How to Implement Functional Derivative?

[yueming liu]

According to the definition, we need the limit of the function when alpha approaches 0. How to implement this on a Function object?

from sympy import *

from sympy.abc import *

from IPython.display import display

init_printing(use_latex='mathjax')

def func_diff(F, f, df):

alpha = Symbol('alpha')

F = F.subs(f, f+alpha*df)

dF = F.diff(alpha)

return simplify(dF.subs(alpha, 0))

u = Function('u')(x,y,z)

du = Function('\delta u')(x,y,z)

dfun = func_diff(u**2, u, du)

display(dfun)

[Francesco Bonazzi]

Maybe it's easier to define some if ... elif ... elif ... ... clauses to dispatch on the structure of F.

[Francesco Bonazzi]

Otherwise try to use multiple dispatch:

http://multiple-dispatch.readthedocs.org/en/latest/

That is, you dispatch on the expression type of F by defining your function many times.

[Aaron Meurer]

SymPy already supports taking derivatives with respect to functions symbolically, like F(f(x), x).diff(f(x)). Is this different from what you are looking for?

Aaron Meurer

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[Francesco Bonazzi]

On Thursday, January 29, 2015 at 10:35:53 PM UTC+1, Aaron Meurer wrote:

SymPy already supports taking derivatives with respect to functions symbolically, like F(f(x), x).diff(f(x)). Is this different from what you are looking for?

In [1]: f(x).diff(f(y))
Out[1]: 0

In a functional derivative, this would have yielded DiracDelta(x-y).

https://en.wikipedia.org/wiki/Functional_derivative