1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 83

Just as we found that the IVP (1) is equivalent to the integral equation (2),

we find the IVP y' = f(x, y); y(x 1 ) = y 1 is equivalent to the integral equation

y(x) = Y1 + 1 ~ f(t, y(t)) dt.

Using the constant function y 1 to approximate y(t) and the constant function

f(x1,Y1) to approximate the integrand f(t,y(t)) on the interval [ x 1 ,x 2 ], we

obtain the following approximation to the exact solution at x 2

Proceeding in this manner, we find the following general recursion for this

numerical approximation method to be

(10)

Equation (10) is known as Euler's method or the tangent line method.

We note that Euler's method is the Taylor series expansion method of
order 1, so the local discretization error is

(11)

where~ E (xn, Xn+i) and hn = Xn+i -Xn· Equation (10) is a simple formula
to use, especially for hand calculation; but, in general, it is not very accurate.
The error of equation (11) is a local error- that is, t he error per single step.
As one moves away from the initial point xo, the total error usually builds
up. The total error at any point is composed of the accumulated local error

and round-off error. It can be shown that if the exact solution of the IVP ( 1),

¢(x), has a continuous second derivative on [xo, xn] and if lfy(x, y)I :::; Land

lyC^2 l(x)I:::; Y for x E [xo,xnJ, then the global discretization error at Xn,

which is Yn - </>(xn), is bounded by

(12)

for a fixed stepsize h. Suppose we have reasonable estimates for L and Y on the
interval [x 0 , Xn] and we wish to maintain a specified accuracy A throughout
the interval [x 0 , Xn]· Then we can require that

Solving the right-most inequality for h, we usually obtain an underestimate
of the stepsize to use throughout the interval [xo, xn] in order to maintain the
specified accuracy. It should be noted that global discretization error bound in