1550078481-Ordinary_Differential_Equations__Roberts_

88 Ordinary Differential Equations

Table 2.3 Improved Euler's approximation to the IVP

(7) y' = y + x; y(O) = 1 on [O, 1] with stepsize h = .1

Yn+l = Yn+

h h

n Xn Yn S1 S2 "2(S1 + S2) 2(S1 + S2)

0 .0 1.0 1.0 1.2 .11 1.11

1 .1 1.11 1.21 1.431 .13205 1.24205

2 .2 1.24205 1.44205 1.68625 .156415 1.39846

3 .3 1.39846 1.69846 1.96831 .183339 1.58180

4 .4 1.58180 1.98180 2.27998 .213089 1.79489

5 .5 1.79489 2.29489 2.62438 .245963 2.04085

6 .6 2.04085 2.64085 3.00494 .282289 2.32314

7 .7 2.32314 3.02314 3.42546 .322430 2.64557

8 .8 2.64557 3.44557 3.89013 .3667^85 3.01236

9 .9 3.01236 3.91236 4.40359 .415797 3.42815

10 1.0 3.42815

Analyzing the form of the recursive formula (14) might lead one to try to

devise a more general recursion of the form

in which the constants a , b, c, and d are to be determined in such a manner
that (16) will agree with a Taylor series expansion of as high an order as
possible. As we have seen the Taylor series expansion for y(xn+l) about Xn is

)

1 2

(17) y( x = y(xn) + f hn + 2 U x + fyf)hn +

~ Uxx + 2fxyf + fyyf^2 + f x fy + J; f)h~ + O(h':,)

where f and its partial derivative are all evaluated at (xn, Yn) and O(h~)
indicates that the error made by omitting the remainder of the terms in the
expansion is proportional to the fourth power of the stepsize.
Let k1 = f(xn,Yn) and k2 = f(xn + chn,Yn + dk1hn)· Using the Taylor
series expansion for a function of two variables to expand k 2 about (xn, Yn),
we obtain