Solving Coupled PDEs Numerically with Unknown Functions u(x,y) and v(x,y)

[MathNerd]

I have two unknown function namely u(x,y) and v(x,y). These functions are part of two coupled partial differential equations. I realize that it will be almost impossible to get a general solution seeing as one on the PDEs is non-linear. But given a set of boundary conditions I wish to solve for these unknown functions numerically. I don’t quite know how to go about this though, so any help would be appreciated. The equations are attached to this thread

 

[suyver]

If I wanted to solve this numerically, I'd use a finite-timestep approach.

Remember:

$$\frac{\partial f(x)}{\partial x} \equiv \lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

So: just make sure that your timestep $\Delta x$ is small compared to the (expected) fluctuations in your solutions. In that case, you can re-write your equations in an iterative form (note that I used $\Delta x=1$):

$$f(n+1) = {\rm some\; function\; of}\; f(n)$$

which you can do for both of your functions. Now, you can start with your boundary values (for time $n=0$) and generate the solutions for $n>0$ with a computer. Computationally intensive, but that should be no problem for your equations...

Succes!

 

[dhris]

It looks to me as though this is a problem from complex analysis. Your second equation is the analyticity condition for a function of a complex variable. Perhaps the first equation is simply expressed in terms of that function.

dhris