Integro-differential equation with 2 variables (but 1D integral)

[c-straub]

Hi everybody,

I'm currently trying to apply DeepXDE to a certain type of integro-differential equation. Although all integrals are 1-dimensional, I'm having difficulties because the sought solution depends on 2 variables.

Concretely, I want to find the solution $u\colon [0,1]^2\ni(x,t)\mapsto u(x,t)\in\mathbb{R}$ of an equation of the form

$$\partial_tu(x,t)=(\partial_x^2u(x,t))\cdot\int_0^te^s\ u(x,s)\ ds + u(x,t)$$

with initial condition $u(x,0)=1$ and vanishing Neumann boundary conditions $\partial_xu(0,t)=0=\partial_xu(1,t)$.

Any help on how to implement this in DeepXDE would be very much appreciated!

Best
Christopher

[lululxvi]

You can first understand the DeepXDE code for IDE, and then you should be able to implement the new code.