Talk:Delay partial differential equations
I think that the article is well written in the spirit of Scholarpedia.
I would only suggest some minor changes as follows.
- page 3: I would mention that equation (3) includes one of the first examples by Hutchinson, where the integral is replaced by a single retarded value u(t-\tau,x). The possible references are [G.E. Hutchinson: Ann. New York Acad. Sci. 50(1948), 221-246] or the book [Y. Kuang: Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993].
- page 4: I would not use the term "delay operator" for (7). Rather, I would call it "solution segment" as in [F. Hartung, T. Krisztin, H.O. Walther, J. Wu: Chapter 5 in Handbook of Differential Equations, A. Canada, P. Drabek & A. Fonda Eds., Elsevier, 2006]. Moreover, since this notation is due to Krasovskii and/or Hale and since it is well established in the literature since a long time, I would not cite the recent book by Bellen & Zennaro here.
- page 8: the factor 2 is missing in the central difference that approximates the second derivative (twice).
- I would use script style to emphasize some words, namely "neutral" at line 8 of page 1; "finite difference method" at line 14, "semi- discrete systems" at line 22, "semi-discretization" and "method of lines" at line 23, "Galerkin finite difference method" al line 27, "pseudospectral methods" at line 29 and "stiff" at line -4 of page 8.
Finally, which is the meaning of the numbers [n] attached to some references? (namely, Wu (1996)  at page 1, Bellen & Zennaro (2003)  at page 4, Bellen & Zennaro (2003)  at page 8 and  and  at page 9).
Very nice examples of applications to automatically controlled furnace, proliferating and maturing cellular populations, single-species population models, cancer cell division cycle, coupled oscillators, and equations describing cell populations.
All these problems exhibit different phenomena and require specifically designed numerical techniques which can resolve different features of these models. The author and her coworkers contributed significantly to this area and some of this work is mentioned in the last section of the paper.
I would like to see this section expanded, but this is perhaps topic for a separate paper. In this paper I would just add comments about specific difficulties releted to specific models discussed in the first part of the paper.
All the problems discussed in this papers require more complicated initial and boundary conditions than those requred for PDEs without functional dependence. The author deserves credit for illustrating these conditions nicely in Figures 1-4.
Moreover, I agree with comments of the referee #1. In particular credit should be given Hale concerning the notation employed in some models.
I still have just a minor recommendation about the reference to the Hutchinson equation at page 3: after "D=0 (no diffusion)" I would add a sentence like: <<or by a single retarded value u(t-\tau,x) for D\neq 0 (Hutchinson equation with diffusion), see Wu (1996).>>
This is an expanded version of the previous version with a selection of very nice examples from many areas of science and engineering. Also the numerical section has been somewhat expanded and there is a better review of the relevant work in this section.
I think the paper should be accepted. I noticed only one small typo - before formula (13) 4)-(5) should be repaled by (4)-(5) (i.e., one parenthesis is missing.