# Eikonal Blog

## 2010.01.24

### Heat conduction equation

Filed under: physics — Tags: — sandokan65 @ 23:38

The problem is defined by following three elements:

• ODE: $\partial_t u = k \partial_x^2 u$,
• Domain: $D=\{(x,t)|x\in{\Bbb R}, t\in{\Bbb R}^{+}\}$,
• Initial condition: $u(x,0)=f(x)$.

Using the Fourier transform one gets to the solution:
$u(x,t) = \frac1{(2\pi)^\mu} \int_{\Bbb R} e^{-ipx-kp^2t}{\bf F}[f](p).$

For example, for $f(x)=A e^{-Bx^2}$ one gets $u(x,t) = \frac{A}{\sqrt{1+4Bkt}} e^{-\frac{Bx^2}{1+4Bkt}}$.