ON THE SOLUTION OF THE VAN DER POL EQUATION

Abstract

Ordinary differential equations play a major and prominent role in all fields of science due to their wide of applications in physics, engineering, and biology as an approximate model or as a result of physical laws for a phenomenon. The nonlinear behavior of the real world makes the nonlinear ordinary differential equations the key that determines the relationship between all the variables that describe that phenomenon. Closed-form solution of nonlinear ordinary differential problems are rarely obtainable or impossible, in this case, mathematician deals with qualitative methods to show stability, periodicity and different properties of solutions without solving these nonlinear ordinary differential equations, another different point of view is to deal with numerical techniques to get an approximate solution, and finally searching for an analytical approximate solution is another way to deal with the subject, which is the main goal of this thesis taking into account the classical well known Van der Pol second order non–linear ordinary differential equation (VDPDE) which appears in the study of nonlinear damping. The Van der Pol oscillator was first introduced by the Dutch engineer and physicist Balthasar van der Pol. It has been used in the analysis of a vacuum-tube circuit among other practical problems in engineering as a basic model for oscillatory processes in physics, electronics, biology, neurology, sociology, and economics. In this work, we will study the behavior of the Van der Pol equation mathematically, and see some analytical approximate methods to solve the Van der Pol equation. Van der Pol oscillator is a non-conservative oscillator with nonlinear damping governed by the following second-order ordinary differential equation d 2x dt2 − ε(1 − x 2 ) dx dt + x = 0, this equation is equivalent to the first order nonlinear system x˙ = y, ˙y = ε(1 − x 2 )y − x. The first chapter is devoted to basic concepts, definitions and some prerequisites in the 3 CONTENTS subject of ordinary differential equations and their solutions, and we give also some necessary tools which are needed for the next chapters of our thesis to solve the Van der Pol equation. In the second chapter,contains three sections the first one we apply the G0 /G expansion method to determine some general solutions to the Van der Pol equation x 00 + ε (x 2 − λ) x 0 + α x = 0 where α, λ and ε are real parameters. Using the change of variable x = √w, we convert the Van der Pol equation into a second order nonlinear differential equation with respect to w. Finally, we apply the G0 /G method to the new equation to find two families of solutions. In the second section we use the first order approximation perturbation method to find the approximate solution by substituting x(t) = x0 + αx1 in the van der Pol equation x” − α(1 − x 2 )x 0 + x = 0, then we collocate the terms with same power of α and we equating the terms which multiplying by higher power α to zero we get two coupled second order differential equations with respect to x0(t) and x1(t) to get the solution of the van der Pol equation on the form x(t) = x0 + αx1. The averaging method is used in the last section to prove that the van der Pol equation has a period solution and studying its stability, furthermore the periodic orbits in this case is an isolated one which means that the Van der Pol oscillator has a limit cycle. In the final chapter the homotopy perturbation method is considered and used to give an approximate analytical solution for the Van der Pol differential equation with different boundary conditions.