I have written a program (coefs.c) that will provide the coefficients for any order and any degree of differentiation or integration with or without smoothing at any point, all without solving any matrices. The solution is obtained using orthogonal polynomials. With neither integration nor differentiation, it yields interpolation coefficients. It is described with examples in Appendix A of this book http://www.amazon.com/dp/B07BS1DN1S which is free on these dates 1/1/2020, 1/9/2020, 1/17/2020, 1/25/2020. The source code is free and can be found in folder examples\coefs in this archive http://dudleybenton.altervista.org/software/Numerical_Calculus_examples.zip

Thanks, but I could not find an answer related to my question in your material. I'm looking to expand the first derivative of a given "functional" (not a function) to higher orders of accuracy.

Finite Differences (as well as other local methods) offer compact ways for representing high order approximations of the derivatives of any order, as long as the function is arbitrarily differentiable. This "Locality" principle reflects in the fact that the matrices representing numerical derivatives with given boundary conditions are "banded" with the number of involved diagonal growing the order of the derivative and with approximation order.

Fractional derivatives are nonlocal and this plays an important role, because even the use of local methods implies a global expansion. As a consequence, even with a low order approximation, the matrices representing numerical fractional derivatives with proper boundary conditions are dense matrices, still presenving the Toeplitz structure or the Locally Toeplitz asymptotic structure.

Of course the dense character of these matrices has an impact in the efficiency of the solution methods.

Dear Stefano,

Thank you for your answer, but still could not find a clear answer to my question plus the fact that I'm a physicist and not a mathematician makes me not understanding what is behind the lines (if there exist). My question is simply :

What should be the higher order derivative expansion for functionals compared to the expansion of functions given by f'(i) =( -11/6 f(i) + 3f(i+1) -3/2f(i+2)+1/3f(i+3))/h which is the 3rd order expansion of first derivative, then what would be the same order expansion for "functionals"?

Stefano Serra-Capizzano

Are these an answer:

https://math.stackexchange.com/questions/2965948/numerical-compuation-of-functional-derivative

https://math.stackexchange.com/questions/2341215/numerical-implementation-of-functional-derivatives

I understand that finite differences depend on the coordinate, so that functional derivatives cannot be directly calculated from finite difference. Your best option seems to be the Euler-Lagrange equation.

Muftah Al-Mahdawi

Dear Muftah, I've already checked the links and they are not useful. Concerning your point about locality, yes functionals are usually integrals and non local but I know the first order expansion of the first derivative is given by something similar to the functions derivative, hence I thought that should be an "finite difference" like expansion for the functional too. Indeed the functional first order derivative is given by : f'(phi)(i) = (f(phi+h*delta(i)) - f(phi))h.

Thank you Bijaya Basnet

It is the nth order derivative and is not practical, myself I'm looking for an expansion of the first derivative as in the question.

Dario Fasino ,

This is exactly what I have asked, an "nth" order first functional derivative. I was wondering would it have a similar expansion like the normal function in spite of the different input and output spaces of both (for functionals the input is function and the output is scaler)? So your answer is yes (given the formula you provided). Then should we use only one direction? I mean should it be F(f + 2h𝜂) or F(f + h𝜂1 + h𝜂2). Thank you again, finally someone understood my question.

Dario Fasino ,

Thank you for the clarification, however I'm a bit not very sure of whether to choose F(f + 2h𝜂) or F(f + h𝜂1 + h𝜂2) in analogy to f(i+2) (for normal functions).