The Computable Differential Equation Lecture ... - Bruce E. Shapiro

152 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS
b 3 (θ) =
b 4 (θ) =
(− 2 3 θ + 1 )
θ 2 (7.109)
( 2
3 θ − 1 2)
θ 2 (7.110)
Details on✬
implementing these methods are given in the reports by Paul and Baker. ✩
3
Algorithm 7.2. Continuous Extension to Method Steps To
solve the delay differential equation
on (t 0 , t f )
y ′ (t) = f(t, y(t), y(t − τ(t, y))) (7.111)
y(t) = g(t), t 0 − τ < t < t 0 (7.112)
1. Locate all discontinuity points ξ 1 , ξ 2 , . . . , ξ m < t f .
2. Set ξ 0 = t 0 , ξ m+1 = t f
3. Solve
y ′ = f(t, y(t), g(t − τ(t))) (7.113)
y(xi 0 ) = g(ξ 0 ) (7.114)
on (ξ 0 , ξ 1 ) using any ODE solver.
4. For i = 1, . . . , m
(a) Compute the continuous extension η(t) on (ξ i−1 , ξ i )
(b) Solve the equation
on (ξ i , ξ i+1 )
y ′ = f(t, y(t), η(t − τ(t))) (7.115)
y(xi 0 ) = η(ξ 0 ) (7.116)
✫
✪
3 C.A.H. Paul & C.T.H. Baker, ”Explicit Runge-Kutta Methods for the Nuermical Solution of
Singular Delay Differential Equations,” Numerical Analysis Report 212, April 1992, University of
Manchester, Dept. of Mathematics, http://citeseer.ist.psu.edu/37968.html
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007