# Numerical solution of two coupled nonlinear PDEs

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• ShayanJ
In summary, my friend needs to numerically solve this two dimensional boundary value problem but has now idea where to begin. Could anybody help?
ShayanJ
Gold Member
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?

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?

jedishrfu said:
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.

Shyan said:
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.

soarce said:
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?

The only advice I can give is to possibly use MATLAB:

https://www.mathworks.com/moler/exm/chapters/water.pdf

non-linear analysis reference (no programingsource though):

https://www.utwente.nl/ewi/aa/people/phd%20Alumni/thesis/thesis-jaapharmwesthuis.pdf

Other choices would be Open Source Physics java framework:

You could use the mathworks article above to figure out to code the java program to solve it numerically.

Last edited by a moderator:
Shyan said:
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.

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.

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.

## 1. What is the purpose of using numerical solutions for two coupled nonlinear PDEs?

Numerical solutions for coupled nonlinear PDEs are used to solve complex mathematical models that cannot be solved analytically. These models often have multiple coupled equations that cannot be solved using traditional methods, making numerical solutions the preferred approach.

## 2. How are two coupled nonlinear PDEs solved numerically?

The numerical solution of two coupled nonlinear PDEs involves discretizing the equations into a system of algebraic equations, which can then be solved using numerical methods such as finite difference, finite element, or spectral methods.

## 3. What are the challenges in solving two coupled nonlinear PDEs numerically?

One of the main challenges is the computational cost, as these models often require a large number of grid points and iterations to achieve accurate solutions. Another challenge is the stability of the numerical methods used, as nonlinear systems can exhibit unstable behavior.

## 4. How do you validate the accuracy of a numerical solution for two coupled nonlinear PDEs?

The accuracy of a numerical solution can be validated by comparing it to known analytical solutions, if available. In cases where analytical solutions are not available, the solution can be compared to results obtained using different numerical methods and grid resolutions.

## 5. What are the applications of numerical solutions for two coupled nonlinear PDEs?

Numerical solutions for coupled nonlinear PDEs have a wide range of applications in various fields, including fluid dynamics, heat transfer, chemical reactions, and population dynamics. They are also used in engineering and scientific simulations to predict and analyze complex systems.

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