Abstract
The Caputo-Fabrizio fractional derivative is analyzed in classical and distributional settings. The integral inequalities needed for application in linear viscoelasticity are presented. They are obtained from the entropy inequality in a weak form. Moreover, integration by parts, an expansion formula, approximation formula and generalized variational principles of Hamilton’s type are given. Hamilton’s action integral in the first principle, do not coincide with the lower bound in the fractional integral, while in the second principle the minimization is performed with respect to a function from a specified space and the order of fractional derivative.
1 Introduction
M. Caputo and M. Fabrizio proposed in [11] a fractional derivative of order α ∈ (0, 1) of an absolutely integrable function f on
where
where * denotes the convolution.
In this paper we reconsider
We comment now a class of fractional derivatives to which (1.1) belongs (cf. [20]). In general, fractional derivative of Riemann-Liouville and Caputo type can be defined as
where k is a function with specified properties. For example, in [20] it is required that the Laplace transform of k satisfies conditions that guarantee certain useful properties of (1.2). Relation (1.2) may also be used to define variable order fractional derivatives as it is proposed in [26]. However, in this case k satisfies additional conditions. Many choices of k, in forms of special functions, with integrals not necessarily of convolution type, for example with k being Meijer G-function, and more generally, Fox H -function of the specific form, lead to generalized fractional calculus presented in [18] and [19]. The classical cases of Riemann-Liouville and Caputo fractional derivatives correspond to k given as
The paper is organized as follows. We present in Section 2 distributional framework for (CFFD) as well as some generalizations including symmetrized complex order (CFFD). Moreover, we introduce the Riemann-Louiville type fractional derivative with CF kernel. Several properties of (CFFD) are derived in Section 3, while Section 4 is related to two variational problems for (CFFD).
2 An intrinsic approach to new fractional derivatives
We assume that f ∈ ACloc([t0, ∞)), t0⩾0, and that f(1) is supported by [t0, ∞). The independent variable t ∈ [ t0, ∞) in (1.1) is dimensionless. In cases when t has a dimension (time for example), we introduce a parameter T having the dimension of t and replace t in kernel of
In this definition one has two parameters, α ∈ (0, 1) and T ∈ (0, ∞). If we take T=1, then (2.1) becomes (1.1).
Remark 2.1
The kernel, initially proposed in [11] was modified in [12] so that the following definition was proposed (see [12], eq. (1)
where T0 has the same meaning as our T. However, the expression (2.2) does not reduce to f(1) in the limiting case α → 1− since
Therefore, the modification of the kernel, in order to make the argument of the exponential function dimensionless, requires that the normalization function (in the terminology of [12]) has to be modified with
2.1 Distributional framework
In order to simplify the notation, we assume T=1 and t0=0. Let H denote the Heaviside function so that it is right continuous at 0, that is H (0)=1. We will consider f ∈ ACloc([0, ∞)) as well as functions f ∈ ACloc((−∞, ∞)) so that their restrictions on [0, ∞) denoted by fH belong to ACloc([0, ∞)).
Let α ∈ (0, 1) and
By the use of distribution theory, one has, for any
which implies
Then, we define the generalization of (1.1) corresponding to Riemann-Liouville of fractional derivative, (RLCFFD), with CF kernel
so that
The properties are stated in the next proposition. The proofs are simple (in the distribution framework) and they are omitted. In the sequel, the assumption f ∈ AC ([0, ∞)) enable us to consider f on (−∞, ∞) being equal zero on (−∞, 0).
Proposition 2.1
Suppose that f ∈ ACloc([0, ∞)). Then the following holds
where the Caputo-Fabrizio fractional integral is defined as
Moreover,
Next, we consider the distributional Laplace transform (see [27], p. 127). Let f be of exponential growth, that is, f is locally integrable on [0, ∞) and ∣f(t)∣ ⩽ Meωt, t > 0 (ω ϵ ℝ). Then, the Laplace transform is defined as
It is a matter of simple calculations to show that
Recall that the Fourier transform is defined by
Changing s with iω, ω ϵ ℝ we obtain the Fourier transforms of both types of fractional derivatives (having in mind that supp f⊂[0, ∞)). Let
Let α=A+iB be a complex number so that A=ℜα ∈ (0, 1). We can define
where 𝕋 is a constant having the dimension of time and can be interpreted as relaxation time.
3 Some properties of t 0 C F D t α f
Since the previous exposition of Section 2 is the same if we consider t0 > 0 instead of t0=0, for the sake of simplicity, we continue with the assumption t0=0.
3.1 A consistency result
In viscoelasticity and heat conduction problems with fractional derivatives, in proving consistency of a model with the Second law of Thermodynamics, the estimate of the following functional is needed
for arbitrary
Proposition 3.1
Suppose that f ∈ ACloc([0, ∞)) is real valued. Then, I(f)⩾0 for every
Proof
By (2.3),
Letting n → ∞, by the Lebesque theorem, we obtain
Since
we obtain I(f)⩾0.
Remark 3.1
The result of Proposition 3.1 OK for the case of Riemann-Liouville fractional derivative, was proved in [25], [15] and [3]. In [24] the problem of estimating an integral of type (3.1) was treated for the case when the lower bound in the integral is −∞ and when the support of the functions involved in the integration is not
The result of Proposition 3.1 NOT: 1 can be generalized in order to derive dynamically consistent models for viscoelastic bodies of Kelvin-Voigt type. We formulate this result as follows.
Proposition 3.2
Suppose that f ∈ ACloc([0, ∞)) and that it is real-valued. Then, for
if the following condition is satisfied:
Proof
Without loss of generality we assume that β ⩾ α. Let
Then, by (2.3),
It follows from (3.3) that
Since
(3.4) leads to (3.2). Thus, as in the previous assertion, we conclude that F is of positive type. Now we proceed in the same way as in the proof of Proposition 3.1 and conclude that the assertion holds true.
3.2 Partial integration
We present integration by parts formula for
where 0 < a<t < b.
Proposition 3.3
Suppose that f ∈ ACloc([a, b)]), 0<α <1, 0 < a<b. Then
The proof is easy and we omit it. Similarly, one can prove the corresponding assertion for the (RLCFFD).
3.3 Approximation formula
Next we derive an approximation formula needed for the development of an appropriate numerical procedure for solving differential equations with fractional derivatives. We follow our approach given in [4,5] for
Proposition 3.4
Suppose that f ∈ ACloc([ 0, ∞)) and 0<α <1. Then, for
where the convergence is uniform on
are moments off(1). In particular, the approximation of
with
and
Proof
We have
Decompose (3.7) as
where QN+1(t) is the remainder. We have the estimate for
Therefore, the statement of the proposition follows, with Vk(f(1))(t) given by (3.6).
Remark 3.2
Using the same procedure, we obtain the approximation formula for
where the series converges in the uniform sense and
We derive now the expression for derivative of
Proposition 3.5
Suppose that f ∈ ACloc([0, ∞)), 0<α<1. Then
Proof
Definition gives (1.1) we have
Therefore, (3.8) follows.
Note that the result equivalent to (3.8) for Riemann-Liouville fractional derivative was obtained in [7] and can be, as above, transferred for the (RLCFFD).
4 Variational principles of Hamilton type with 0 C F D t α y ( t )
In this section we present the necessary conditions for an extremum in the case when a Lagrangian density contains
4.1 Problem 1
Find necessary conditions for the existence of minimum of a functional
where y belongs to a prescribed set 𝓊 ⊂ ACloc ([0, ∞)) described below, [ A, B] ⊂ (a, b) ⊂ [0, ∞). In this exposition we follow our paper [6].
Let y* ∈ 𝓊 exist so that
Suppose that
Proposition 4.1
Additionally to (4.2) and (4.3), assume that the set of admissible functions 𝓊 is
Then, for t ∈ (a, B), y* in (4.1)has to satisfy
Proof
Suppose that (4.1) holds and let y=y*+ε f. From y(a)=y0, we conclude that f(a)=0. Then, by the standard procedure, the condition
Integration by parts formula (3.5), applied to the interval [ a, B], leads to
After the use of boundary conditions defined by (4.4), we obtain
Integration by parts formula for the interval [ a, A] leads to
Again, the boundary terms vanish so that
Since f is arbitrary, the fundamental lemma of variational calculus (see [1], Lemma 3.31 and [16]) leads to (4.5).
Remark 4.1
Let a=A. Then (4.5) reduces to the known equations (see [2]) and the elements of the set of admissible functions satisfy the natural boundary conditions. The interpretation of (4.5)2 subjected to y(a)=0, is that y(t) represents the history of the process for t ∈ (a, A) which contributes to the extremum of the action integral I in the interval (A, B).
4.2 Problem 2
Here we consider the minimization problem of a functional when both y and α are independent variables in the functional. Thus, we consider
where 𝓊 is given by (4.4) and A=[α0, α1], with 0<α0<α1 < 1. We assume that
Proposition 4.2
Suppose that (4.8) and (4.9) hold. If
Proof
Let y=y*+ε1f, α=α*+ε1Δ α, where f and Δα are fixed. Substituting this in (4.7), we obtain
Conditions
By (3.8) in (4.10)2 and the fact that f is arbitrary, we obtain the necessary conditions of the proposition. □
Remark 4.2
The application of
Acknowledgements
This work was supported by the grants of the Ministry of Education, Science and Technological Development of Republic of Serbia, 174005 (TMA and DZ), and 174024 (SP). It is also under the working program of the bilateral project between Serbian Academy of Sciences and Arts and Bulgarian Academy of Sciences.
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