# analytic solution 2D only diffusion

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 March 1, 2012, 13:08 analytic solution 2D only diffusion #1 New Member   diego avesani Join Date: Dec 2010 Location: Trento Posts: 12 Rep Power: 9 hi every one, I have 2D diffusion equation. Are there any analitic solution for bidimensional diffusion equation? The initial intial conditon has not to be puntual thanks a lot

 March 2, 2012, 19:15 #2 New Member   Mazhar Iqbal Join Date: Mar 2009 Posts: 10 Rep Power: 10 You got to explain it a litle more. You can find series solution.

 March 2, 2012, 22:09 #3 Super Moderator     Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 287 Blog Entries: 6 Rep Power: 11 You can cook up many examples by separation of variables. Here is one in the unit square (0,1) x (0,1) u_t = u_xx + u_yy Code: ```function ue = exact_solution(t) global X Y ue = exp(-2*4*pi^2*t) * sin(2*pi*X) .* sin(2*pi*Y) ... + exp(-2*16*pi^2*t) * sin(4*pi*X) .* sin(4*pi*Y);``` Put t=0 to get initial condition. Boundary condition is zero.

 March 3, 2012, 08:40 #4 Member   Join Date: Oct 2011 Posts: 77 Rep Power: 8 Hello ! If you have the possibility to add source terms to you discretized equations, then you can generate infinite exact solutions. You choose an arbitrary solution that you inject into the PDE, then you add the extra terms to balance the equations into you solver. Choose analytic functions infinitely differentiable and which derivatives do not vanish (to maintain all terms in the truncation error) like trigo functions. Then put Dirichlet conditions from the exact solution and you'll be able to test your code (except the BC of course) Tou can look for the Method of Manufactured Solutions for more details. Good luck ! François

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