate the weakly third and fourth order nonlinear systems with slowly varying coefficients and damping effects. Firstly, second order damped nonlinear autonomous differential systems are considered and He's homotopy perturbation and the KBM methods have been extended to Duffing type strongly nonlinear physical problems with small damping effects. Then the method has been applied to find the analytical approximate solution of damped oscillatory nonlinear systems with slowly varying coefficients with strong nonlinearity. Further, this method has been developed to solve second order strongly nonlinear oscillating processes in biological system with small damping effects. We have also extended the homotopy perturbation technique to find the second approximation of second order strongly nonlinear differential systems with damping effects. We have extended the KBM method to determine the second approximation of third order weakly nonlinear damped oscillatory systems under some special conditions. Lastly, a unified KBM method has been presented to obtain the analytical approximate solution of a fourth order ordinary weakly nonlinear differential equation with varying coefficients and large damping, when a pair of eigen-values of the unperturbed equation is a multiple of the other pair or pairs. The methods have been illustrated by several examples.