# Solving the 3-d heat equation

by nickthequick
Tags: equation, heat, solving
 P: 48 Hi, I've been trying to solve $\frac{\partial \vec{u}(x,y,z,t)}{\partial t} = \nu \nabla^2 \vec{u}(x,y,z,t)$ Where the Laplacian is 3-d. My initial conditions are $\vec{u}(x,y,z,0)=\vec{u_o}(x,y,z)$. And my BC is that $(u_z,v_z,w)= 0$ at z=0. where $\vec{u} = (u,v,w)$ I want to solve for $\vec{u}(x,y,0,t)$ i.e. at z=0. Here, $(x,y) \in \mathbb{R}, z \in (-\infty,0), t\in \mathbb{R}^+$. 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, Nick Nick

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