Im trying to solve the heat equation in 2dim on a plate.(adsbygoogle = window.adsbygoogle || []).push({});

0=<x=<L, 0=<y=<L. With homogenous dirichlet conditions on the boundary and the initial condition:

T(x,y)=T0sin(pi*x/L)sin(pi*y/L)

With separation of variables i get the solution

[tex]

T(x,y,t)=\sum_{m=0}^\infty\sum_{n=0}^\infty B_{mn}*exp[\frac{-(m^2+n^2)\pi^2kt}{L^2}]*sin[\frac{m\pi x}{L}]*sin[\frac{m\pi y}{L}]

[/tex]

m,n integers and k the constant from the heat equation.

Now the initial condition determine the constants B_mn

[tex]

B_{mn} = \frac{4T_0}{L^2}*\int_{0}^{L}\int_{0}^{L} sin[\frac{\pi x}{L}]*sin[\frac{\pi y}{L}] sin[\frac{m\pi x}{L}]*sin[\frac{n\pi y}{L}] dx dy

[/tex]

But an integral like

[tex]

\int_{0}^{L}sin\frac{\pi x}{L}sin\frac{m\pi x}{L}dx

[/tex]

is zero for m not equal to 1.

So m=1 and n=1...???

and with this i dont get a fourier series as the solution...

What am i doing wrong ?

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# Heat equation in 2dim

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