Functional analysis applications

[cfddjk]

Can anyone tell me the Engineering applications of Functional analysis with a real world example, If possible?

Thanks in advance.

 

[DrRocket]

Can anyone tell me the Engineering applications of Functional analysis with a real world example, If possible?

Thanks in advance.

You might take a look at Luenberger's book Optimization by Vector Space Methods. You can also find some applications in theoretical control theory and information theory. But the examples are fairly esoteric. Most examples will be researchy and pretty theoretical.

A good response to your question requires some appreciation of what you mean by "real world example". Quite frankly most real world engineering requires zero sophisticated mathematics. There are exceptions.

One of the most mathematically sophisticated engineers that I know insisted on giving me his copy of Kreyzig's book Functional Analysis because he found it impenetrable and useless. It is in fact a lousy book, and there are much better ones. But in truth he had no need of functional analysis in his work (he was an excellent mechanical engineer, complete with PhD).

One the other hand a deeper understanding than most engineers have of Fourier transforms and Fourier series depends on functional analysis. So if you consider this real world, here goes:

I once ran into a problem where were looking at EMP signals on a missile. The data was taken and there was software to take the Fourier transform. In fact one could filter the data and take either the FFT or an integral transform. One guy was quite upset because when he took the Fourier transform and then inverted back to the time domain, he found an acausal signal (i.e. a response occurring prior to time zero). This created a big hulabaloo and a splinter meeting. At the splinter meeting I explained that 1) Fourier transforms and inverse transforms are basically the same thing, so they follow the same theorems. 2) The Paley-Weiner theorem (the tie to Functional Analysis) shows that the Fourier transform of a function with compact support is analytic 3) All functions in a computer are of compact support 4) No analytic function can be zero except at isolated points. So when you invert such a function using a numberical approximation for the integral Fourier transform you get an analytic function that must of necessity be acausal. So in that case a bit of Functional Analysis resolved a lot of consternation and we could get back to solving real problems.

Such a specific example is pretty rare. In truth functinal analysis is more likely to just give you a deeper understanding of some of the tools used more routinely and not necessarily be a tool used very often. But one never knows.

 

[Studiot]

Engineers like numbers.
Functionals are functions that map from some vector space to numbers.

Engineers are very concerned with solving the differential equations and boundary value problems of various field theories. In all but the simplest situations numerical methods are employed.

Functional analysis provides the theoretical basis for many of these numerical methods.

For someone looking to explore the connection I would recommend looking at

Applied Functional Analysis and Variational Methods in Engineering by J N Reddy

Many practical examples are developed in this text.

 

[radou]

Functional analysis provides the theoretical basis for many of these numerical methods.

Exactly. For example, a full theoretical understanding of the FEM (finite element method) can't be obtained without knowing what Banach, Hilbert and Sobolev spaces are.