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}}.

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