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

144 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS
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Algorithm 7.1. Method of Steps To solve the delay differential
equation
on the interval (t 0 , Kτ), K ∈ Z + .
y ′ (t) = f(t, y(t), y(t − τ)) (7.66)
y(t) = g(t), t 0 − τ < t < t 0 (7.67)
1. Input n, number of points in the mesh; set h = τ/n.
2. For i = 0, . . . , K − 1,
(a) Set p(t) = y(t − τ) (either as a function or an array of
data to index into).
(b) Solve the following IVP using any solver you choose:
y ′ (t) = f(t, y(t), p(t)) (7.68)
y(t 0 + iτ) = p(t 0 ) (7.69)
on the interval (t 0 + iτ, t 0 + (i + 1)τ).
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We choose to store our initial data as a set of points evaluated at the delayed times
t 0 − τ, t 0 − τ + h, t 0 − τ + 2h, . . . , t 0 in an array. The conversion between the array
index i and the time t is given by the formula
i = 1 + n(1 + t/τ) (7.70)
This formula works find so long as t falls on a mesh point; if not, we have two choices:
interpolation (more accurate); and rounding (simpler). Since this is an illustrative
implementation, we choose the simpler method, and force the value of i to be an
integer in the following code:
index[t , tau , n ] := Module[{},
If[t < -tau , Return[$Failed]];
Return[Floor[1 + n(1 + t/tau)]];
];
The function index[t, tau, n] returns the value of i corresponding to a given
value of t , based on the mesh size n and delay tau . Assuming that our initial
data is stored in the array data, to get the value of f(t − τ) would would call
data[[index[t-tau, tau, n]]]
and to implement our function f(t) = −y(t − τ)
f[t ] := -1.0*data[[index[t - tau, tau, n]]];
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007