5 - Solution of Differential Equations

Abstract

This chapter describes and develops algorithms for the solution of initial value ordinary differential equations and for the solution of systems of initial value ordinary differential equations. The Euler, trapezoidal, and Runge–Kutta methods are discussed and implemented in Matlab and compared. A range of predictor–corrector methods are given together with error estimates. Standard Matlab functions are applied to the solution of systems of equations. Specific applications include Zeeman's heart model, the Lorenz equations, the predator–prey problem, and neural networks. The chapter includes a discussion of the stability and accuracy of the methods used. It is shown how higher-order differential equations may be reduced to systems of first-order differential equations. The problem of stiff differential equations is considered and methods for their solution considered and applied. Problems and solutions are provided.