Abstract
Delay differential equations differ from ordinary differential equations in that they may need their initial conditions specified on an interval, not just at a finite set of points. The influence of discontinuities propagates forward in time as the solution progresses. As is the case with ODE, however, a good numerical solution gives the exact solution to a nearby problem. The same technique as used in previous chapters, namely, computing the residual, works here as well to verify that the computed solution is good in this sense. One then has to wonder, as usual, about the conditioning of the problem. ⊲
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Notes
- 1.
In Brunner (2004), this propagating discontinuity is called a primary discontinuity (following notation of earlier authors).
- 2.
After this example was constructed, we realized that it could be interpreted as a method-of-lines solution to the delayed PDE \(u_{t}(t,x) = u_{x}(t - 1,x) - u(t - 1,x)\). Replacing u(t, x) by a vector of values \(y_{k}(t) = u(t,\tau _{k})\) on the grid \(x =\tau _{k} {=\cos }^{\pi (k-1)}/_{(n-1)}\) for 1 ≤ k ≤ n, then the x-derivative could be replaced by using the Chebyshev spectral differentiation matrix. We do not pursue this here, but simply point out that this kind of DDE isn’t all that unrealistic.
- 3.
See the original paper by Battles and Trefethen (2004) and the literature and code available at www.chebfun.org.
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Corless, R.M., Fillion, N. (2013). Numerical Solution of Delay DEs. In: A Graduate Introduction to Numerical Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8453-0_15
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