Application of Runge-Kutta methods for solving Nonlinear Systems of Ordinary Diffferential Equations to Unemployment Model.

Abstract

Youth unemployment is on the rise globally and it poses huge economic, social and political challenges which results to unsustainable growth of the country’s economy. The main aim of this thesis is to focus on youth unemployment in Kenya by developing a mathematical model consisting of rst order systems of Ordinary Di erential Equations (ODEs) and solve it numerically and to give results from mathematical perspective.Firstly, we give introduction to Unemployment problem, ODEs, meaning of a solution to an ODE and also conditions for a problem to satisfy for it to be referred as well posed. We then discuss in brief analytical solution and also the numerical approach of nding a solution to an ODE. A consideration of error analysis,convergence,consistency and stability of numerical methods in general is discussed. We consider Runge-Kutta (RK) methods of di erent orders, derivation of Euler methods, second order,third order and RK method of fourth order are obtained and a brief formula for fth order is given.Afterwards,error analysis for RK methods as well as the stability analysis are discussed. After describing the methods for a single equation, we focus on RK methods for solving systems of ODEs since it is our area of interest. We also analyze the stability for one-step and multi-step methods for ODEs. Once we are fully equipped with the necessary tools,we now obtained the numerical solution the model using RK method of fourth order and discuss the results.