M. Caputo and M. Fabrizio proposed in  a fractional derivative of order α ∈ (0, 1) of an absolutely integrable function f on , for any , f ∈ ACloc([t0, ∞) (which means that the first derivative of f is integrable on for every in the form
where and M(α) is a normalization function such that M(0)=M(1)=1. We take M(α)=1 in the analysis which is to follow. Thus, we consider (see , p. 89) (1.1)
where * denotes the convolution.
In this paper we reconsider , in the distributional setting, even introduce a Riemann-Liouville version of such fractional derivative and examine several properties of that are important in applications. One can simply derive the similar properties of the Riemann-Liouville version of this notion. By the use of the Caputo-Fabrizio fractional derivative (CFFD), we shall formulate the Hamilton principle of analytical mechanics and its generalization corresponding to the case when a memory of a system and a time of a process do not coincide. The results presented here follow the corresponding ones for “standard” Caputo fractional derivative , cf. [8,14,17,22] and references given therein.
where k is a function with specified properties. For example, in  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 . 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  and . The classical cases of Riemann-Liouville and Caputo fractional derivatives correspond to k given as . The analysis presented here corresponds to the case when , called CF kernel. We refer to Section 2 for the explanation of the introduction of parameter T. Note that in papers ,  in the definition of (CFFD), it was assumed that T=1.
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 , by t/T. The modified definition of (CFFD) proposed in this paper is (2.1)
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 ) has to be modified with instead of .
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 ,
Then, we define the generalization of (1.1) corresponding to Riemann-Liouville of fractional derivative, (RLCFFD), with CF kernel
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).
Suppose that f ∈ ACloc([0, ∞)). Then the following holds
where the Caputo-Fabrizio fractional integral is defined as
Next, we consider the distributional Laplace transform (see , 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 . For the later use, we note that the same arguments give, for (2.3)
Let α=A+iB be a complex number so that A=ℜα ∈ (0, 1). We can define by (1.1). Since we prefer to work with real valued functions after fractional differentiation, we follow our approach presented in  and define combination of (CFFD) of complex order as:
where 𝕋 is a constant having the dimension of time and can be interpreted as relaxation time.
3 Some properties of
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 (3.1)
for arbitrary . In order to estimate I, we shall examine the properties of the kernel in (1.1).
Suppose that f ∈ ACloc([0, ∞)) is real valued. Then, I(f)⩾0 for every .
By (2.3), . The Bochner-Schwartz theorem (, p. 331) implies that e−λ∣τ∣ is a function of positive type (positive definite). Recall that a function f is of positive type if for any test function θ ϵ D(ℝ) there holds 〈f(t), θ * θ*〉 ≥ 0 where This result holds in a more general case for positive definite tempered distributions which is not needed here. The derivative of f in is an integrable function so we consider f(1) as a locally integrable function in [0, ∞) equal zero outside . Let us denote by ϕn, n ϵ ℕ, a sequence of smooth functions, supported by , which converges to f(1) in . Since f is real valued, we take ϕn, n ϵ ℕ to be real valued. By the Bochner-Schwartz theorem, for every n,
Letting n → ∞, by the Lebesque theorem, we obtain
we obtain I(f)⩾0.
The result of Proposition 3.1 OK for the case of Riemann-Liouville fractional derivative, was proved in ,  and . In  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 but . Our result presented here can be simply transferred to the similar assertion with the (RLCFFD).
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.
Suppose that f ∈ ACloc([0, ∞)) and that it is real-valued. Then, for ,
if the following condition is satisfied: (3.2)
Without loss of generality we assume that β ⩾ α. Let
Then, by (2.3), (3.3)
It follows from (3.3) that , if (3.4)
(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 . First, we define the right (CFFD) of real order 0<α<1 for f ∈ ACloc([0, ∞)) as
where 0 < a<t < b.
Suppose that f ∈ ACloc([a, b)]), 0<α <1, 0 < a<b. Then (3.5)
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 and continue to consider functions in ACloc([0, ∞)).
Suppose that f ∈ ACloc([ 0, ∞)) and 0<α <1. Then, for ,
where the convergence is uniform on and (3.6)
are moments of f(1). In particular, the approximation of may be written as
We have (3.7)
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).
Using the same procedure, we obtain the approximation formula for as
where the series converges in the uniform sense and
We derive now the expression for derivative of with respect to α. We state this as:
Suppose that f ∈ ACloc([0, ∞)), 0<α<1. Then (3.8)
4 Variational principles of Hamilton type with
In this section we present the necessary conditions for an extremum in the case when a Lagrangian density contains . We start with the generalization of the classical Hamilton principle that we state as the following two problems.
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 .
Let y* ∈ 𝓊 exist so that (4.1)
Suppose that (4.2) (4.3)
Then, for t ∈ (a, B), y* in (4.1) has to satisfy (4.5)
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 , gives (4.6)
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
Let a=A. Then (4.5) reduces to the known equations (see ) 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 (4.7)
where 𝓊 is given by (4.4) and A=[α0, α1], with 0<α0<α1 < 1. We assume that (4.8) (4.9)
Let y=y*+ε1f, α=α*+ε1Δ α, where f and Δα are fixed. Substituting this in (4.7), we obtain
Conditions imply (4.10)
The application of in Viscoelasticity for the Stress relaxation and the Creep in viscoelastic body of Hooke-Newton type, Kelvin-Voigt and Generalized Zener model will be given in our paper in preparation.
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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Published Online: 2018-03-13
Published in Print: 2018-02-23