How Are BV Functions Applied in Physics and Engineering?

[Hjensen]

I am reading about a branch of mathematics which does not allow separable spaces. The author of the text gives the space of functions of bounded variation as an example of a non-separable space, which is fine - except for the fact that he goes on to claim that "this space is relevant to both physicists and engineers" without giving any further elaboration.

So my question is this: Do any of you have a few examples of BV-functions being used in physics or engineering? I don't need a detailed explanation, I just need to convince myself that the argument given in my book is actually important.

 

[dlgoff]

The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation. The book (Hudjaev & Vol'pert 1985) details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description.

http://en.wikipedia.org/wiki/Bounded_variation"

You might want to check out "Analysis in classes of discontinuous functions and equations of mathematical physics":
http://books.google.com/books?id=lA...tinuous+functions"&hl=en#v=onepage&q&f=false"

 

[Hjensen]

http://en.wikipedia.org/wiki/Bounded_variation"

You might want to check out "Analysis in classes of discontinuous functions and equations of mathematical physics":
http://books.google.com/books?id=lA...tinuous+functions"&hl=en#v=onepage&q&f=false"

I did have a look at Wikipedia before writing here. However, all it states is that BV functions have uses in mechanics, physics and chemical kinetics. I would have liked something a bit more specific. As for the book, it goes through the theory of BV - which I am already familiar with - but I can't find any physical applications of it. If you know a page number on which I could find this, I would appreciate it. Thanks for your time.

 

[dlgoff]

Page 326 talks about eigenvalues and eigenfunctions that are used in quantum physics.

 

[pmacias]

Functions of bounded variation (BV) are a fundamental concept in the field of mathematical analysis, and they have a wide range of applications in physics and engineering. One of the main reasons for their importance is that they provide a natural framework for studying functions with discontinuities, such as shock waves in fluid dynamics or phase transitions in thermodynamics.

In physics, BV functions are commonly used to model physical phenomena that exhibit sudden changes or jumps, such as in the case of discontinuous solutions to partial differential equations. For example, in fluid mechanics, the velocity of a fluid can be described by a BV function, as it may experience sudden changes in direction or magnitude at certain points. BV functions are also used in quantum mechanics to describe the behavior of particles at boundaries, where their wave functions may exhibit discontinuities.

In engineering, BV functions are used in the study of control systems, where they are used to model the behavior of systems with discontinuous inputs or outputs. They are also relevant in signal processing, where they can be used to analyze signals with abrupt changes or spikes.

Furthermore, BV functions have applications in optimization problems, where they can be used to characterize the regularity of a function and its variations. This is useful in engineering applications, where one may want to minimize the variations of a function in order to achieve a more stable and predictable system.

In summary, BV functions may seem like a purely theoretical concept, but they have numerous practical applications in physics and engineering. They provide a powerful tool for studying and understanding functions with discontinuities, and their relevance in these fields cannot be overstated.