What is Functional Derivation and How is it Used to Solve Complex Problems?

[etzzzz]

I have a problem with this simple (?) derivation

[tex] u(f(t),t) = \frac{\partial }{\partial f(t)}
\int_0^T \ g(f(t),f(t'),t,t') \ dt'
[/tex]

 

[Klaus_Hoffmann]

Hi 'etzzzz' the derivative you have put has some bit different notation.

The expression you gave is just a 'Frechet' derivative on an infinite dimensional space if we note

[tex] g( f(t) , f(t') ,t,t')=F [/tex]

considering that we are varying the function f(t') but not the function f(t) the Euler-Lagrange equations are.

[tex] \frac{ \partial g}{\partial f(t)} [/tex]

in case the derivative respect to t' of f(t') do not appear

 

[tacman]

Functional derivation is a mathematical technique used to find the derivative of a function with respect to another function. In this case, we are finding the derivative of the function u with respect to the function f(t). This can be useful in solving problems involving multiple variables and functions.

The expression given, u(f(t),t), represents the function u with arguments f(t) and t. The right-hand side of the equation, \frac{\partial }{\partial f(t)} \int_0^T \ g(f(t),f(t'),t,t') \ dt', is the derivative of the integral with respect to f(t). This means that we are finding the rate of change of the integral with respect to f(t).

It is important to note that this is not a simple derivation, as it involves finding the derivative of an integral, which can be a complex process. However, with the proper understanding of functional derivation and integration techniques, this problem can be solved.

In conclusion, functional derivation is a powerful technique in mathematics that allows us to find the derivative of a function with respect to another function. While this problem may seem complex at first, with the right approach and understanding, it can be solved effectively.