Numerical solution of two coupled nonlinear PDEs

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One of my friends needs to numerically solve this two dimensional boundary value problem but has now idea where to begin. Could anybody help?

## [(K H )(f g_x-gf_x)]_x+[(K H )(f g_y-gf_y)]_y=0 ##


## K H G^2 (f^2+g^2)+\frac 1 2 [KH (f^2+g^2)_x]_x+\frac 1 2 [K H (f^2+g^2)_y]_y-K H[((f_x)^2+(g_x)^2)+((f_y)^2+(g_y)^2)]=0##

Where K,H and G are known functions of x and y and the unknown functions are f and g.
The boundary conditions are:
## f_x=- G \beta g ## and ## g_x=G\beta f ## at x=0.
## f_x=G g ## and ## g_x=G(2A-f) ## (A=const) at ##x\to \infty ##.
## f_y=g_y=0 ## for both ## y\to \pm \infty ##.

Is there any hope?
 

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  • #2
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Is there any context that others should know like is this a pure math problem, did it come from some physics book? or is this a MATLAB type of problem?
 
  • #3
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Is there any context that others should know like is this a pure math problem, did it come from some physics book? or is this a MATLAB type of problem?
This is a problem from an engineering doctoral dissertation about wave propagation in water.
 
  • #4
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This is a problem from an engineering doctoral dissertation about wave propagation in water.
Solitons?

What is the physical origin of the two diferential equations? E.g. Two coupled waves, media properties and wave propagation.
 
  • #5
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Solitons?

What is the physical origin of the two diferential equations? E.g. Two coupled waves, media properties and wave propagation.
I don't know anything more than what I said about the context.

Is it possible to somehow linearize the equations in some limit? Or any other method to make the problem simpler?
What methods can he use?
 
  • #6
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  • #7
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I don't know anything more than what I said about the context.
Is it possible to somehow linearize the equations in some limit? Or any other method to make the problem simpler?
What methods can he use?
Depends on the magnitude of the nonlinear term, if it is only a small adjustments of the linear solution then one may start examining first the linearized system.

There are several methods to numerically solve coupled nonlinear differential equations (relaxation methods, shooting method, fixed point method, imaginary time method -- some of these methods are described in Numerical Recipes in C, the chapter on boundary value problems). However, depending on the type of the equations one method would be more appropriate in comparison to others. Also is important to know whether the solutions you look for have any symmetry, e.g. radial symmetry, this may lead to a simplified set of equations.
Examining the second equation, it seems that one can split it into two nonlinear equations corresponding to f and g functions, this is true?

In general, without detailed information about the equations (i.e. physical system) is difficult apply a numerical method.
 
  • #8
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I would tackle this problem using finite element methods. There is some work to set up the problem, but not technically out of reach of a doctoral student.
 
  • #9
phyzguy
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This looks like a set of elliptical equations, so you should be able to attack it by an iterative method like successive over relaxation (SOR). You set up a grid in x and y, and discretize the given set of equations so that you can write f and g at the (i,j) grid point in terms of f and g at the (i+1,j), (i-1,j), (i,j+1), and (i,j-1) points. Then you start with initial guesses for f and g and iterate through the array calculating new values of f and g at each point in terms of the old values. You keep iterating until f and g stop changing.

Since you can't extend the grid to infinity, you will have to study the solutions in the limit as x-> infinity and y-> infinity to know where to cut things off.
 

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