1550078481-Ordinary_Differential_Equations__Roberts_

100 Ordinary Differential Equations

Table 2 .7 Adams-Bashforth approximations to the solution of the IVP

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

Runge-Kutta Adams-Bashforth formulas

starting m= 1 m=2 m=3 Exact

Xn values Yn Yn Yn solution

.0 1.0 1.0

.1 1.11034 1.11034

.2 1.24280 1.24189 1.24281

.3 1.39971 1.39766 1.39963 1.39972

.4 1.58021 1.58345 1.58364 1.58365

.5 1. 79236 1.79711 1.79742 1.79744

.6 2.03720 2.04374 2.04420 2.04424

.7 2.31816 2.32682 2.32745 2.32751

.8 2.63903 2.65018 2.65100 2.65108

.9 3.00397 3.01804 3.0 1911 3.01921

1.0 3.41762 3.43509 3.43644 3.43656

2 .4.2.2 Nystrom Meth o d s

A set of formulas called Nystrom formulas results by selecting r = n, p = 1,
and q = 1. After calculating the interpolating polynomial and integrating one
obtains the foll owing formulas.

rri Nystrom formulas

(^0) Yn+l = Yn-1 + 2hj n
(^1) Yn+l = Yn-1 + 2hj n
2
h(7 Jn - 2j n-1 + J n-2)
Yn+l = Yn-1 + 3
3
h(8j n - 5j n-1 + 4j n-2 - f n-3)

Yn+l = Yn-1 + 3

These formulas were derived by the Finnish mathematician E. J. Nystrom and
published in 1925. The formula for m = 0- which is the same as the formula