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I have the following system of second order linear PDEs.

1) $\frac{\partial^2 w}{\partial x^2} = C_1 \cdot M_{1} + C_2 \cdot M_{2} + C_3 \cdot \frac{\partial Q_{1}}{\partial x}$

2) $\frac{\partial^2 w}{\partial y^2} = C_4 \cdot M_{1} + C_5 \cdot M_{2} + C_6 \cdot \frac{\partial Q_{2}}{\partial y}$

3) $\frac{\partial^2 w}{\partial x\partial y} = C_7 \cdot M_{3} + C_8 \cdot \frac{\partial Q_{1}}{\partial y} + C_9 \cdot \frac{\partial Q_{2}}{\partial x}$

4) $\frac{\partial^2 M_{1}}{\partial x^2} -2 \frac{\partial^2 M_{3}}{\partial x\partial y} + \frac{\partial^2 M_{2}}{\partial y^2} = C_{10} \cdot \frac{\partial^2 w}{\partial x^2} + C_{11} \cdot \frac{\partial^2 w}{\partial y^2} + C_{12} \cdot \frac{\partial^2 w}{\partial x\partial y}$

5) $ Q_{1} = \frac{\partial M_{1}}{\partial x} - \frac{\partial M_{3}}{\partial y}$

6) $ Q_{2} = \frac{\partial M_{2}}{\partial y} - \frac{\partial M_{3}}{\partial x}$

with all of the $w$, $M_{1}$, $M_{2}$, $M_{3}$, $Q_{1}$, $Q_{2}$ being functions of $(x, y)$ of course and all the $C_i$s known and constant.

I am not posting the boundary conditions as I am not looking for a complete solution of the above, but I would like some guidance regarding how to approach it. Any help\hint would be highly appreciated.

More specifically, I am interested in knowing how one would go about implementing a numerical method.


For some background info in case that is in any way helpful\interesting, the problem has to do with the buckling of composite plates.

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  • $\begingroup$ @hardmath I hope the edit helps clear the question. $\endgroup$ – Ev. Kounis Oct 2 '18 at 7:16
  • $\begingroup$ Thanks for the clarification. The application to buckling of a composite plate would suggest the use of finite element discretization, and it would be helpful to know the region over which the PDEs are to hold. $\endgroup$ – hardmath Oct 3 '18 at 16:33

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