# 2D heat equation

• Apr 17th 2010, 03:58 AM
raheel88
2D heat equation
hello all.

does anyone know the general solution to the 2d heat equation, namely

$\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}$

subject to

$u(x_1,x_2,\tau = 0) = u_0(x_1,x_2)$

if anyone could help me I'd be well chuffed!
• Apr 17th 2010, 10:02 AM
Jester
Quote:

Originally Posted by raheel88
hello all.

does anyone know the general solution to the 2d heat equation, namely

$\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}$

subject to

$u(x_1,x_2,\tau = 0) = u_0(x_1,x_2)$

if anyone could help me I'd be well chuffed!

Taken directly from David Colton's book - if $u_0$ is continuous and bounded in absolute then the solution is

$
u(x_1,x_2,\tau) = \frac{1}{4 \pi \tau} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} u_0({\xi}_1,{\xi}_2)\, \text{exp} \left[ - \frac{(x_1-{\xi}_1)^2+(x_2-{\xi}_2)^2}{4 \tau} \right] d {\xi}_1 d {\xi}_2
$
• Apr 17th 2010, 11:17 AM
raheel88
many thanks danny

exactly what i was looking for!
• Apr 17th 2010, 11:28 AM
raheel88
Quote:

Originally Posted by Danny
Taken directly from David Colton's book - if $u_0$ is continuous and bounded in absolute then the solution is

$
u(x_1,x_2,\tau) = \frac{1}{4 \pi \tau} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} u_0({\xi}_1,{\xi}_2)\, \text{exp} \left[ - \frac{(x_1-{\xi}_1)^2+(x_2-{\xi}_2)^2}{4 \tau} \right] d {\xi}_1 d {\xi}_2
$

by the way danny, exactly which of his books is this taken from?
need a reference you see...thanks again!
• Apr 17th 2010, 11:41 AM
Jester
Quote:

Originally Posted by raheel88
by the way danny, exactly which of his books is this taken from?
need a reference you see...thanks again!

Yes, sorry, "Partial Differential Equations - An Introduction."