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

[etzzzz]

I have a problem with this simple (?) derivation

$$u(f(t),t) = \frac{\partial }{\partial f(t)} <br /> \int_0^T \ g(f(t),f(t'),t,t') \ dt'$$

 

[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

$$g( f(t) , f(t') ,t,t')=F$$

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

$$\frac{ \partial g}{\partial f(t)}$$

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