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# 22 maximum pointwise errors in the solution of

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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:...
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