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

CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS 143
The plot is then generated with
methodOfSteps2p[g, t, 1, 10]
We observe that the solution is a decaying oscillation, whereas the corresponding
initial value problem without deal y would give a pure exponential decay. Thus the
two solutions are qualitatively diffferent.
7.3 Numerical Implementation of Method of Steps
In this section we illustrate the numerical solution of a delay differential equation
using the method of steps via several examples. These examples illustrate that the
solution of the DDE can vary radically by minor changes in the initial data or the
parameters of the DDE, such as the size of the delay.
We will begin with the example from the previous section:
y ′ (t) = −y(1 − t) (7.64)
y(t) = 1, −τ < t < 0 (7.65)
One key drawback of the method of steps is that the step size must be an integral
divisor of the delay. Thus for some integer n, we must have nh = τ. So long as there
is only a single delay and the delay is not a function of time, we can get around this
by automatically setting the mesh size based on the number of points we want to
have in the mesh. For a fixed step size, this would give us h = n/τ with n an input
parameter; for a variable step size, we would be limited to multiples of this step.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge