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Solving the 3-d heat equation

by nickthequick
Tags: equation, heat, solving
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Mar12-12, 11:16 PM
P: 52

I've been trying to solve

[itex]\frac{\partial \vec{u}(x,y,z,t)}{\partial t} = \nu \nabla^2 \vec{u}(x,y,z,t)[/itex]

Where the Laplacian is 3-d. My initial conditions are


And my BC is that

[itex](u_z,v_z,w)= 0 [/itex] at z=0. where [itex]\vec{u} = (u,v,w)[/itex]

I want to solve for [itex]\vec{u}(x,y,0,t)[/itex] i.e. at z=0.

Here, [itex] (x,y) \in \mathbb{R}, z \in (-\infty,0), t\in \mathbb{R}^+ [/itex].

Is there an analytic way to do this? So far I have only come across numerical schemes or examples for lower dimensions.

Any suggestions would be appreciated,


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