The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro
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166 CHAPTER 8. BOUNDARY VALUE PROBLEMS<br />
✬<br />
Algorithm 8.2. Quasilinearization with the Midpoint Method<br />
To solve the boundary value problem<br />
✩<br />
y ′ = f(t, y)<br />
g(y(a), y(b)) = 0<br />
1. Input f, f y (the Jacobian), g(u, v), the Jacobians g u , g v , a mesh definition;<br />
an initial guess y 0 (t); and a convergence tolerance ɛ<br />
2. For k = 0, 1, . . . until ‖y k+1<br />
n<br />
− y k n‖ < ɛ, repeat the following<br />
(a) Calculate each of the S n and R n using<br />
(<br />
)<br />
A(t n−1/2 ) = f y t n−1/2 , yk n + yn−1<br />
k<br />
2<br />
(<br />
)<br />
q(t n−1/2 ) = f<br />
t n−1/2 , yk n + y k n−1<br />
2<br />
− yk n − y k n−1<br />
h<br />
(8.105)<br />
(8.106)<br />
(b) Calculate<br />
B a = g u (y k 0, y k N) (8.107)<br />
B b = g v (y k 0, y k N) (8.108)<br />
d = −g(y k 0, y k N) (8.109)<br />
✫<br />
(c) Solve 8.102<br />
✪<br />
Math 582B, Spring 2007<br />
California State University Northridge<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007
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