The Functional Equation Headquarter
What is a Functional
The concept of functional generalizes that of function. While a function is an object of a certain variable which gives as result a value a functional is an object that has functions as variables and gives functions as results. The derivation is a functional since it takes a function and gives a function as a result. Here is an example of linear functional: the variable twist functional.
Functionals can be generally expresses in integral form which is a technique to make them fall onto the ground of integral operators and eigenfunctions. For example, the translation functional
which is also a linear functional can be expressed as an integral operator in the following way
where I have used the Dirac's Delta function under the integration symbol. Of course there are a lot of other functionals arising in different problems of physics and engineering. Some of these have notable properties as the following parity functional
whose square (this is also true for the variable twist functional) is equal to the identity functional.
What are Functional Equations
Functional equations are equations where the unknown, a function, has to be found. This function must satisfy the functional relations contained in the LHS and in the RHS. As an example, differential equations are functional equations since (as just said) the derivation is a functional. However the classification of functional equations goes much further and includes, integral, Boltzmann, partial derivative and many other equations actually treated in very different ways in mathematics. In functional equations, for example, you may have a function whose time-behavior is given by the action of some functional as in the following case
which can, as always, be expresses in integral or extended form depending on your specific needs.
My Aim
I intend to generalize the techniques I have worked out for the Boltzmann equation to all functional equations. This is very easy to do in the linear case but I still don't know how to proceed in the non-linear case. This is however the most interesting one and I intend to work out something. Actually, as an example, partial derivative equations are much easier if viewed from a functional point.
How to Use this Page
If in your research work you have encountered a strange or uncommon equation you might want to know about its solvability with functional techniques. To do so you can fill out a form here with the description of the equation and eventually (best) put the Tex/LaTex code in the apropriate text box. I will examine your equation and email you telling about the possibility of solving. Since this is an experimental free service, I will split multiple requests and put those exceeding the first in queue. Please, do not expect timely replying since this is not a professional service.