Euler's Method  Numerical O.D.E.'s  Background

Notation

Given the initial value problem (5.1), there are two courses of action that we are concerned with. One is to find a closed-form expression for the true solution, y(t), which satisfies the differential equation as well as the specified initial condition. Second is to find a computed approximation to the true solution based upon the inputs f(t,y) and  .

    For the latter approach, some additional notation is required. The techniques discussed here to approximate the true solution will do so by approximating y(t) only at a discrete set of points,

where the true solution that we seek exists for each t in some specified interval  .

    For simplicity, the methods discussed in subsequent sections will consider this discrete set of points to be equally spaced. Specifically

represents the ``step size'' that will be taken along the time axis. At each point of this collection, an approximate solution is obtained, which will be denoted by

In the sections which follow, methods which generate approximate solutions  are proposed and analyzed.
 

Wed Oct 28 11:42:13 EST 1998