I can't say I know exactly how to solve your problem, but I think Chapter 7 of Crank's The Mathematics of Diffusion might be useful.
I've been using the Laplace transform to solve the above equation with certain boundary conditions. Ive done this successfully when D is constant. However, I have now figured out that D is infact not constant in the system im looking at, it depends on t. The exact form of D(t) is not known, but I know which value it starts for and that it decreases towards another value. My thinking was to choose a D(t) that made the problem solvable. My initial guess was:
which leads me to figure out
Is there any easy way to do this, for example some rules that let me isolate