How to make a substitution-based equation solver?

[OffTheRecord]

I have a fluid dynamics problem I need to solve. There are 6 equations, and 6 unknowns, but the form of the equations are non-linear. Some unknown variables are not only multiplied by coefficients, but also by other unknown variables. Thus, I can't just do a Gaussian elimination matrix. I'm trying to come up with a way to write a program that uses substitution to solve non-linear algebraic equations. (I'm using Matlab, but I'm also somewhat familiar with Python, if need be).

My question is this: is this a topic that has a wealth of information on it? Has it been done before, is it common practice? I need to know if I'm reinventing a wheel.

If I am reinventing the wheel that is a substitution-based equation solver, then does anyone know where I can find, say, a sample code that does this?

 

[chiro]

Hey OffTheRecord and welcome to the forums.

Maybe something like this? (using Maple)

http://www.maplesoft.com/support/help/Maple/view.aspx?path=solve/system

 

[OffTheRecord]

Hmm. Well, not having maple at hand, I may never know. Anything similar in Matlab? Any way to create a solve function like that one?

 

[chiro]

MATLAB is not like Maple in that Maple is a symbolic platform, where-as MATLAB is more to do with a numerical computational platform.

You would have to check the libraries for MATLAB to see if they have general multi-equation (systems of equations or inequalities) solver routines. Personally, I think you might have to implement your own routines that do solutions for very specific kinds of relationships (polynomials etc), because again MATLAB is not designed to be a core symbolic computational platform like Maple.

What you might be able to do is to send commands to the Maple engine to execute and then pipe the results back to MATLAB. You can do this with a bit of programming know how and understanding of calling the Maple Engine, and to be frank, if you need to do this thing a lot for a wide range of functions, this would be the way to go if you have access to Maple already.

 

[blue_raver22]

I understand your need for a reliable and efficient method to solve your fluid dynamics problem. Substitution-based equation solvers are indeed a commonly used method in solving non-linear algebraic equations. There is a wealth of information available on this topic, and it has been extensively studied and implemented in various programming languages, including Matlab and Python.

There are a few things to consider when developing a substitution-based equation solver. First, you need to carefully examine your equations and determine the appropriate substitution variables. This will depend on the specific problem you are trying to solve and may require some trial and error. Additionally, it is important to consider the stability and convergence of your solver, as non-linear equations can be sensitive to initial conditions.

Fortunately, there are many resources available to guide you in developing a substitution-based equation solver. There are textbooks, online tutorials, and even open-source codes that you can use as a reference or starting point for your own program. Some popular codes include the Newton-Raphson method, the Broyden method, and the Levenberg-Marquardt method.

In terms of finding sample codes, a simple internet search for "substitution-based equation solver Matlab" or "substitution-based equation solver Python" will yield numerous results. I would also recommend reaching out to your colleagues or professors who may have experience with this type of problem and can provide valuable insights and resources. Additionally, many universities have software libraries or coding resources that may have examples of substitution-based equation solvers.

In conclusion, while substitution-based equation solvers are a well-studied and commonly used method, it is always important to carefully examine your specific problem and choose the appropriate solver and approach. With the wealth of information and resources available, I am confident that you will be able to successfully develop a program to solve your fluid dynamics problem.