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

IIn summary, the conversation discusses two unknown functions, u(x,y) and v(x,y), that are part of two coupled partial differential equations. The equations are non-linear, making it difficult to find a general solution, but the person is hoping to solve for the functions numerically with a set of boundary conditions. The conversation also mentions using a finite-timestep approach and generating solutions with a computer. Lastly, there is speculation that the problem may involve complex analysis.
  • #1
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
 

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  • #2
If I wanted to solve this numerically, I'd use a finite-timestep approach.

Remember:

[tex]\frac{\partial f(x)}{\partial x} \equiv \lim_{\Delta x \rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}[/tex]

So: just make sure that your timestep [itex]\Delta x[/itex] 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 [itex]\Delta x=1[/itex]):

[tex] f(n+1) = {\rm some\; function\; of}\; f(n)[/tex]

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

Succes!
 
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  • #3
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
 

1. How do you define coupled PDEs with unknown functions u(x,y) and v(x,y)?

Coupled PDEs with unknown functions u(x,y) and v(x,y) involve two or more partial differential equations that are interrelated and have unknown functions u and v as solutions. These equations are usually nonlinear and cannot be solved analytically, hence the need for numerical methods.

2. What are some common numerical methods used to solve coupled PDEs with unknown functions u(x,y) and v(x,y)?

Some common numerical methods used to solve coupled PDEs with unknown functions u(x,y) and v(x,y) include finite difference methods, finite element methods, and spectral methods. These methods involve discretizing the PDEs and solving them as a system of algebraic equations.

3. How do you choose the appropriate numerical method for solving coupled PDEs with unknown functions u(x,y) and v(x,y)?

The choice of numerical method depends on various factors such as the type of PDEs, boundary conditions, and the desired accuracy. Finite difference methods are suitable for simpler PDEs, while finite element and spectral methods are better for more complex PDEs or problems with irregular boundaries.

4. What are some challenges in solving coupled PDEs with unknown functions u(x,y) and v(x,y) numerically?

Solving coupled PDEs with unknown functions u(x,y) and v(x,y) numerically can be challenging due to the complexity of the equations and the need for accurate discretization. Convergence issues, stability concerns, and high computational costs are also common challenges in numerical solutions of coupled PDEs.

5. Can coupled PDEs with unknown functions u(x,y) and v(x,y) be solved analytically?

In most cases, coupled PDEs with unknown functions u(x,y) and v(x,y) cannot be solved analytically. However, for certain special cases, such as linear PDEs with simple geometries and boundary conditions, analytical solutions may exist. In general, numerical methods are the most effective approach for solving coupled PDEs with unknown functions u(x,y) and v(x,y).

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