# Analytical solution to the diffusion equation with variable diffusivity

## Main Question or Discussion Point

Hi, I'm trying to find an analytical solution (if one exists) to the 1d diffusion equation with variable diffusivity κ(x);

$\partial_t u(x,t) = \partial_x[\kappa(x) \partial_x u(x,t)]$

Could someone point me in the right direction to solve this if its possible to do so analytically. I've tried separation of variables after using the product rule to expand out the diffusive term but the equation isn't of the correct form.
Kieran

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hunt_mat
Homework Helper
Greens function perhaps?

Mute
Homework Helper
Why doesn't separation of variables work? $u(x,t) = X(x) T(t),$ so

$$X\frac{dT}{dt} = T\frac{d}{dx}\left(\kappa(x) \frac{dX}{dx}\right).$$

You can divide through by XT and the left hand side will have only terms dependent on t and the right hand side will have only terms dependent on x, so both sides are some constant, k. Solving the x ordinary differential equation for general $\kappa$ may be tricky, but separation of variables otherwise works.