1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = f(x, y); y(c) = d 97

e. Find the expli cit solution of the IVP y' = y/x + 2; y(l) = l. On

what interval does the solution exist?

f. Produce a table of values comparing the numerical solution values

generated in parts a-d with the exact solution values on the interval

[l , 2].

2.4.2 Multistep Methods

Let z (x) be a function which is defined on some interval containing the

points xo, x 1 , ... , Xn· It is well known that there exists only one polyno-

mial, p(x), of degree less than or equal to n for which p(xi) = z (xi) for

i = 0, 1, ... , n. The polynomial p(x) is called the interpolating polyno-

mial. There are many ways to write an expression for the interpolating poly-

nomial; however , we shall not present any of those expressions here. We are

only interested in the fact that an interpolating polynomial exists and that it

is unique.

Consider again the IVP (1) y' = f(x, y); y(x 0 ) = y 0. Suppose that we

have used some single-step method to produce the approximations Yi to the

exact solution ¢ (xi) for i = 1, 2, ... , n, where the Xi's are equally spaced. Each

single-step method computes y~ = f(xi, Yi) in order to produce Yi · Henceforth,

let y~ = f(xi, Yi) = k In deriving single-step methods, we integrated the

differential equation y' = f(x, y) of the IVP (1) over the interval [xn, Xn+i] to

obtain the integral equation

Y(Xn+i) = y(xn) + 1 :n+i J(t, y(t)) dt

and then we approximated f on the interval [xn, Xn+i] and integrated. Mul-

tistep methods are usually derived by integrating the differential equation

y' = f(x, y) from Xn-p to Xn+q where p, q ~ 0 and by approximating the

integrand f on the interval [xn-p, Xn+q] by the interpolating polynomial p(x)

which interpolates fat them+ 1 points Xr-m, Xr-m+1, ... , Xr-1, Xr, where

r =nor r = n + l. See Figure 2.13. If r = n, the resulting formula is said to
be open; whereas, if r = n + 1, the r esulting formula is called closed. Open
formulas are explicit formulas for Yn+l · Closed formulas, on the other hand,
are implicit formulas for Yn+l ·

2.4.2.1 Adams-Bashforth Methods

The English mathematician and astronomer John Couch Adams (1819-

1892) introduced multistep methods prior to the introduction of Runge-Kutta

methods. Adams is p erhaps better known as an astronomer than a mathe-

matician. In 1845, he predicted the existence and orbit of a new planet - the

planet Neptune. His prediction was based upon an analysis of the perturba-

tions of the orbit of Ura nus. In 1882 , J. C. Adams and Francis Bashforth

published the Adams-Bashforth methods for numerical integration in an ar-

ticle on the theory of capillary action.