Computing a functional derivative using the VariationalMethods package

Question

To test the VariationalMethods` toolbox, I tried to compute $\frac{\delta I[y]}{\delta y(x)}$ for the following functional, assuming $w(x)$ is some well-behaved function:

$\qquad I[y(x)]=\int y(x) g\left(\int w(t) y(t-x)dt\right)dx$

The exact answer is:

$\qquad \frac{\delta I[y]}{\delta y(x)}= g\left(\int w(t) y(t-x)dt\right) + \int w(x)y(x)g'\left(\int w(t) y(t-x)dt\right)dx$,

where $g'(x)$ is the derivative of $g(x)$.

Mathematica did not find the above result.

These are the commands I run:

<< VariationalMethods`
VariationalD[y[x]*g[Integrate[w[t]*y[t - x], t]], y[x], x]

The the above code gives only the first term of the exact answer. What am I doing wrong? Is it even possible to compute such variations with Mathematica?

Answer 1

You could use FunctionalD from my answer to Non-trivial functional derivatives, which I repeat here:

FunctionalD[expr_, v:(f_[_]|{f_[_],_Integer}) .., OptionsPattern[]] := Internal`InheritedBlock[{f},
    f /: D[f[x_], f[y_], NonConstants->{f}] := DiracDelta[x-y];
    f /: D[f, f[y_], NonConstants->{f}] := DiracDelta[#-y]&;

    D[expr, v, NonConstants->{f}]
]

Then, for your example, we have:

Assuming[
    (s|x) ∈ Reals,
    FunctionalD[
        Integrate[y[x] g[Integrate[w[t] y[t-x], {t, -∞, ∞}]], {x, -∞, ∞}],
        y[s]
    ]
] //TeXForm

$\int_{-\infty }^{\infty } \left(\delta (x-s) g\left(\int_{-\infty }^{\infty } w(t) y(t-x) \, dt\right)+y(x) w(s+x) g'\left(\int_{-\infty }^{\infty } w(t) y(t-x) \, dt\right)\right) \, dx$

You still need to distribute integrand, and resolve the Dirac delta function, but both terms are there.