1550078481-Ordinary_Differential_Equations__Roberts_

m

0

1

2

3

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

Adams-Bashforth formulas

Yn+i = Yn +hf n

h(3J n - fn-1)

Yn+l = Yn + 2

h(23J n - l6J n-1 + 5J n-2)

Yn+l = Yn + 12

h(55fn - 59fn-l + 37 fn-2 - 9fn-3)

Yn+l = Yn + 24

Error

h2y(2) (~)

2

5h3y(3)(~)

12

9h4y(4)(0

24

251h^5 y(^5 ) (~)

720

99

In each case ~ E (xm-n, Xn)· The formula for m = 0 is a single-step

method-Euler's method. Notice that each method utilizes f evaluated at

m + 1 points and has a local discretization error of order hm+^2. Hence, each
formula requires m + 1 starting values.

EXAMPLE 5 Adams-Bashforth Approximations of the Solution

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

Find an approximate solution to the initial value problem

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

on the interval [O, l] using a constant stepsize h = .1 and Adams-Bashforth
formulas for m = 1, 2, 3. Use starting values obtained from the fourth order
Runge-Kutta method.

SOLUTION

Table 2.7 contains the Adams-Bashforth approximations form= 1, 2, 3 to
the solution of the IVP (7) on the interval [O, 1] obtained using a constant
stepsize of h = .1. For m = 1 the initial condition, y(O) = 1, and one
fourth order Runge-Kutta value y 1 = 1.11034 were used to start the Adams-
Bashforth method. For m = 2 the initial condition, y(O) = 1, and two fourth
order Runge-Kutta values y 1 = 1.11034 and Y2 = 1.24280 were used to start
the Adams-Bashforth method.