@Rouben Rostamian  The outer boundary value L is arbitrary and is not central to the problem I have. However, I have now clarified the boundary values on the script. Both the integral and the Pde need to be computed on a domain between 0 and L, L being defined arbitrarily as 20. Thank you for checking it. 

@Carl Love No worries, thank you for checking it. 

@Carl Love I am not sure what is your line of reasoning.

If the kernel inside the integral was a delta function, i.e., delta(x-y), (instead of an exponential) the equation would be simply a diffusion equation with a drag term.  

@Rouben Rostamian  Thank you for your feedback. I have now attached the worksheet and corrected the typo. The variable v should be x.