I'm looking for a way to get analytical solution of diffusion with time and space-varying diffusivity, i.e.
$$\frac{\partial }{\partial x} k(x,t)\frac{\partial T}{\partial x}= \frac{\partial T}{\partial t}$$
where in this case diffusivity $k$ change with space and time.
Is there a way to solve this type of equation in analytical way? I have figured out how to solve it when $k$ is solely a function of time but stucked when $k$ is a space fundtion as welll.
Any help is appreciated.