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Consider the delay differential equation:

$ y_x(x) = \sqrt{y(x-\bar{x})} $

where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter.

This equation does not seem to have a known closed-form solution.

Would anyone know how to get a series solution for $y(x)$?

Thanks!

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why do you want a series solution ? You could instead use qualitative tools. A reference book on delay differenctial equation is that of Jack Hale. –  Denis Serre Oct 11 '12 at 20:02
    
Thank you, well ideally I'd like to see an approximate formula for $y(x)$, with $\bar{x}$ a parameter (or for the special case $\bar{x}=1$ or for the special case where $\bar{x}$ small), that would help in understanding the solution. My math is very limited, a series solution is merely the first thing I thought about. –  PatrickT Oct 13 '12 at 15:42
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1 Answer

As mentioned already by Denis Serre, there is a rich literature investigating delay equations.

If you make an experiment, and fix $\bar{x}=1$, then you see that you need as an initial value the complete past on $[-1,0]$. To play a bit, tak as an initial function the constant function $y(s)=1$ for $s\in[-1,0]$. Then you can calculate the solution explicitly for $x\in[0,1]$, then using this you can calculate the solutuion on $[1,2]$, etc. We see that it is far from being analytic. Hence, no chance for a series sepresentation of a solution.

If you are interested in classical stuff, then Bellman and Cooke is an excellent book. An other good reference is the one by Hale and Verduyn Lunel.

ADDED: If it is a delay equation (i.e., $\bar{x}>0$), then the initial condition has to be a function (you have to know the whole past). Then the iteration procedure I described works always. This gives you a possible approximation formula, most numerical methods also work this way.

You are right about analyticity: series representation does it. Smoothness is a consequence. The example I presented to you is only once differentiable at $x=1$, twice at $x=2$, etc. Hence, cannot be analytic.

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Thanks! I have seen the books you cite and read parts of them, but I didn't see anything there that might be directly applicable to this problem. I had simulated the solution following the scheme you suggest, for various choices of initial conditions, but I'm not entirely sure how to write the initial conditions; I was hoping an approximate formula might help me understand how to set the initial conditions. Based on related problems solved without reduction to a DDE, I expect the solution is continuous and differentiable for strictly positive values of $x$. The case $\bar{x}<0$ may be easier. –  PatrickT Oct 13 '12 at 15:54
    
Thanks for pointing out the analyticity problem. I'm not sure what "analytic" means, to be quite honest. How can you tell when a function is or isn't? I seem to remember something like: non-analytic functions don't admit a (any?) series representation, but I'm quite ignorant beyond that. The derivative of $y(x)$ exists, doesn't it? since it's the right-hand side of the DDE, provided $y(x)$ itself exists... or is that a fallacy? How about other kinds of series? Thanks again! –  PatrickT Oct 13 '12 at 16:01
    
And by the way, I'm such a newbie that I can't upvote or do anything useful like that, sorry... –  PatrickT Oct 13 '12 at 16:03
    
Thanks András for the additional information. –  PatrickT Oct 16 '12 at 14:05
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