I am looking for some help finding a numerical solution to a pde of the form:
$$C_t=f(x)C_x+\alpha C_{xx}$$
with initial condition for $C(x,t)$:
$$C(x,0)=\delta(x)$$
and boundary condition
$$C(\pm\infty,0)=0$$
I am using Mathematica's NDSolve function, but it doesn't seem to be able to find a solution for some chosen $f(x)$ and $\alpha$. I think this may be due to the condition's at infinity? Another idea I have is to turn it into a ODE via separation of variables. Any advice on resources or methods would be helpful.
You can use a standard method, like the Crank-Nicolson method, on a truncated spatial domain. The conditions on this artificial boundary can be approximated by setting the solution to be zero, or some function of $(x,t)$, if some adicional information on the asymptotics is available.