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Remixex
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Homework Statement
$$\frac{\partial U}{\partial t}=\nu \frac{\partial^{2} U}{\partial y^{2}}$$
$$U(0,t)=U_0 \quad for \quad t>0$$
$$U(y,0)=0 \quad for \quad y>0$$
$$U(y,t) \rightarrow {0} \quad \forall t \quad and \quad y \rightarrow \infty$$
Homework Equations
This is a diffusion problem on fluid mechanics, but it's more of a math problem so i posted it here.
The Attempt at a Solution
I'm trying to solve this via separation of variables (the textbook uses a "similarity" method I've never seen before, and concludes the function U must be erf) is it even possible to reach an analytic result via SV?
The first boundary condition is what gets me, I tried
$$U_{0} e^{{k^{2}t}} e^{{-\frac{k}{\sqrt{\nu}}y}}$$
But it clearly doesn't work for any boundary condition except the last.
I don't think sinusoidal is the answer here either because it must eventually converge to zero, for every t.
Is there really no analytic answer?