1. Introduction
In the framework of the current discussions regarding the “right” fractional derivatives [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12], the main suggested approach was to define the fractional derivatives via the Fundamental Theorem of Fractional Calculus (FC), i.e., as the left-inverse operators to the corresponding fractional integrals that satisfy the index low, interpolate the definite integral, and build a family of the operators continuous in a certain sense with respect to the order of integration.
According to a result that was derived in [
13], under the conditions mentioned above, the only family of the fractional integrals defined on a finite interval are the Riemann-Liouville fractional integrals [
14]. Until recently, three families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals were discussed in the literature: the Riemann-Liouville fractional derivatives [
14], the Caputo fractional derivatives [
15], and the Hilfer fractional derivatives [
16]. However, in [
7], infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann–Liouville fractional integrals, were introduced and called the
nth level fractional derivatives. These derivatives satisfy the Fundamental Theorem of FC, i.e., they are the left-inverse operators to the Riemann–Liouville fractional integrals on the appropriate nontrivial spaces of functions that justifies calling them the fractional derivatives.
In [
7], some basic properties of the
nth level fractional derivatives were studied, including a description of their kernels. However, the question of their applicability to some real world problems remained open. In this paper, we provide a first evidence of their usefulness for applications on an example from the linear viscoelasticity. More precisely, we show that the solution to the Cauchy problem for the fractional relaxation equation with the
nth level fractional derivative is a completely monotone function that can be represented in form of a linear combination of the Mittag–Leffler functions with some power law weights. As discussed in [
17], the property of complete monotonicity is characteristic for any relaxation process. Only in this case, it can be interpreted as a superposition of (infinitely many) elementary, i.e., exponential, relaxation processes. In linear viscoelasticity, the assumption of complete monotonicity of the solutions to the relaxation equations that model the relaxation processes is usually supposed to be fulfilled, see, e.g., [
17] and references therein.
The rest of this paper is organized as follows: in
Section 2, we discuss some new properties of the
nth level fractional derivative, including the explicit formulas for the projector of the
nth level fractional derivative and for its Laplace transform.
Section 3 addresses the Cauchy problem for the relaxation equation with the
nth level fractional derivative and properties of its solution, including complete monotonicity. In particular, we focus on an important particular case of the Cauchy problem for the relaxation equation with the second level fractional derivative.
3. Fractional Relaxation Equation with the nth Level Fractional Derivative
We start this section with a short discussion of the simplest fractional differential equation with the
nth level fractional derivative of order
, namely, the one in the form
In fact, we already considered this equation in the previous section because its solution is the kernel of
. Depending on the parameters
, the kernel dimension ranges from
n to 1, because the
nth level fractional derivative can degenerate to the fractional derivatives with the level
to 1. In the case, the conditions (
9) are satisfied, the kernel of
is
n-dimensional and the general solution to the Equation (
39) is as follows:
being arbitrary constants. To guarantee uniqueness of solution to the Equation (
39),
n initial conditions
are required. The initial-value problem for the Equation (
39) with the initial conditions (
41) then has the unique solution
This situation is rather unusual for the fractional differential equations with a fractional derivative of the order and has consequences for their possible applications, as we will see in the further discussions.
Now, we consider the fractional relaxation equation with the nth level fractional derivative and prove the following result:
Theorem 3. The fractional relaxation equationwith the initial conditions given by (41) has a unique solution given by the formula In the case, the initial conditions are non-negative ( in (41)) and the conditionshold true, the solution (44) is completely monotone. In the formulation of the theorem,
stands for the two-parameters Mittag–Leffler function that is defined by the following convergent series:
We apply the Laplace transform method to prove the theorem. First, we do this formally, but then verify that the Laplace transform of the obtained solution does exist. Using the Formula (
32), the initial-value problem (
41) and (
43) can be transformed to the Laplace domain:
The solution in the Laplace domain is as follows:
Now, we employ the well-known formula
and immediately arrive at the Formula (
44) for the solution to the fractional relaxation Equation (
43) with the initial conditions (
41).
As already mentioned in the previous section, under the conditions (
9), the exponents
fulfill the inequalities
. For
, the Mittag–Leffler function has the following asymptotics as
:
These both facts ensure that the Laplace transform of the function at the right-hand side of (
44) does exists and, thus, it is indeed the unique solution to the initial-value problem (
41) and (
43).
Now, let us prove that the solution (
44) is a completely monotone function provided the conditions (
45) hold true and the initial conditions are all non-negative.
For the reader’s convenience, we recall that a non-negative function is called completely monotone if it is from and for all and .
The Mittag–Leffler function
is completely monotone if and only if
and
[
20]. The power law function
is a Bernstein function, because its derivative
is completely monotone. A composition of a completely monotone function and a Bernstein function is completely monotone [
21]. Therefore, the function
is completely monotone for
,
, and
. The product of two completely monotone functions is again a completely monotone function [
21] that leads to the complete monotonicity of the function
under the conditions
The functions
from the solution Formula (
44) have the form (
51). Moreover, their parameters
,
, and
fulfill the conditions (
52), because of the conditions (
1), (
9), and (
45) we posed on the order
and the type
of the
nth level fractional derivative. Thus, the functions
are all completely monotone as well as their linear combination with non-negative coefficients that builds the solution (
44). The proof of the theorem is completed.
Remark 4. Taking into account the Formula (50) for the asymptotics of the Mittag–Leffler function, the behavior of the solution (44) as has the following form:where the coefficients depend on both the initial values , the order α, and the type . Thus, the fractional relaxation Equation (43) can be employed to model the relaxation processes with the asymptotic behavior of type (53). Moreover, the free parameters can be used for optimal fitting of the measurements data for a concrete relaxation process with a power law asymptotics. Remark 5. In [18], uniqueness and existence of solutions to the Cauchy problems for the linear and non-linear fractional differential equations with the Djrbashian–Nersesian operators that are similar to the nth level fractional derivatives (see Remark 2) were addressed. In particular, an analogy of the Formula (44) on a finite interval was deduced while using the method of power series. However, no analysis of the solution properties, including their complete monotonicity was presented there. It is worth mentioning that, in [22], the eigenfunctions and the associated functions for some boundary value problems for the special equations containing the Djrbashian–Nersesian operators were constructed in explicit form. Moreover, these functions were interpreted as the bi-orthogonal systems of vector functions and used for the interpolation expansions for some Hilbert spaces of entire functions. In the rest of this section, we illustrate Theorem 3 on the case of the fractional relaxation equation with the truly second level derivative (the conditions (
9) with
hold true)
and with the initial conditions
According to Theorem 3, its solution
is completely monotone if the initial conditions
are non-negative and the following restrictions on the order
and type
of the second level fractional derivative hold true:
The points of the
-plane that satisfy the conditions (
57) are graphically represented in the plot of
Figure 1.
They build a triangle and, in what follows, we shortly discuss the particular cases of the fractional relaxation Equation (
43) that correspond to the vertexes and edges of this triangle. As already mentioned in the previous section, the second level fractional derivatives with
(upper edge of the triangle) are reduced to the first level derivatives (the Riemann–Liouville, the Caputo, and the Hilfer fractional derivatives). Still, it is instructive to include them into considerations.
We start with the vertexes and write down the types of the corresponding fractional derivatives, their form, and the solutions to the fractional relaxation Equation (
43) with these derivatives:
The Riemann-Liouville fractional derivative:
- –
,
- –
,
- –
.
The Caputo fractional derivative:
- –
,
- –
,
- –
.
A truly second level fractional derivative:
- –
,
- –
,
- –
.
Whereas, the first two cases (the fractional relaxation equations with the Riemann-Liouville and the Caputo fractional derivatives) are well-known, the third case seems to be new. The solution to the corresponding relaxation equation is a linear combination of the solutions to the fractional relaxation equations with the Riemann–Liouville and with the Caputo fractional derivatives.
Now, let us inspect the edges of the triangle from
Figure 1.
The Hilfer fractional derivative:
- –
,
- –
,
- –
.
A truly second level fractional derivative:
- –
,
- –
,
- –
.
The solution to the relaxation equation with the second level fractional derivative is a linear combination of the solutions to the fractional relaxation equations with the Caputo fractional derivative and with the Hilfer fractional derivative with the type .
A truly second level fractional derivative:
- –
,
- –
,
- –
.
The solution to the relaxation equation with the second level fractional derivative is a linear combination of the solutions to the fractional relaxation equations with the Riemann–Liouville fractional derivative and with the Hilfer fractional derivative with the type .
In all other cases, the solution to the fractional relaxation Equation (
54) is given by the Formula (
56). It is worth mentioning that (
56) can be interpreted as a linear combination of solutions to the fractional relaxation equations with the Hilfer fractional derivatives of order
and with the types
and
, respectively.
Finally, let us mention that the second Fundamental Theorem of FC for the
nth order fractional derivative (Remark 3) can be used for the analysis of more complicated and even nonlinear fractional differential equations. Say, the fractional differential equation
subject to the initial conditions in form (
41) can be transformed to the following Volterra-type integral equation of the second kind by applying the Riemann–Liouville fractional integral to the Equation (
58) and by using the Formula (
18):
The integral Equation (
59) can be analyzed by the standard method of the fix point iterations. This problem will be considered elsewhere.