Hello, 


I am wondering whether it is possible to convert the following integral equation to a partial differential equation. 


[Integral Equation here http://ima.epfl.ch/~lechen/images/integralEq.jpg](http://ima.epfl.ch/~lechen/images/integralEq.jpg)


where $J\_0(t,x)$ is some given nonnegative function and $\nu>0$ is a constant. It is clear $t\ge 0$. 


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. 


My current solution is 


[PDE http://ima.epfl.ch/~lechen/images/PDE.jpg](http://ima.epfl.ch/~lechen/images/PDE.jpg)


But I am not sure whether it is right or not.


Thanks for any comments or hints!



From an integral equation to a differential equation
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
$$ f\_t - f\_{xx} = (\partial\_t-\partial^2\_{xx}) J\_0^2 + f.$$


EDIT: you seem to know already the answer, so I stop here :) You edited your question when I was writing my answer...


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


