Coupled partial differential equations

[nickthequick]

Hi,

I'm trying to solve the following coupled PDE's

$$u_{tt}-gHu_{xx} - gHv_{xy} = -2\frac{g^2}{\omega} \left\{k\frac{\partial^2 |A|^2}{\partial x^2} + \ell \frac{\partial^2 |A|^2}{\partial x\partial y} \right\}$$$$v_{tt}-gHv_{yy}- gHu_{xy} = -2\frac{g^2}{\omega} \left\{ \ell\frac{\partial^2 |A|^2}{\partial y^2} + k \frac{\partial^2 |A|^2}{\partial x\partial y} \right\}$$

Where A= A(x,y,t), g, H, k $$\ell, \omega$$ are given. The form of the forcing on the RHS of these equations is not so important.

I've been playing around with some things but if anyone has any analytic (or even numerical) insights into solving these equations they'd be most appreciated.Thanks!

Nick

 

[jvicens]

Dear Nick,

Thank you for sharing your coupled PDE's with us. It seems like you are dealing with a challenging problem and I would be happy to offer some insights and suggestions to help you solve it.

First of all, it is important to understand the physical meaning of these equations. From the given form, it seems like you are dealing with a wave propagation problem in a two-dimensional medium. The variables u and v represent the displacements in the x and y directions, respectively, while A is the amplitude of the wave. The parameters g, H, k, l, and ω are related to the properties of the medium and the frequency of the wave.

To solve these equations, you can use standard techniques such as separation of variables, Fourier transform, or numerical methods like finite difference or finite element methods. However, since these equations are coupled, it might be more efficient to use a numerical method that can handle coupled systems, such as the method of lines.

One approach to solving these equations is to first linearize them by assuming small displacements and amplitudes. This will simplify the equations and allow you to use standard techniques to solve them. Once you have a solution for the linearized equations, you can then use perturbation methods to find a solution for the full nonlinear equations.

Another important aspect to consider is the boundary conditions. Depending on the physical system you are studying, you may need to impose appropriate boundary conditions for the variables u and v. These conditions will play a crucial role in determining the behavior of the system.

In terms of numerical insights, it might be helpful to use adaptive time-stepping methods to handle the different time scales in the problem. Also, try to use a higher-order numerical scheme to improve the accuracy of the solution.

Lastly, I would recommend consulting with experts in the field or looking for similar problems in the literature to get a better understanding of the problem and possible solution techniques.

I hope these insights will be helpful in your research. Good luck with your work!