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

CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS 149
and so forth.
This method will converge for DDE’s with the same order as the corresponding
RK method for IVPs does (i.e., the RK method with the same coefficients). To
determine the stability of the method for solving delay differential equations we
introduce the following generalization of the test equation to include delays,
y ′ (t) = λy(t) + µy(t − 1) (7.85)
for some constants λ and µ. The substitution y(t) = e γt yields the “characteristic
equation,”
γ − λ − µe −γ = 0 (7.86)
Theorem 7.3. The solutions of the test equation 7.85 remain stable as t → ∞ if all
the roots of the characteristic equation remain in the left-half of the complex plane.
The region of stability is very difficult to calculate and we will not consider it
further here. 2
7.5 Continuous Methods
One of the restrictions of the method of steps is that we must always have an integral
number of mesh-points within each delay interval; otherwise there will be an
error due to the overlap between the mesh and the delay. When the delay is timedependent,
or there are multiple delays, the problem becomes more complicated.
For constant delays, the method of steps is generally sufficient and it is not necessary
to resort to continuous methods. Instead, one can just adjust the step size to
match the interval.
Any numerical method can be extended to a continuous form by replacing the
coefficients with functions that depend on a parameter so long as we enforce appropriate
continuity conditions on the mesh. This allows us to evaluate the method
at points in-between mesh points. Clearly there can be many possible extensions
of any given method; ideally we would like an extension that matches the desired
solution as well as possible. This implies that we need some kind of first-guess to a
solution before we can even compute a continuous extension, and need to do some
sort of prediction-correction iteration before we can proceed from one step to the
next.
An additional problem is the propagation of discontinuity. For a method of
order p to work it is necessary to include among the mesh all discontinuities of
2 See Hayes, N.D. “Roots of the Transcendental Equation Associate with a Certain Difference-
Differential Equation” Journal of the London Mathematical Society, 25:226-232 (1950); C.A.H. Paul
and C.T.H. Baker, ”Stability Boundaries Revisited - Runge-Kutta Methods for Delay Differential
Equations,” Numerical Analysis Report No. 205, University of Manchester, Dept. of Mathematics
(1991); R. Bellman and K.L. Cooke “Differential-Difference Equations,” Academic Press (1963).
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge