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

CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS 147
ListPlot[results, PlotJoined -> True, PlotLabel -> ‘‘tau’’
ToString[tau], Frame->True, AspectRatio-> 0.3]
Figure 7.3 illustrates the results when this DDE is solved using τ = 1, 4, 7, 15, 20,
showing a constant steady state, damped oscillations, stable regular oscillations,
irregular oscillations, and chaotic oscillations. The point to be gained from this is
that even with constant initial conditions (g(t) = 1) the behavior of the solution
can change radically with different size delays.
7.4 Runge-Kutta Methods for Delay Differential Equations
Delay differential equations can be solved using Runge-Kutta methods. Suppose
that we have an equation with a single delay that we wish toslve,
It is convenient to use a fixed step size such that
y ′ (t) = f(t, y(t), y(t − τ)) (7.73)
y(t) = g(t), t 0 − τ < t < t 0 (7.74)
τ = kh (7.75)
for some integer k. Then the corresponding Runge-Kutta method is
K (n)
i
γ (n)
j
=
= y n + h
ν∑
j=1
a ij f(t j + c j h, K (n)
j
, γ (n)
j
) (7.76)
{
γ (n)
j
= g(t n + c j h − τ) if n < ν
K (n−ν)
j
y n = y n−1 +
ν∑
j=1
if n ≥ ν
(7.77)
b j f(t n−1 + c j h, K (n−1)
j
, γ (n−1)
j
) (7.78)
This is equivalent to solving a different set of ordinary differential equations on each
interval: on the interval [t 0 , t 0 + τ], solve
On the interval [t 0 + τ, t 0 + 2τ], solve
On the interval [t 0 + 2τ, t 0 + 3τ], solve
u ′ (t) = f(t, u(t), g(t − τ)) (7.79)
u ′ (t) = f(t, u(t), v(t)) (7.80)
v ′ (t) = f(t − τ, v(t), g(t − 2τ)) (7.81)
u ′ (t) = f(t, u(t), v(t)) (7.82)
v ′ (t) = f(t − τ, v(t), w(t)) (7.83)
w ′ (t) = f(t − 2τ, w(t), g(t − 3τ)) (7.84)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge