nonlinear delay differential equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:36:52Z http://mathoverflow.net/feeds/question/109410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109410/nonlinear-delay-differential-equation nonlinear delay differential equation PatrickT 2012-10-11T19:46:22Z 2012-10-13T17:04:29Z <p>Consider the delay differential equation:</p> <p>$y_x(x) = \sqrt{y(x-\bar{x})}$</p> <p>where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter. </p> <p>This equation does not seem to have a known closed-form solution.</p> <p>Would anyone know how to get a series solution for $y(x)$? </p> <p>Thanks!</p> http://mathoverflow.net/questions/109410/nonlinear-delay-differential-equation/109412#109412 Answer by András Bátkai for nonlinear delay differential equation András Bátkai 2012-10-11T20:24:24Z 2012-10-13T17:04:29Z <p>As mentioned already by Denis Serre, there is a rich literature investigating delay equations.</p> <p>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.</p> <p>If you are interested in classical stuff, then <a href="http://books.google.hu/books/about/Differential_difference_equations.html?id=j4LwpwGPXzwC&amp;redir_esc=y" rel="nofollow">Bellman and Cooke</a> is an excellent book. An other good reference is the one by <a href="http://books.google.hu/books?id=ZNLjAJQMhqwC&amp;lpg=PP1&amp;hl=de&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">Hale and Verduyn Lunel</a>.</p> <p><strong>ADDED:</strong> 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.</p> <p>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.</p>