Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 6

SYSTEMS OF DIFFERENTIAL EQUATIONS

In this chapter we motivate, by means of a simple observation, the generalization to systems of equations of what we have learned for scalar differential equations.

Consider the initial value problem for a system of differential equations

(6.1)

where y is an N-component vector and f is an N-component vector-valued function:

The simplest systems are linear systems with constant coefficients:

(6.2)

Here A is a constant N x N-matrix and F has N components.

Let us assume that A has a complete set of eigenvectors; that is, there exists a transformation S that transforms A into a diagonal matrix λ:

Exercise 6.1 The exponential of a matrix A ^NxN is defined via the Taylor series,

(a) Show that exp(A) is well defined (i.e., the series is always convergent). ...