1550078481-Ordinary_Differential_Equations__Roberts_

76 Ordinary Differential Equations

finite rectangle in the xy-plane will guarantee that the IVP (1) has a unique
solution ¢(x) on some interval I containing xo. Recall that the solution <f;(x)
must be defined, continuous, and differentiable at least once on the interval I;
that ¢(x) must satisfy the differential equation <f;'(x) = f(x, <f;(x)) on I ; and
that <f;(x) must satisfy the initial condition <f;(xo) =Yo·
A "numerical solution" of the IVP (1) is a discrete approximation of the
solution. A numerical solution is, in fact, a finite set of ordered pairs of rational
numbers, (xi, Yi) for i = 0, 1, 2, ... , n , whose first coordinates, Xi, are distinct
points in the interval I and whose second coordinates, Yi, are approximations
to the solution <Pat Xi- that is, Yi ~¢(xi)· A numerical method for solving

the IVP (1) is an a lgorithm which chooses points Xi in I such that Xo < X1 <

x 2 < · · · < Xn and determines corresponding values Yo, Yi, Y2, ... , Yn such that

Yi approximates ¢(xi)· Figure 2.11 illustrates the relationship between the
solution ¢( x) of the IVP ( 1) and a numerical approximation F = { (Xi, Yi) I i =

0, 1, ... , n}. The solution <f;(x) is represented by the so lid curve which appears

in Figure 2.11. The numerical approximation is represented by the set of dots

{(xi, Yi) Ii = 0, 1, ... , n} which appear in the graph.

(X2, Y2)

Figure 2.11 Solution ¢(x) of the IVP(l) and a Numerical Approximation

Several different measures of error are used when specifying the accuracy

of a numerical so lution. Three are defined below.

DEFINITIONS Absolute Error, Relative Error, and

Percentage Relative Error

The absolute error at Xi is I¢( Xi) - Yi I·

The relative error at Xi is l<P(xi) - Yil/l<P(xi)I provided ¢(xi)# 0.

The percentage relative error at Xi is lOOx 1 ¢(xi)-Yil/1¢(xi)I provided

¢(xi) # 0.