A good starting point for transient calculation is the old text by Carslaw and Jaeger, Conduction of Heat in Solids. That covers many transient heat transfer cases with a wide range of boundary conditions, then use the Heat / Mass transfer analogue, as in many cases mass transfer is analogous to heat transfer. The more modern approach is transient numerical solutions, but these are less easily validated than a pure maths solution to the problem, In my opinion, numerical solutions should follow the best analytical solution (removing the over-simplifications usually needed for analytical solution); this allows some validation of the numerical solution.

Well, it all depends on the type of differential equation and the initial-boundary conditions that how one should proceed to deal with such problems. It is quite common in the field of fluid dynamics and conservation laws that the initial condition is a function of space and/or the boundary condition is a function of time.

# One of the best approaches is to proceed with the numerical methods (like FDM) and check whether you are able to find the solution.

# You can also try to get bound on the function that depends on the time, i.e. its maximum and minimum, and then can further simplify your boundary condition. It will surely give you a steady-state solution. However, it will only be an approximation and its accuracy will depend on your function, your interval of bound, time domain of problem, and most importantly on the physics of the problem.

I hope it will help you.

Thank you.