From an integral equation to a differential equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:04:00Z http://mathoverflow.net/feeds/question/38800 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38800/from-an-integral-equation-to-a-differential-equation From an integral equation to a differential equation Anand 2010-09-15T10:01:59Z 2010-09-15T11:40:42Z <p>Hello, </p> <p>I am wondering whether it is possible to convert the following integral equation to a partial differential equation. </p> <p><img src="http://ima.epfl.ch/~lechen/images/integralEq.jpg" alt="Integral Equation here"></p> <p>where $J_0(t,x)$ is some given nonnegative function and $\nu>0$ is a constant. It is clear $t\ge 0$. </p> <p>The aim is to solve this equation. To convert it to PDE is just one possible way to solve it, since latter we can use the hopefully the fundamental solutions. </p> <p>My current solution is </p> <p><img src="http://ima.epfl.ch/~lechen/images/PDE.jpg" alt="PDE"></p> <p>But I am not sure whether it is right or not.</p> <p>Thanks for any comments or hints!</p> http://mathoverflow.net/questions/38800/from-an-integral-equation-to-a-differential-equation/38809#38809 Answer by Piero D'Ancona for From an integral equation to a differential equation Piero D'Ancona 2010-09-15T11:33:34Z 2010-09-15T11:40:42Z <p>Of course. When $\nu=1$, if you apply the operator $\partial_t-\partial^2_{xx}$ to the last integral you obtain precisely $f(t,x)$ so the equation is $$f_t - f_{xx} = (\partial_t-\partial^2_{xx}) J_0^2 + f.$$</p> <p>EDIT: you seem to know already the answer, so I stop here :) You edited your question when I was writing my answer...</p> <p>By the way, if you want to solve the PDE just set $f(t,x) = e^{t} g(t,x)$ and the equation in $g$ is a homogeneous heat equation. This sounds like some textbook exercise, I musr say</p>