BAIL 2006 Book of Abstracts - Institut für Numerische und ...

H. TIAN: Uniformly Convergent Numerical Methods for Singularly Perturbed Delay
Differential Equations
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where we assume that −fy(t, y, z) ≥ α > 0 for all t ≥ 0 and all real y, z. We now linearize (5)
and introduce the Newton sequence {ys(t)} ∞
s=0 for the initial guess y0(t) satisfying the initial
condition y0(t) = φ(t), t ∈ [−1, 0]. This is done by defining ys+1(t), for all s ≥ 0, to be the
solution of the linear problem
�Lsys+1(t) ≡ ε dys+1(t)
dt − fy(t, ys(t), ys(t − 1))ys+1(t)
−fz(t, ys(t), ys(t − 1))ys+1(t − 1)
= f(t, ys(t), ys(t − 1)) − fy(t, ys(t), ys(t − 1))ys(t)
−fz(t, ys(t), ys(t − 1))ys(t − 1), t > 0,
ys+1(t) = φ(t), −1 ≤ t ≤ 0.
We may show that not only the convergence of this sequence is quadratic, but also its proportionality
constant is independent of s and ε. If the initial guess y0(t) is sufficiently close to y(t),
converges to y(t).
then the Newton sequence {ys(t)} ∞
s=0
3 Numerical experiments
Consider
We take initial guess as
y0(t) =
�
εy ′ (t) = −y(t) + y 2 (t − 1), t ≥ 0,
y(t) = 2, t ∈ [−1, 0].
t
− 4 − 2e ε , t ∈ [0, 1),
16 − (8 + 16 t−1
t−1
t−1
−2
ε )e− ε − 4e ε , t ∈ [1, 2).
The true solution and the numerical solution using the optimal scheme after two iterations are
plotted in Figure 1. This figure indicates that the uniformly convergent scheme works well also
for nonlinear problem.
References
y
18
16
14
12
10
8
6
4
Numerical and analytic solutions:ε y’(t)=−y(t)+y 2 (t−1),ε=10 −2
Numerical solution
True solution
2
0 0.2 0.4 0.6 0.8 1
t
1.2 1.4 1.6 1.8 2
Figure 1: Comparison between numerical and analytical solutions.
[1] M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, Bifurcation gap in a hybrid optical
system, Phys. Rev., A, 26(1982)3720–3722.
[2] M. C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science,
197(1977)287–289.
Speaker: TIAN, H. 148 BAIL 2006
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