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Properties of the Caputo-Fabrizio fractional derivative and its distributional settings

Teodor M. Atanacković / Stevan Pilipović
  • Department of Mathematics and Informatics Faculty of Sciences, University of Novi Sad, “Trg D. Obradovića”, 4 Novi Sad – 21000, Serbia
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/ Dušan Zorica
  • Mathematical Institute, Serbian Academy of Arts and Sciences, “Kneza Mihaila”, 36 Beograd – 11000, Serbia
  • Department of Physics, Faculty of Sciences, University of Novi Sad “Trg D. Obradovića”, 4 Novi Sad – 21000, Serbia
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Published Online: 2018-03-13 | DOI: https://doi.org/10.1515/fca-2018-0003

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.

MSC 2010: Primary 26A33; Secondary 46F10; 70H25

Key Words and Phrases: Caputo-Fabrizio fractional derivative; variational calculus; linear viscoelasticity

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 [t0,T˜], for any T˜>t00, fACloc([t0, ∞) (which means that the first derivative of f is integrable on [0,T˜) for every T˜>0) in the form t0CFDtαf(t)=M(α)1αt0texp(α1α(tτ))f(1)(τ)dτ,t>t0,

where f(1)=ddτf 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 [21], p. 89) t0CFDtαf(t)=11αt0texp(α1α(tτ))f(1)(τ)dτ=11αexp(α1αt)f(1)(t),t(t0,),(1.1)

where * denotes the convolution.

In this paper we reconsider t0CFDtα,α(0,1), in the distributional setting, even introduce a Riemann-Liouville version of such fractional derivative and examine several properties of t0CFDtαf(t) 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 t0CDtαf(t), cf. [8,14,17,22] and references given therein.

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 RLDtαy(t)=ddtt0tk(tτ)y(τ)dτ,CDtαy(t)=t0tk(tτ)dy(τ)dτdτ,(1.2)

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 k(t)=tαΓ(1α),t>0,0<α<1. The analysis presented here corresponds to the case when k(t)=1(1α)Tαexp(α1αtT),t>0,0<α<1,T>0, called CF kernel. We refer to Section 2 for the explanation of the introduction of parameter T. Note that in papers [10], [13] 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 fACloc([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 t0CFDtαf(t), by t/T. The modified definition of (CFFD) proposed in this paper is t0CFDtαf(t)=1Tα(1α)t0texp(α1αtτT)f(1)(τ)dτ,t(t0,).(2.1)

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) Dtαf(t)=1(1α)t0texp(α1αtτT0)f(1)(τ)dτ,t>t0,(2.2)

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 limα1Dtαf(t)=T0f(1)(t),t>t0.

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 1Tα(1α) instead of 11α.

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 fACloc([0, ∞)) as well as functions fACloc((−∞, ∞)) so that their restrictions on [0, ∞) denoted by fH belong to ACloc([0, ∞)).

Let α ∈ (0, 1) and φα(t)=11αexp(α1αt),t=(,).

By the use of distribution theory, one has, for any θC0(), Hϕα,θ=11α0eατ1αθ(τ)dτ=0eαsθ((1α)s)ds,

which implies HϕαδinS()asα1.

Then, we define the generalization of (1.1) corresponding to Riemann-Liouville of fractional derivative, (RLCFFD), with CF kernel 0RLCFDtαf(t)=((Hφα)(Hf))(t),t(,),

so that 0CFDtαf(t)=0RLCFDtαf(t)f(0)φα(t),t(0,).

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 fAC ([0, ∞)) enable us to consider f on (−∞, ∞) being equal zero on (−∞, 0).

Proposition 2.1

Suppose that fACloc([0, ∞)). Then the following holds 0RLCFDtαf(t)=11αf(t)α1αCF0Itαf(t),t(0,),

where the Caputo-Fabrizio fractional integral is defined as 0CFItαf(t)=(Hφα)(Hf)(t),t(0,).

Moreover, limα0+I0CFt1αf(t)=f(t),t0,limα1I0CFt1αf(t)={0tf(u)du,t>0,0,t<0,limα1D0RLCFtαf(t)=(Hf)'(t)=(Hf')(t)+f(0)δ(t),t(,)limα0+0CFDtαf(t)=(Hf'(t)=f'(t),t>0,(f'islocallyintegrableon[0,))limα0+D0RLCFtαf(t)=f(t),t>0limα0+0CFDtαf(t)=f(t)f(0),t>0

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 Lf(s)=f(t),est=0estf(t)dt,s,s>ω.

It is a matter of simple calculations to show that L(D0CFtαf(t))(s)=1α+s(1α)[sL(f)(s)f(0)],s>ω, L(D0RLCFtαf(t))(s)=1α+s(1α)sL(f)(s),s>ω.

Recall that the Fourier transform is defined by F(f)(ω)=eiωtf(t)dt,ω,fL1().

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 Φ(t)=11αexp(α1α|t|),t. For the later use, we note that the same arguments give, for F(Φ)(ω)=2αα2+(1α)2ω,ω.(2.3)

Let α=A+iB be a complex number so that A=ℜα ∈ (0, 1). We can define t0CFDtαf(t) by (1.1). Since we prefer to work with real valued functions after fractional differentiation, we follow our approach presented in [9] and define combination of (CFFD) of complex order as: 0CFD¯tA,Bf(t)=12[TiB0CFDtA+iBf(t)+TiB0CFDtAiBf(t)],t>0,

where 𝕋 is a constant having the dimension of time and can be interpreted as relaxation time.

3 Some properties of t0CFDtαf

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 I(f)=0T˜f(1)(t)0CFDtαf(t)dt=11α0T˜f(1)(t)[φαf(1)](t)dt,(3.1)

for arbitrary T˜>0. In order to estimate I, we shall examine the properties of the kernel in (1.1).

Proposition 3.1

Suppose that fACloc([0, ∞)) is real valued. Then, I(f)⩾0 for every T˜>0.

Proof

By (2.3), F(eλ|τ|)(ω)>0,ω. The Bochner-Schwartz theorem ([23], p. 331) implies that eλτ 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 θ*(τ)=θ(τ)¯,θ*θ*(t)=θ(τ)θ*(tτ)dτ. This result holds in a more general case for positive definite tempered distributions which is not needed here. The derivative of f in [0,T˜] is an integrable function so we consider f(1) as a locally integrable function in [0, ∞) equal zero outside [0,T˜]. Let us denote by ϕn, n ϵ ℕ, a sequence of smooth functions, supported by [0,T˜], which converges to f(1) in L1([0,T˜]). Since f is real valued, we take ϕn, n ϵ ℕ to be real valued. By the Bochner-Schwartz theorem, for every n, 0T˜0T˜exp(λ|tτ|)ϕn(t)ϕn(τ)dτdt0.

Letting n → ∞, by the Lebesque theorem, we obtain 0T˜0T˜exp(λ|tτ|)f(t)f¯(τ)dτdt0.

Since 0T˜0T˜exp(λ|tτ|)ϕn(t)ϕn(τ)dτdt =0T˜ϕn(t)[0texp(λ(tτ))ϕn(τ)dτ]dt +0T˜ϕn(τ)[tT˜exp(λ(τt))ϕn(t)dt]dτ =20T˜ϕn(t)[0texp(λ(tτ))ϕn(τ)dτ]dt,

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 [0,T˜] but [,T˜]. 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.

Proposition 3.2

Suppose that fACloc([0, ∞)) and that it is real-valued. Then, for α(0,1),β(0,1),a,b,T˜>0, I(f,a,b)=0T˜f(1)(t)[a0CFDtαf(t)+b0CFDtβf(t)]dt,

if the following condition is satisfied: aα(1β)2+b(1α)2β>0.(3.2)

Proof

Without loss of generality we assume that βα. Let Fα,β(τ)=a1αexp(α1α|τ|)+b1βexp(β1β|τ|),τ.

Then, by (2.3), F(Fα,β)(ω)=2[aαα2+(1α)2ω2+bββ2+(1β)2ω2],ω.(3.3)

It follows from (3.3) that F(Fα,β)(ω)>0,ω, if aαβ2+ω2(1β)2α2+ω2(1α)2+bβ>0.(3.4)

Since min{β2+ω2(1β)2α2+ω2(1α)2;ω}=(1β)2(1α)21,

(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 0CFDtβf(t). First, we define the right (CFFD) of real order 0<α<1 for fACloc([0, ∞)) as tCFDbαf(t)=11αtbexp(α1α(τt))f(1)(τ)dτ,

where 0 < a<t < b.

Proposition 3.3

Suppose that fACloc([a, b)]), 0<α <1, 0 < a<b. Then abf(t)aCFDtαg(t)dt=abg(t)tCFDbαf(t)dt+f(b)abg(τ)φα(bτ)dτg(a)abf(τ)φα(τ)dτ.(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 0CDtαf and continue to consider functions in ACloc([0, ∞)).

Proposition 3.4

Suppose that fACloc([ 0, ∞)) and 0<α <1. Then, for T˜>0, 0CFDtαf(t)=φα(t)k=0αkVk(f(1))(t)(1α)kk!,t(0,T˜],

where the convergence is uniform on [0,T˜] and Vk(f(1))(t)=0tτkf(1)(τ)dτ,t[0,T˜],(3.6)

are moments of f(1). In particular, the approximation of 0CFDtαf(t) may be written as 0CFD¯tαf(t)φα(t)k=0NαkVk(f(1))(t)(1α)kk!,

with 0CFD¯tαf(t)=0CFDtαf(t)QN+1(t),t>t0,

and limN||QN+1(t)||L([0,T˜])=0.

Proof

We have 0CFDtαf(t)=11αexp(αt1α)0tk=0αkτk(1α)kk!f(1)(τ)dτ=11αexp(αt1α)k=0αkVk(f(1))(t)(1α)kk!,t[0,T˜].(3.7)

Decompose (3.7) as 0CFDtαf(t)=11αexp(αt1α)k=0NαkVk(f(1))(t)(1α)kk!+QN+1(t),t[0,T˜],

where QN+1(t) is the remainder. We have the estimate for QN+1(t),t[0,T˜] QN+1(t)=11αexp(αt1α)k=N+1αk0tτkf(1)(τ)dτ(1α)kk!11αexp(αt1α)f(1)L1([0,T˜])k=N+1αktk(1α)k(k)!.

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 tCFDT˜αf(t) as tCFDT˜αf(t)=φα(t)k=0(1)kαkV˜k(f(1))(t)(1α)kk!,t(0,T˜],

where the series converges in the uniform sense and V˜k(f(1))(t)=tT˜τkf(1)(τ)dτ,t[0,T˜]

We derive now the expression for derivative of 0CFDtαf(t) with respect to α. We state this as:

Proposition 3.5

Suppose that fACloc([0, ∞)), 0<α<1. Then 0CFDtαf(t)α=11α[D0CFtαf(t)1(1α)([t(ϕα)(t)]f(1))(t)],t(0,T˜].(3.8)

Proof

Definition gives (1.1) we have 0CFDtαf(t)α=1(1α)20CFDtαf(t)1(1α)20t(tτ)ϕα(tτ)f(1)(τ)dτ,t(0,T˜].

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 0CFDtαy(t)

In this section we present the necessary conditions for an extremum in the case when a Lagrangian density contains 0CFDtαy(t). 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 I(y)=ABL(t,y(t),aCFDtαy(t))dt,α(0,1),

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 minyUI(y)=I(y).(4.1)

Suppose that LC1((a,b)××),(4.2) tL(t,y(t),aCFDtαy(t))yisintegrableon(a,b),tL(t,y(t),aCFDtαy(t))aCFDtαy(t)AC(a,b)for everyyAC(a,b).(4.3)

Proposition 4.1

Additionally to (4.2) and (4.3), assume that the set of admissible functions 𝓊 is U={y:yyAC(a,b),y(a)=y0,L(t,y(t),aCFDtαy(t))aCFDtαy(t)|t=A=0,L(t,y(t),aCFDtαy(t))aCFDtαy(t)|t=B=0}.(4.4)

Then, for t ∈ (a, B), y* in (4.1) has to satisfy L(t,y(t),aCFDtαy(t))y+tCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)=0,t(A,B)tCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)tCFDAαL(t,y(t),aCFDtαy(t))aCFDtαy(t)=0.t(a,A)(4.5)

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 dI(y+εf)dε|ε=0=0, gives AB[L(t,y(t),aCFDtαy(t))yf(t)+L(t,y(t),aCFDtαy(t))aCFDtαy(t)aCFDtαf(t)]×dt=0.(4.6)

Integration by parts formula (3.5), applied to the interval [ a, B], leads to aBL(t,y(t),aCFDtαy(t))aCFDtαy(t)aCFDtαf(t)dt=aBtCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)f(t)dt+L(t,y(t),aCFDtαy(t))aCFDtαy(t)|t=BaBf(τ)φα(Bτ)dτf(a)aBL(t,y(t),aCFDtαy(t))aCFDtαy(t)φα(t)dt.

After the use of boundary conditions defined by (4.4), we obtain ABL(t,y(t),aCFDtαy(t))aCFDtαy(t)aCFDtαf(t)dt=ABtCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)f(t)dt+aAtCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)f(t)dtaAL(t,y(t),aCFDtαy(t))aCFDtαy(t)aCFDtαf(t)dt.

Integration by parts formula for the interval [ a, A] leads to aAL(t,y(t),aCFDtαy(t))aCFDtαy(t)aCFDtαf(t)dt=aAtCFDAαL(t,y(t),aCFDtαy(t))aCFDtαy(t)f(t)dt+L(t,y(t),aCFDtαy(t))aCFDtαy(t)|t=AaAf(τ)φα(Bτ)dτf(a)aAL(t,y(t),aCFDtαy(t))aCFDtαy(t)φα(t)dt.

Again, the boundary terms vanish so that AB[L(t,y(t),aCFDtαy(t))y+tCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)]f(t)dt+aA[DtCFBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)tCFDAαL(t,y(t),aCFDtαy(t))aCFDtαy(t)]×f(t)dt=0.

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 minyU,αAI(y,α)=01L(t,y(t),0CFDtαy(t))dt,(4.7)

where 𝓊 is given by (4.4) and A=[α0, α1], with 0<α0<α1 < 1. We assume that L(t,y,0CFDtαy,α)C1((a,b)×××A),(4.8) tL(t,y(t),aCFDtαy(t))yisintegrableon(a,b),yAC(a,b),αA, tL(t,y(t),aCFDtαy(t))aCFDtαy(t)AC(a,b),yAC(a,b),αA.(4.9)

Proposition 4.2

Suppose that (4.8) and (4.9) hold. If minyU,αAI(y,α)=I(y*,a*), =I(y*, α*), then y* and α* satisfy L(t,y(t),aCFDtαy(t))y+tCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)=0,t(0,1), 01L(t,y(t),aCFDtαy(t))aCFDtαy(t)×[D0CFtαy(t)1(1α)2(texp(α1αt)y(1))(t)]dt=0.

Proof

Let y=y*+ε1f, α=α*+ε1Δ α, where f and Δα are fixed. Substituting this in (4.7), we obtain I(y+ε1f,α+ε2Δα)=01L(t,y(t)+ε1f(t),0CFDtα+ε2Δαy(t))dt.

Conditions Iε1|ε1=ε2=0=Iε2|ε1=ε2=0=0 imply 01[L(t,y(t),aCFDtαy(t))y+tCFDBαL(t,y(t),aCFDtαy(t))aCFDtαy(t)]f(t)dt=0, Δα01[L(t,y(t),aCFDtαy(t))aCFDtαy(t)0CFDtαf(t)α]dt=0.(4.10)

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 0CFDtαf(t) 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.

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.

References

  • [1]

    R. Adams, J.J.F. Fournier, Sobolev Spaces. Elsevier, Oxford (2003). Google Scholar

  • [2]

    O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272 (2002), 368–79. CrossrefGoogle Scholar

  • [3]

    T.M. Atanackovic, D. Dolicanin, S. Konjik, S. Pilipovic, Dissipativity and stability for a nonlinear differential equation with distributed order symmetrized fractional derivative. Appl. Math. Lett. 24 (2011), 1020–1025. Web of ScienceCrossrefGoogle Scholar

  • [4]

    T.M. Atanackovic, B. Stankovic, On a numerical scheme for solving differential equations of fractional order. Mech. Res. Commun. 35 (2008), 429–438. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    T.M. Atanackovic, M. Janev, S. Konjik, S. Pilipovic, D. Zorica, Expansion formula for fractional derivatives in variational problems. J. Math. Anal. Appl. 409 (2014), 911–924. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    T.M. Atanackovic, S. Konjik, S. Pilipovic, Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A: Math. Theor. 41 (2008), ID # 095201. Web of ScienceGoogle Scholar

  • [7]

    T.M. Atanackovic, Lj. Oparnica and S. Pilipovic, On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328, (2007), 590–608. Web of ScienceCrossrefGoogle Scholar

  • [8]

    T.M. Atanackovic, S. Pilipovic, B. Stankovic, D. Zorica, Fractional Calculus with Applications in Mechanics. ISTE, Hoboken (2014). Google Scholar

  • [9]

    T.M. Atanackovic, S. Konjik, S. Pilipovic, D. Zorica, Complex order fractional derivatives in viscoelasticity. Mech. Time-Depend. Mater. 20 (2016), 175–195. Web of ScienceCrossrefGoogle Scholar

  • [10]

    D. Baleanu, A. Mousalou, S. Rezapour, A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative. Adv. Differ. Eq. 2017 (2017), 51–1–12. CrossrefWeb of ScienceGoogle Scholar

  • [11]

    M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1 (2015), 73–85. Google Scholar

  • [12]

    M. Caputo, M. Fabrizio, On the notion of fractional derivative and applicatios to the hysteresis phenomena. Mecc. 52, No 13 (2017), 3043-3052; . CrossrefGoogle Scholar

  • [13]

    A. Coronel-Escamilla, J.F. Gomiz-Aguilar, D. Baleanu, R.F. Escobar-Jimémez, V.H. Olivares-Pregrino, A. Abundez-Pliego, Formulation of Euler-Lagrange and Hamilton equations involving fractional operators with regular kernel. Adv. Differ. Eq. 2016 (2016), # 283, 17 pp.; . CrossrefWeb of ScienceGoogle Scholar

  • [14]

    K. Diethelm, The Analysis of Fractional Differential Equations. Springer, Berlin (2010). Google Scholar

  • [15]

    D. Dolicanin-Djekic, On a new class of constitutive equations for linear viscoelastic body. Fract. Calc. Appl. Anal. 20, No 2 (2017), 521–536; ; https://www.degruyter.com/view/j/fca.2017.20.issue-2/issue-files/fca.2017.20.issue-2.xml. CrossrefGoogle Scholar

  • [16]

    I.M. Gelfand, S.V. Fomin, Calculus of Variations. Dover, New York (2000). Google Scholar

  • [17]

    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elesvier, Amsterdam (2006). Google Scholar

  • [18]

    V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman, Longman, Harlow & Wiley, New York (1994). Google Scholar

  • [19]

    V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput. Math. Appl. 59 (2010), 1128–1141. Web of ScienceCrossrefGoogle Scholar

  • [20]

    A.N. Kochubei, General fractional calculus, evolution equations, and renewal processes. Integr. Equ. Oper. Theor. 71 (2011), 583–600. Google Scholar

  • [21]

    J. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1 (2015), 87–92. Google Scholar

  • [22]

    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models. Imperial College Press, London 2010. Google Scholar

  • [23]

    M. Reed, B. Simon, Methods of Modern Mathematical Physics, I: Functional Anaalysis. Academic Press, New York (1980). Google Scholar

  • [24]

    I. Schäfer, S. Kempfle, B. Nolte, Linear fractional PDE, uniqueness of global solutions. Fract. Calc. Appl. Anal. 8, No 1 (2005), 53–62; at http://www.math.bas.bg/complan/fcaa. Google Scholar

  • [25]

    B. Stankovic, T.M. Atanackovic, On an inequality arising in fractional oscillator theory. Fract. Calc. Appl. Anal. 7, No 1 (2004), 11–20; at http://www.math.bas.bg/complan/fcaa. Google Scholar

  • [26]

    S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order-function and diffusion with changing modes. Zeitschrift für Analysis und iher Anwendungen 28 (2009), 431–450. Google Scholar

  • [27]

    V.S. Vladimirov, Methods of the Theory of Generalized Functions. Taylor & Francis, London (2002). Google Scholar

About the article

Received: 2017-10-28

Published Online: 2018-03-13

Published in Print: 2018-02-23


Citation Information: Fractional Calculus and Applied Analysis, Volume 21, Issue 1, Pages 29–44, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2018-0003.

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