Real eigensolutions to the diffusion equation.

Question

How can I find the real eigensolutions to the diffusion equation

$u_t$ = $\left(x^2 u_x\right)_x,$

modeling diffusion in an inhomogeneous medium on the half-line $x>0?$

And which solutions satisfy the Dirichlet boundary conditions $u(t,1)=u(t,2)=0?$

Answer 1

We can separate the variables. Suppose $u(t,x)=X(x) T(t),$ and use $\dot{T}=\dfrac{dT}{dt}$ and $X'=\dfrac{dX}{dx}$ for temporal and spatial derivatives, respectively. Then we have \begin{align*} u_t&=2xu_x+x^2u_{xx}\\ X\dot{T}&=2xX'T+x^2X''T \quad\text{divide by } XT:\\ \frac{\dot{T}}{T}&=\frac{2xX'}{X}+\frac{x^2X''}{X}=k\\ \dot{T}&=kT\\ 2xX'+x^2X''&=kX. \end{align*} These are both fairly easily solvable. We obtain \begin{align*} T&=Ae^{kt}\\ X&=x^{-\frac{1}{2}-\frac{1}{2} \sqrt{4 k+1}} \left(c_1+c_2 x^{\sqrt{4 k+1}}\right)\\ U(t,x)&=Ae^{kt}x^{-\frac{1}{2}-\frac{1}{2} \sqrt{4 k+1}} \left(c_1+c_2 x^{\sqrt{4 k+1}}\right). \end{align*} Imposing the boundary conditions, I'm afraid, yields only the trivial solution $X=0.$ So what I have shown is that, of all separable solutions (which might be the only solutions, given that the PDE is linear - you might be able to prove existence and uniqueness of solutions), only the trivial solution works for your boundary conditions.