Puzzled by A coupled system of PDEs

[Dr_Noface]

Sorry about the format, bit I have no knowledge of LateX.

A,B - are real constants

U=(U_x,U_y,U_z)

I have a system of three coupled linear second order differential equations

(d_i)^2(U_i) +A*Laplacian(U_i)+ B*d_i[Divergence(U)]

Note: The first term is not a sum.

0<z<H, while x & y can be any real number. I have some more boundary conditions, but I feel as if I'm nowhere close to that stage.

I'm pretty stumped. I tried Fourier transforming (in x & y) and end up with a system of six coupled linear ODEs. I can find the eigenvalues (they're the roots of a cubic equation), but solving for the eigenvectors is an awful algebraic exercise. Is there anything I'm missing?

 

[jackmell]

I'm pretty stumped. I tried Fourier transforming (in x & y) and end up with a system of six coupled linear ODEs. I can find the eigenvalues (they're the roots of a cubic equation), but solving for the eigenvectors is an awful algebraic exercise. Is there anything I'm missing?

I'd first try to directly solve the coupled PDEs explicitly via Mathematica's DSolve function. Probably won't do. Next, I'd then try to directly solve numerically the coupled PDEs via Mathematica's NDSolve function. May be some issues with providing acceptable boundary and initial values to NDSolve there. Next, I'd use Mathematica's Eigenvectors function to compute the eigenvectors of your coupled ODE system. In general, if the objective is the solution and not practice doing it manually, then I'd rely heavily on using Mathematica to solve it.

If you're not familiar with Mathematica then really need to try to become so. If you post the matrix in a way I can understand it, I'll run Eigenvalue/Eigenvector for you in Mathematica.