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

150 CHAPTER 7. DELAY DIFFERENTIAL EQUATIONS
y, y ′ , y ′′ , . . . , y (p+1) ; however, we may not even know where all the discontinuities
are. For a system with time-dependent delay,
Discontinuities can arise because
y ′ = f(t, y(t), y(t − τ(t, y))) (7.87)
y(t) = g(t), t ≤ t 0 (7.88)
• The initial function g(t) may have discontinuities.
• The initial function g(t) may not link smoothly to the solution at t = t 0 . This
will lead to a discontinuity in y ′′ at the next step, y ′′′ at the following step,
and so on.
• Discontinuities in the derivatives of f, τ, and g.
In particular, one can construct pathological examples where the distance between
discontinuities gets smaller and smaller, approaching zero as t → ξ for some number
ξ. This vanishing delay can lead one to require infinitely small grid spacing, which
is not possible. The subject of discontinuity is subtle and non-trivial. We will not
consider the details here (see [3] for details) except to point out that when the delay
is not constant they can arise in unexpected places. When the delay is constant,
the functions f and g are smooth and the solution matches g at t = 0 then the only
discontinuities correspond to multiples of the delay.
Definition 7.3. Let
y n+1 =
k∑
α n,i y n+1−i + h n+1 φ(y n , . . . , y n−k+1 ) (7.89)
i=1
be a general multistep method for the initial value problem y ′ = f(t, y), y(t 0 ) =
y 0 . Then a continuous extension of the method is a piecewise polynomial
interpolant η(t)
η(t n + θh n+1 ) =
j n+i
∑ n+1
i=1
β n,i (θ)y n+jn−i+1 + h n+1 φ(y n+jn , . . . , y n−in ) (7.90)
η(t n ) = y n (7.91)
η(t n+1 ) = y n+1 (7.92)
that is computed on an interval I = [t n−in , t n+jn+1], where [t n , t n+1 ] ⊂ I.
A continuous Runge-Kutta method, for example, is
y(t n + θh) = y n + h
Y ni = y n + h
2∑
b i (θ)f(t n , Y ni (7.93)
i=1
2∑
a ij Y nj (7.94)
j=1
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007