Testwiki:Reference desk/Archives/Mathematics/2016 April 3

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April 3

Inhomogeneous heat equation on a finite interval

Could I get an example of solving the 1D heat equation $u_t=k{u}_{xx}+f(x,t),x\in (0,L)$? The article has an example over the entire real axis (Heat_equation#Inhomogeneous_heat_equation)- how can I apply Green's function over a finite interval (I'm guessing not just changing the limits of integration from $(-\mathrm{\infty },\mathrm{\infty })$ to $(0,L)$)? 24.255.17.182 (talk) 20:56, 3 April 2016 (UTC)

If there are no boundary conditions on $x=0$ and $x=L$, then you can get a solution of this equation just by extending the forcing term $f(x,t)$ by zero outside of $(0,L)$ (so, effectively, yes, changing some limits of integration). I'm not sure how to get general boundary conditions. Some special cases can be gotten using Jacobi theta functions (for zero boundary conditions) or the method of images. Sławomir
Biały 00:22, 4 April 2016 (UTC)