1550078481-Ordinary_Differential_Equations__Roberts_

80 Ordinary Differential Equations

Substituting these expressions into equation (5) with m = 3 and hn = .1,

we obtain the recursive formula

Yn+l = Yn + (Yn + Xn)(.l) + Yn + Xn + 1 ( ) Yn + Xn + 1 ( )

2

.0 1 +

6

.00 1

=. 00516667 +. 105167Xn + l.10517Yn·

In this case, each constant was rounded to six significant digits. The

following table is the third order Taylor series approximation to the

IVP (7) on the interval [O, 1] obtained using a constant stepsize of h = .1.

All calculations were performed using six significant digits.

Third order Taylor series approximation to

the IVP: y' = y + x ; y(O) = 1 on [O, 1] with

stepsize h = .1

Xn Yn

.o 1.00000

.1 1.1103 4

.2 1.24279

.3 1.39969

.4 1.58361

.5 1.79739

.6 2.04417

.7 2.32742

.8 2.65098

.9 3. 0190 8

1.0 3.43641

b. The differential equation of the IVP (7) is linear and, in this instance, we

can find the exact so lut ion. Therefore, we could use the exact solut ion

when estimating the maximum local discretization error. However, since

we will not normally b e able to obtain the exact solution, we will do what

one must usually do in practice. We will use the information obtained

from the numerical approximation to estimate the error. Examining

the Taylor series numerical approximation values above, we see t hat

IYI < 3.45 for x E [O, l]. For x E [O, 1], we assume that IYI < 7 (which is

slightly more than twice the largest y value appearing above) and using

the triangle inequality, we see that

(8) 1JC^3 l(x,y)I ~ IYI + lx l + 1<9 for x E [O, l].

Using this upper bound in equation (6) with m = 3 and hn = .1, we

obtain the following estimate for the maximum local discretization error