1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 79

and the discretization error, truncation error, or formula error for this

method is given by

(6)

E = j(m) (C y(0) hm+l

n (m+l)! n

where ~ E (xn, Xn+i)·

If all calculations were performed with infinite precision, discretization error
would be the only error present. Local discretization error is the error

that would be made in one step, if the previous values were exact and there

were no round-off error. Ignoring round-off error, global discretization
error is the difference between the solution ef>(x) of the IVP (1) and the

numerical approximation at Xn- that is, the global discretization error is en =

Yn - ef>(xn)·

A derivation of the series that bears his name was published by the En-
glish mathematician Brook Taylor (1685-1731) in 171 5. However, the Scottish
mathematician James Gregory (1638-1675) seems to have discovered the se-
ries more than forty years before Taylor published it. And Johann Bernoulli
had published a similar result in 1694. The series was published without
any discussion of convergence and without giving the truncation error term-
equation (6).

EXAMPLE 1 Third Order Taylor Series Approximation to the

Solution of the IVP: y' = y + x ; y(O) = 1

a. Find an approximate solution to the initial value problem

(7) y' = y + x = f(x, y); y(O) = 1

on the interval [O, 1] by using a Taylor series expansion of order 3 and a

constant stepsize hn = .1.

b. Estimate the maximum local discretization error on the interval [O, l].

SOLUTION

a. From the differential equation in (7), we see that

yCll = j(x, y) = y + x, SO J (xn, Yn) = Yn + Xn.

Differentiating the equation yCll = f(x, y) = y + x three times, we find

jCll(x,y)=y(l)+l=y+x+l, so j(ll(xn,Yn)=Yn+Xn+l

j(^2 ) (x, y) = y(l) + 1 = y + X + 1, so JC^2 ) (xn, Yn) = Yn + Xn + 1

jC^3 l(x,y)=y(ll+l=y+x+l, so jC^3 l(xn,Yn)=yn+xn+l.