Find approximate solution of equation

Dear Mathworks Community,

does anyone know a method (In best case implemented in Matlab and its Toolboxes) how to approximate an equation?

Background: The solution of a differential equation, that models the behaviour of systems in engineering, is very often so large, that it is uninterpretable by humans. Of course, this solution is valid for all parameter values. However, due to knowledge of the system, the relation of parameters to each others is known (in specific borders), as well as absolute maximum and minimum can be assumed. So, is there a method to find an approximate solution?

Kind Regards,

David

Answer 1

[Shreshth]

Hello David,

I can deduce that you want to know a method to find a simplified, approximate solution for solving differential equations. Additionally, you want the operation to execute using MATLAB’s toolbox.

For this purpose, there are several ways of doing it using MATLAB. Some of the most effective ways are:

1. Using numerical solvers for differential equations like the built in functions (ode45, ode23, ode113 etc.)
2. Control system toolbox: Identify a suitable approximation method for complex equations within MATLAB's capabilities.
3. Curve fitting toolbox: Identify a suitable approximation method for complex equations within MATLAB's capabilities.

To further align with your requirement of modelling a large system and including higher number of parameters, Control System Toolbox seems the best fit. Although it is used for Linier control systems but by the process of Linearization it can be reduced to a linear form. This toolbox can be particularly useful if the differential equations describe a linear time-invariant (LTI) system.

To further understand about the functionalities of Control System Toolbox, you can refer to the below MathWorks documentation:

www.in.mathworks.com/help/control/getting-started-with-control-system-toolbox.html

Hope it helps.

Regards,

Shubham Shreshth

Answer 2

[John D'Errico]

Is there some general method, to ALWAYS be able to approximate the solution of a large system, and have a reliable result? No. Of course not. At least not without potential large inaccuracies. For example, using such models, we could use them to predict the weather. But clearly you cannot just approximate any such large system with a simple low order model, and be confident about your ability to predict the behavior of that system.

In fact, it is often a special skill of some to learn how to reduce a large model, by knowing which terms you can safely discard, and still have a viable model, instead of an unwieldy, unusable thing.

That does not say that one cannot ever reduce the complexity of a model. But when you do so, you almost always end up losing something. Will it be something important? Very possibly. When you linearize things, for example, you implicitly make an assumption the process will go on forever, increasing or decreasing in a very simple way. Is that adequate? Possibly. But possibly not. And that all depends on the mathematics of the system.