# Solving the 3-d heat equation

1. ### nickthequick

52
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