# From an integral equation to a differential equation

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

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

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

Thanks for any comments or hints!

• Anand: You can directly type $\LaTeX$ when you're asking or answering a question on MathOverflow, with the usual dollar signs (e.g., $x + 3 = y$). – Tom LaGatta Sep 15 '10 at 15:48
• @Tom, I tried once with a big formula. But it failed. So I edit latex and convert to jpg file. I will try next time. Thank you. :-) – Anand Sep 15 '10 at 16:22

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 $$f_t - f_{xx} = (\partial_t-\partial^2_{xx}) J_0^2 + f.$$
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
• Dear Prof. D'Ancona, what kind of initial condition should we pose on this problem? I think it might be $f(0,x)=J^2_0(0,x)$. However, in some cases, it is reasonable to ask when $J_0(0,x)=\delta_0(x)$. Then in this case, what does it mean for $\delta^2_0$? Thank you for your help! – Anand Sep 16 '10 at 13:12