Diffusion / Random Walk with two semi-infinite composite materials.

Question

I'm trying to think through a 1-d diffusion / random walk problem. There are two semi-infinite regions with different diffusion constants joined together at the origin, such that the diffusion constant D1 holds from the origin to infinity and D2 holds from the origin to negative infinity. An impulse (delta source) is given at t=0 some distance from the origin on either side of the origin. What is the solution for the distribution at some later time t? In other words, how is the typical Gaussian solution of an impulse in an infinite medium changed, with this composite setup? Can anyone point me in the right direction / resource to think through how to solve this? Certainly one of the issues is the appropriate boundary condition at the origin. In this regard, what I'd like to get closest to is a random walker suddenly getting "stickier" feet as they cross the origin.

Answer 1

You just have a smoothness criterion at the origin, $u(t,0^+)=u(t,0^-)$ and $u_x(t,0^+)=u_x(t,0^-)$. Without this condition, mass accumulates at $x=0$ (assuming $u$ has the interpretation of density). Then you have just two separate diffusion equations on the separate spatial domains. One way to solve them is by taking the Laplace transform with respect to $x$.