Mathematical Methods for Physicists: A concise introduction - Site Map

NUMERICAL METHODS
approximate value obtained for y 1 , we obtain an improved value, denoted by
…y 1 † 1
:
…y 1 † 1 ˆ y 0 ‡ 1 2 f f …x 0; y 0 †‡f …x 0 ‡ h; y 1 †g:
…13:21†
This process can be repeated until there is agreement to a required degree of
accuracy between successive approximations.
The three-term Taylor series method
The rationale for this method lies in the three-term Taylor expansion. Let y be the
solution of the ®rst-order ordinary equation (13.18) for the initial condition
y ˆ y 0 when x ˆ x 0 and suppose that it can be expanded as a Taylor series in
the neighborhood of x 0 .Ify ˆ y 1 when x ˆ x 0 ‡ h, then, for suciently small
values of h, wehave
y 1 ˆ y 0 ‡ h
dy ! !
‡ h2 d 2 y
dx
0
2! dx 2 ‡ h3 d 3 y
3! dx 3 ‡: …13:22†
Now
and
d 3 y
dx 3 ˆ
0
dy ˆ f …x; y†;
dx
d 2 y
dx 2 ˆ @f
@x ‡ dy @f
dx @y ˆ @f
@x ‡ f @f
@y ;
@
@x ‡ f @ @f
@y
@x ‡ f @f
@y
ˆ @2 f
@x 2 ‡ @f @f
@y @y ‡ 2f @ 2 f
@x@y ‡ f
@f 2
‡ f 2 @ 2 f
@y @y 2 :
Equation (13.22) can be rewritten as
y 1 ˆ y 0 ‡ hf …x 0 ; y 0 †‡ h2 @f …x 0 ; y 0 †
‡ f …x
2 @x
0 ; y 0 † @f …x
0; y 0 †
;
@y
where we have dropped the h 3 term. We now use this equation as an iterative
equation:
y n‡1 ˆ y n ‡ hf …x n ; y n †‡ h2 @f …x n ; y n †
‡ f …x
2 @x
n ; y n † @f …x
n; y n †
: …13:23†
@y
That is, we compute y 1 ˆ y…x 0 ‡ h† from y 0 , y 2 ˆ y…x 1 ‡ h† from y 1 by replacing x
by x 1 , and so on. The error in this method is proportional h 3 . A good approxima-
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