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My main problem to solve is anisotropic wood (2D) with tree rings paralel and all in one direction.

I'm having problem solving the equation

$$au_{xx} + bu_{yy} = -c.$$

It's a non time dependent part of temperature profile in 2D. I've already used seperation to solve that.

$$T(x,y,t) = u(x,y)\exp{(-wt)}$$

I'm trying to solve non-stationary anisotropic heat equation. The first equation above is already rotated in eigensystem of my material.

My real question is: Is this by any chance analiticaly solveable (probably not)? If not, is it possible to get the equation in shape to solve it numericaly?

I was already trying with Wolfram Mathematica 8.0 but can't get the boundary conditions to programs liking.


a,b,c > 0 are positive constanst

My original equation is heat equation:

a1*Txx + a2*Txy + a3*Tyy = a4*Tt

a1,a2,a3,a4 > 0 are positive constants, x,y, are coordinates, t is time

indexes are representing partial derivation by the variable.

To get to the upper eqution I had the use seperation T(x,y,t) = U(x,y)*w(t). After that I had to rotate the system in to eigensystem.

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Would you mind telling us what the original PDE is? – Cameron Williams Aug 14 '13 at 19:08
Are $a,b,c$ constants? How about the signs of $a$ and $b$? – Shuhao Cao Aug 14 '13 at 19:16
Thanks for your quick reply. I've edited the post, if anything else is unclear don't hesitate to say so. – Bostjan Aug 15 '13 at 10:26

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