1550078481-Ordinary_Differential_Equations__Roberts_

78 Ordinary Differential Equations

2.4.1 Single-Step Methods

For single-step methods, it is convenient to symbolize the numerical approx-
imation to the exact solution y = ¢(x) of the IVP (1) y' = f(x, y); y(xo) =Yo
by the recursive formula

(3)

where hn = Xn+l - Xn· The quantity hn is call ed the stepsize at Xn and it

can vary with each step. However, for computations performed by hand it is

usually best to keep the stepsize constant- that is, set hn = h , a constant,

for n = 0, 1, ....

2.4.1.1 Taylor Series Method

If y(x) has m+ 1 continuous derivatives on an interval I containing xo, then
by Taylor's formula with remainder,

(4)

yC2l(x )

y(x) = y(xn) + yCll(xn)(x - Xn) +

2

n (x - Xn)^2 + · · ·

(m) ( ) (m+l) (C)

+ Y Xn (x - X )m + Y <,, (x - X )m+l

m! n (m + 1)! n

where~ is between x and Xn· In particular, if y(x) is a so lution to the IVP (1)
and y( x) has m + 1 continuous derivatives, then from t he differential equation
in (1)

and by repeated implicit differentiation and use of the chain rule, we obtain

yC^2 l(xn) = J(ll(xn, y(xn)) = fx + fyy(l) = fx + fyf

yC^3 l(xn) = f(^2 ) (xn, y(xn)) = fx x + fxyY(l) + (fyx + fyyY(l))y(l) + fyY(^2 )

= fx x + 2fxyf + fyyf^2 + f x fy + J;J

where jCll denotes df / dx, where jC^2 l denotes d^2 f / dx^2 , and where f and its
partial derivatives are all evaluated at (xn,y(xn)). We could continue in this
manner and eventu ally write any derivative of y evaluated at Xn, y(k) (xn), up
to and including order m+l, in terms off and its partial derivatives evaluated
at (xn, y(xn)). However, it is apparent t hat the evaluation of each successive
higher order derivative by t his technique usually becomes increasingly difficult
unless the function f is very simple. Hence, one chooses min equation (4) to
be reasonably small and approximates y(xn+l) by

(5) f

(l) ( ) J(m-1) ( )

Yn+l = Yn + f( Xn, Yn )h n + Xn, Yn h2 Xn, Yn hm

2 n + .. · + m. I n ·

This single-step method of numerical approximation to the solution of the

IVP (1) is known as the Taylor series expansion method of order m