Unformatted text preview: vely.
The implementation of the collocation scheme is usually done by eliminating the
unknowns k , k = 1 2 : : : J , appearing in (9.2.7b) on each subinterval and then solving
for the nodal values Y = Y(x ), i = 0 1 : : : N . We'll illustrate this for a linear system
; ; ; ik i i f (x y) = A(x)y + b(x)
g (y(a)) = Ly(a) ; l g (y(b)) = Ry(b) ; r: L R 18 (9.2.8a)
(9.2.8b) Nonlinear problems are solved using Newton's method to linearize them to the form of
(9.2.8).
For a linear problem, (9.2.7b) becomes k = A( ) Y 1 + h
ik ik i; X
J i
j =1 a k ] + b( )
kj ij k = 1 2 : : : J: ik (9.2.9a) This system can be written in matrix form as Wk =VY 1+q
i where 2 a A(
6 a12 11A(
W = I;h 6
6 ...
4
i i i i i; (9.2.9b) i a1 2 A( 1)
a2 2 A( 2) 1)
2) a1 A( 1)
a2 A( 2) i J i ...
...
...
a 1 A( ) a 2 A ( )
a A( )
i J 2k 3
6 k 12 7
k = 6 .. 7
6.7
45 i iJ J J iJ i (9.2.9c) iJ i i i i i A( ) iJ 3
7
7
7
5 2 b( ) 3
6 b( 12) 7
q = 6 .. 7 :
6.7
4
5 i i i JJ 2 A( ) 3
6 A( 12) 7
V = 6 .. 7
6.7
4
5 i k i (9.2.9d) b( ) iJ iJ Let us write (9.2.7a) in the form Y = Y 1 + h Dk
i where i; i 2bI
6 1 b2I
D=6
6
4 ... (9.2.10a) i bI 3
7
7:...
View
Full Document
 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, Collocation, Ly, collocation points

Click to edit the document details