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

CHAPTER 1. CLASSIFYING THE PROBLEM 13
Then we have
F (t, y, z) = 0 (1.73)
but the Jacobian matrix becomes nonsingular:
∂F (t, y, z)
∂(y, z)
=
( ) I 0
0 0|
(1.74)
The general theory of differential algebraic equations is still being developed. A
system of the form
Ay ′ (t) + By(t) = f(t) (1.75)
where A and B are square matrices and A is singular, is called a linear constant
coefficient DAE. If the determinant
det λA + B ≠ 0 (1.76)
is not identically zero (considered as a function of the variable λ), then 1.75 is
solvable.
1.6 Delay Differential Equations
Definition 1.8. (Functional differential equation) If a differential systems can be
written explicitly as a function of a function of the abscissa variable, noteably
y ′ = f(t, y(t), y(u(t)) (1.77)
then it is know as a functional differential equation (FDE). If the it can be
written as
y ′ = f(t, y(t), y(t − τ 1 , t − τ 2 , . . . )) (1.78)
then it is known as a delay differential equation (DDE).
Example 1.8. A typical functional differential equation is
A typical DDE is given by
y ′ = y(t) + y(sin(t)) (1.79)
y ′ = y(t) + y(t − 3) + 7y(t − 5) (1.80)
DDEs arise frequently in control theory and population growth. The theory of
functional differential equations is not as well advanced as other types of DEs and
there are fewer numerical methods for solving them. The best understood functional
equations are delay equations. Delay equations are similar to initial value problems,
in that the value of the function must be known at the starting point. They are
more complicated, however, in that the value must also be known over an entire
interval preceding t 0 , of duration given by the greatest delay in the equation. Thus
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge