The idea goes back to so-called variational (functional) derivatives [
Let 
be a functional over function
. Then ratio
is named a variational (or functional) derivative. It is obvious that this derivative is also a functional that depends on function 
and point t (as a parameter). In this way we can define the second functional derivative of 
with respect to 
at point
:
.
This is again a functional with respect to 
that depends on the couple of parameters 
and so on.
In the case when functional 
depends on functional 
(this is the most interesting case for us) the functional derivative satisfies chain rule:
.
And with
, we have
.
Also notice that the functional derivative of functional 
with respect to function 
satisfies
and this is very convenient for differentiation procedure.
For following purposes we take the functional 
. Then we will obtainaccording to above formulas:
if
.
Functional Taylor’s series for functional 
over function η(τ) about a point η ≈ 0 looks like [
where functional shift operator 
is understood in the sense of expansion over infinite integration limits.
In [1-3] we used the special summation procedure of diagrams of a certain type and received the diffusion equation with fractional derivatives. In this section, we will exploit another method for to receive the self-contained systems of governing equations for diffusion problems that we have used also in [
.
Averaging with respect to realizations 
of the random porosity yields
This equation contains the unknown
, and also 
and
. However, the Furutsu-Novikov formula connects the new unknowns to functional derivatives of 
itself, since [4,6-8] the concentration 
is a functional of
. Indeed, we have:
Here
, 
, and so on,
are functional derivatives of increasing order k or of functional
, with respect to
, at points
. The functions 
above are cumulant of random field
. A similar expression can also be derived for functional 
instead of 
in the equation above
.
By substituting all above into (1), we derive:
We see here that together with averaged concentration 
we have also averaged values of functional derivatives for 
of different orders:
. By taking the functional derivative of (1) with respect to 
at point
, we obtain
Here 
in turn is a random functional 
, with two free coordinates which are
. By averaging (2) over random realizations of
, we receive the following equation for
:
This equation contains the unknown function 
and also new unknown functions, which are 
and
. If we apply Furutsu-Novikov formula to functionals 
and 
instead of
, then substituting the results into (3), we find: (see below (4))
We should have written here “and so on”, since after that we should take the functional derivative over 
and then average the obtained equations over the realizations of 
and apply Furutsu-Novikov’s formula again. As a result, we would obtain an equation for 
. Iteratively using the procedure would lead us to an infinite system of equations for the sequence of functions
, if we set
.
The structure of the thus obtained equations is such that the k-th equation in the hierarchy, with unknown 
on the left hand-side has
, 
, and the concentration 
in its right hand-side. And we are led to set the natural question “When can we break this chain of equations?”. It turns out that under quite reasonable assumptions, this problem can be solved.
Here we consider random media, where cumulant functions entering Equation (4) are of the even order 
where 
denotes the amplitude of the telegraph or normal random field
. Here we should assume that in a porous media model, and due to the definition of the porosity 
we should take for granted that
. Considering the fluctuations to be weak, we neglect cumulant of the order higher than two. Moreover, each time 
is a centered Gaussian process, all the cumulant are equal to zero, except 
which also is to the two-point correlation function of process
. That is why in (4) and as well as in subsequent equations, which are not given here, we can leave only the addends containing the second order correlation functions. In this case, instead of above we get, respectively:
We mentioned that if porosity represents itself a normal random field, then (5) is exact because all cumulant except one are zero. As we consider small porosity fluctuations, then in the right-hand side of (5) the third and the fourth addend in the right-hand part, containing
, will be neglected because they are proportional to
. Since also the functional derivative of
, computed respect to 
itself at 
is 
we arrive at the following system of equations, which turns out to be closed and has the following form:
where 
is the averaged concentration, while 
is the first functional derivative of the concentration, with respect to
. And 
denotes the correlation function of second order of random field
. System (6)-(7) will be solved in the unbounded domain
, starting from the following initial condition:
This leads us immediately to the following initial condition for
:
Indeed, we have 
> since the initial concentration 
is determined and independent of
.
In order to give the integrals on the right hand-side of (6) a concrete meaning, we now specialize the Gaussian process 
by assigning a definite expression to its correlation function.
is homogeneous and isotropic. Then, without any concrete definition of the correlation function
, let us note that the latter depends only on the modulus of its arguments’ difference
.
. The procedure of getting such as equation seems evident. It is clearly seen from (7) that 
are generated by the concentration that appears in the right-hand part of this equation. The function 
can be found using the Green’s function of operator
. Then, after substituteing the obtained solution for 
into (6), we get the final integro-differential equation for
.
- representation. In (6) and (7) it is more convenient to turn to 
-representation, i.e. to use Laplace transform with respect to time (q-parameter) and Fourier transform over with respect to space (k-parameter). After all the necessary calculations instead of (6) and (7), we get, respectively:
of 
and 
is omitted for the sake of simplicity. We also have
.
in (11) instead of argument
, then we get the following equation for function 
appearing in (10):
. Substituting in (10) the thus obtained expression, we get an algebraic equation with respect to
:
-representation. Therefore, Equation (12) in 
representation is equivalent to basic equation [