Generic Solution of a Coupled System of 2nd Order PDEs

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Summary:

We are looking for the generic solution of this coupled system of 2nd order PDEs.
Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it.

I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C are constants, and (if it helps) A, B > 0.
pde.PNG
 
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Answers and Replies

  • #2
pasmith
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It's linear, so you can try looking for separable solutions. If you set [itex]u = Ue^{kx + ly}[/itex] and [itex]v = Ve^{kx + ly}[/itex] then you get the following eigenvalue problem for [itex]k[/itex] and [itex]l[/itex]:
[tex]
k^4(1 + B^2C) + k^2 l^2 (2 + C(A^2 + B^2)) + l^4(1 + A^2C) = 0.[/tex]

EDIT: This factorises as [tex]
(k^2 + l^2)(k^2(1 + CB^2) + l^2(1 + CA^2)) = 0.[/tex]
 
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wrobel
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Laplace transform; Fourier transform...
 
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