A Cauchy-type problem involving a weighted sequential derivative in the space of integrable functions

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A Cauchy-type problem involving a weighted sequential derivative in the space of integrable functions Khaled M. Furati ∗ Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

article

abstract

info

Keywords: Fractional derivatives Riemann–Liouville fractional derivative Sequential fractional derivative Fractional differential equation

A Cauchy-type nonlinear problem for a class of fractional differential equations involving sequential derivatives is considered. Some properties and composition identities are derived. The equivalence with the associated integral equation is established. The existence and uniqueness of global solutions in the space of Lebesgue integrable functions are proved. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction We consider a Cauchy-type problem associated with the equation Dαa (x − a)r Dβa y(x) = f (x, y),





x > a, 0 < r < α < 1, 0 < β < 1,

β

(1)

where Dαa and Da are the Riemann–Liouville fractional derivatives. In recent years there has been considerable interest in the theory and applications of fractional differential equations. See for example [1–11] and references therein. The fractional calculus approach has been introduced in many models. Fractional models provide a tool for capturing and understanding complex phenomena in many areas, see for example [12–15]. Indeed, some of these models are supported by experimental evidence and yield results that agree with the observed behavior [16]. As a result, there is a lot of recent work on boundary value problems and Cauchy-type problems associated with different classes of fractional differential equations. For example, Ahmad and Nieto in [17] obtained existence and uniqueness results for a class of Riemann–Liouville fractional differential equations of order α ∈ (1, 2] with fractional boundary conditions. They considered solutions in a weighted space of continuous functions. When r = 0, the left-hand side in (1) reduces to the sequential derivative introduced by Miller and Ross [18] and considered by Podlubny [19]. Compositions of two weighted Riemann–Liouville integrals or derivatives have been considered by Samko et al. in [20]. On the other hand, the weighted sequential derivative in (1) is an interesting example of Kiryakova’s weighted compositions of Erdélyi–Kober and Riemann–Liouville operators, called operators of the generalized fractional calculus [21,22]. In a series of papers, [23–25], Glushak studied the uniform well-posedness of a Cauchy-type problem with two fractional derivatives and a bounded operator. He also proposed a criterion for the uniform correctness of an unbounded operator. The author in [26,27] considered a Cauchy-type problem in the space of continuous functions. In this paper we prove an existence and uniqueness result for a nonlinear Cauchy-type problem associated with the equation (1) in the space of Lebesgue integrable functions, L(a, b).

∗

Tel.: +966 503872196; fax: +966 38602340. E-mail address: [email protected].

0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2013.01.044

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We start with some preliminaries in Section 2. In Section 3 we develop some properties and composition identities. In Section 4 we set up the Cauchy-type problem and establish the equivalence with the associated integral equation. Finally, in Section 5 we prove the existence and uniqueness of the solution. 2. Preliminaries Definition 1. Let r ≥ 0. Then as in [28], we introduce the following space Lr (a, b) := f ∈ L(a, b) : (x − a)−r y(x) ∈ L(a, b) ,





where L(a, b) is the set of measurable functions y such that b



|f (x)| dx < ∞.

∥f ∥L(a,b) := a

The space Lr is a Banach space with respect to the norm

∥y∥Lr (a,b) = ∥(x − a)−r y(x)∥L(a,b) . The left-sided Riemann–Liouville fractional integrals and derivatives are defined as follows. Definition 2. Let f ∈ L(a, b). The integral α

Ia f (x) :=

x



1

Γ (α)

a

f (s) ds, (x − s)1−α

x > a, α > 0,

is called the left-sided Riemann–Liouville fractional integral of order α of the function f . Definition 3. The expression Dαa f (x) := DIaα−1 f (x),

x > a, 0 < α < 1, D =

d dx

,

is called the left-sided Riemann–Liouville fractional derivative of order α of f provided the right-hand side exists. We use the notation D0a f = Ia0 f = f . The following lemma follows by direct calculations using the Dirichlet formula. Lemma 4. Let α ≥ 0, β ≥ 0, and f ∈ L(a, b). Then Iaα Iaβ f = Iaα+β f almost everywhere on [a, b]. It was proved in [20] that Dαa is the left inverse operator of Iaα . Lemma 5. If α > 0 and f ∈ L(a, b), then Dαa Iaα f (x) = f (x) almost everywhere on [a, b]. For power functions we have the following formulas. Lemma 6. For x > a, Iaα (t − a)β−1 (x) =





Dαa (t − a)



 α−1

Γ (β) (x − a)β+α−1 , Γ (β + α)

(x) = 0,

α ≥ 0, β > 0.

0 < α < 1.

Unless otherwise stated, for the rest of this paper we let −∞ < a < b < ∞. We denote the space of an absolutely continuous function on [a, b] by AC [a, b]. The following lemma follows from the fundamental theorem of integral calculus proved in [29]. Lemma 7. Let 0 < α < 1. Let y ∈ L(a, b) be such that Dαa y exists everywhere on (a, b] and is in L(a, b). Let Ia1−α y(a+ ) exist. Then Ia1−α y ∈ C (a, b] and bounded in (a, b] and a.e

Iaα Dαa y(x) = y(x) −

Ia1−α y(a+ )

Γ (α)

(x − a)α−1 .

If, in addition, Ia1−α y(a) = Ia1−α y(a+ ), then Ia1−α y ∈ AC [a, b].

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The following lemma gives the composition in Lemma 7 but under stronger sufficient conditions. The space CL(a, b) denotes L(a, b) ∩ C (a, b]. Lemma 8. Let 0 < α < 1. (a) If y ∈ CL(a, b) and Dαa y ∈ CL(a, b), then Iaα Dαa y(x) = y(x) −

I 1−α y(a+ )

Γ (α)

(x − a)α−1

on (a, b]. (b) If Ia1−α y ∈ AC [a, b] then Iaα Dαa y(x) = y(x) −

I 1−α y(a)

Γ (α)

(x − a)α−1

almost everywhere on (a, b]. The following lemma states that the Riemann–Liouville fractional integral Iaα is bounded in L(a, b). See [20,30]. Lemma 9. The fractional integral Iaα , α > 0, is bounded in L(a, b) with

∥Iaα f ∥L(a,b) ≤

( b − a) α ∥f ∥L(a,b) . Γ (α + 1)

Lemma 10. Let 0 < r < α < 1 and f ∈ L(a, b). Then Iaα f (x) ∈ Lr (a, b), and

∥(x − a)−r Iaα f (x)∥L(a,b) ≤

(b − a)α−r ∥f ∥L(a,b) . α−r

Proof. We have the relations

 x     (x − t )α−1 f (t ) dt    Γ (α) a  x 1 = (x − t )α−r −1 (x − t )r |f (t )| dt Γ (α) a  x (x − a)r (x − t )α−r −1 |f (t )| dt ≤ Γ (α) a 1

|Iaα f (x)| :=

=

Γ (α − r ) (x − a)r Iaα−r |f (x)|. Γ (α)

Now, from Lemma 9

∥(x − a)−r Iaα f (x)∥L(a,b) =

 Γ (α − r )  I α−r f (t ) a L(a,b) Γ (α)

≤

Γ (α − r ) (b − a)α−r ∥f ∥L(a,b) Γ (α) Γ (α − r + 1)

=

(b − a)α−r ∥f ∥L(a,b) .  α−r

For the existence and uniqueness result we will use the following Banach fixed point theorem. Theorem 11. Let (U , d) be a nonempty complete metric space and let T : U → U be the map such that, for any u, v ∈ U, the relation d(Tu, T v) ≤ w d(u, v),

0 ≤ w < 1,

holds. Then the operator T has a unique fixed point u∗ ∈ U.

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3. Properties and composition identities In this section we define the sequential derivative and integral that we consider and develop some of their properties. In particular, we derive the composition identities. α,β

Definition 12. Let α > 0, β > 0, r ≥ 0. Let f ∈ L(a, b). Define the sequential integral Jr ,a f and the sequential derivative

α,β Dr ,a f

by α −r β Jrα,β ,a f (x) = Ia (x − a) Ia f (x),

and α r β Drα,β ,a f (x) = Da (x − a) Da f (x),

if the right-hand sides exist. Lemma 13. Let α > 0, β > 0, r ≥ R. If

ρ > max{−1, β − r − 1} = −1 + max{0, β − r }, then for x > a, ρ Drα,β ,a (x − a) =

Γ (ρ + 1) Γ (ρ + r − β + 1) (x − a)ρ+r −β−α . Γ (ρ − β + 1) Γ (ρ + r − β − α + 1)

(2)

Proof. From Lemma 6 we have ρ α r β ρ Drα,β ,a (x − a) := Da (x − a) Da (x − a)

  Γ (ρ + 1) Dα (x − a)r (x − a)ρ−β Γ (ρ − β + 1) a Γ (ρ + 1) Dα (x − a)ρ+r −β = Γ (ρ − β + 1) a Γ (ρ + r − β + 1) Γ (ρ + 1) (x − a)ρ+r −β−α .  = Γ (ρ − β + 1) Γ (ρ + r − β − α + 1) =

The following lemma follows from Lemmas 6 and 13. Lemma 14. (a) Let α > 0, 0 < β < 1, r ∈ R. Then for x > a, β−1 Drα,β = 0. ,a (x − a)

(3)

(b) Let 0 < α < 1 and β > 0. Let r ≥ 0 be such that r < α + β . Then for x > a, α+β−r −1 Drα,β = 0. ,a (x − a)

(4)

Property 15 (Left Inverse). Let 0 < r < α < 1 and 0 < β < 1. If f ∈ L(a, b) then β,α Drα,β ,a Jr ,a f (x) = f (x)

almost everywhere in [a, b]. Proof. From Lemma 10 we have Iaα f ∈ Lr (a, b). Thus we can apply Lemma 5 twice to get the result.



Lemma 16. Let 0 < r < α < 1 and 0 < β < 1. Let f ∈ L(a, b). Then

Jrβ,α ,a f ∈ L(a, b), and

∥Jrβ,α ,a f ∥L(a,b) ≤

(b − a)α+β−r ∥f ∥L(a,b) . (α − r )Γ (β + 1)

(5)

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Proof. From Lemmas 9 and 10 we have β −r α ∥Jrβ,α ,a f ∥L(a,b) = ∥Ia (x − a) Ia f ∥L(a,b)  (b − a)β  (x − a)−r I α f  ≤ a L(a,b) Γ (β + 1) α−r β (b − a) (b − a) ∥f ∥L(a,b) ≤ Γ (β + 1) α − r (b − a)α+β−r = ∥f ∥L(a,b) . (α − r )Γ (β + 1)

This yields the result.

 α,β

Lemma 17. Let 0 < α < 1, 0 < β < 1, and r ≥ 0. Let y ∈ L(a, b) be such that Dr ,a y exists everywhere on (a, b] and in β L(a, b). Let Ia1−α (x − a)r Da y (a+ ) exist. Then β

I 1−α [(x − a)r Da y](a+ ) Dβa y(x) = (x − a)−r Iaα Drα,β (x − a)α−r −1 ,a y(x) +

(6)

Γ (α)

almost everywhere on (a, b]. β

Proof. With the given hypothesis we can apply Lemma 7 to (x − a)r Da y and obtain the result. β

The next lemma states that for 0 < r < α < 1, the sequential derivative

α,β Dr ,a y



cannot be Lebesgue integrable unless

Da y is Lebesgue integrable and thus we do not need to impose this as a condition. α,β

Lemma  18. Let 0 < r< α < 1 and 0 < β < 1. Let y ∈ L(a, b) be such that Dr ,a y exists everywhere on (a, b] and in L(a, b). β β 1−α Let Ia (x − a)r Da y (a+ ) exist. Then Da y ∈ L(a, b). Proof. From Lemma 17 we have β

I 1−α [(x − a)r Da y](a+ ) (x − a)α−r −1 . Dβa y(x) = (x − a)−r Iaα Drα,β ,a y(x) +

Γ (α)

α,β

α,β

Since Dr ,a y ∈ L(a, b), then by Lemma 10, Iaα Dr ,a y ∈ Lr (a, b). This proves the result.



Lemma 19. Let 0 < r < α < 1 and 0 < β < 1. Let y ∈ L(a, b) satisfy the following hypothesis. α,β

i. Dr ,a y exists everywhere on (a, b] and in L(a, b), β

ii. Da y exists everywhere on (a, b], β 1−β iii. Ia1−α [(x − a)r Da y](a+ ) and Ia y(a+ ) exist. Then 1−β

a.e

α,β Jrβ,α ,a Dr ,a y(x) = y(x) −

[ Ia

y](a+ )

(x − a)β−1 −

Γ (β)

β

[Ia1−α (x − a)r Da y](a+ ) Γ (α + r ) (x − a)α+β−r −1 , Γ (α) Γ (α + β − r )

(7)

almost everywhere on (a, b]. β

Proof. It follows from the hypothesis and Lemma 18 that (x − a)r Da y ∈ L(a, b) and exists everywhere. By applying Lemma 7 twice we obtain the result.  The following lemma follows from Lemma 8. α,β

β

Lemma 20. (a) If Dr ,a y ∈ CL(a, b) and Da y ∈ CL(a, b), then a.e

α,β Jrβ,α ,a Dr ,a y(x) = y(x) −

1−β

[ Ia

y](a+ )

Γ (β)

β

1−β

(b) If Ia1−α (x − a)r Da y ∈ AC [a, b] and Ia a.e

α,β Jrβ,α ,a Dr ,a y(x) = y(x) −

1−β

[ Ia

[Ia1−α (x − a)r Dβ y](a+ ) Γ (α + r ) (x − a)α+β−r −1 . Γ (α) Γ (α + β − r )

(8)

y ∈ AC [a, b], then

y](a)

Γ (β)

(x − a)β−1 −

(x − a)β−1 −

[Ia1−α (x − a)r Dβ y](a) Γ (α + r ) (x − a)α+β−r −1 . Γ (α) Γ (α + β − r )

(9)

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4. Cauchy-type problem and equivalency Consider the Cauchy-type problem associated with (1),

Drα,β ,a y(x) = f (x, y(x)),

x > a, 0 < r < α < 1, 0 < β < 1,

(10)

(x) = c0 ,

lim Ia1−β y x→a+

(11)

lim Ia1−α (x − a)r Dβa y (x) = c1





(12)

x→a+

where c0 and c1 are real numbers. Based on the composition in Lemma 17, we consider the equivalence with the following reduced in order fractional integro-differential equation: Dβa y = (x − a)−r Iaα f (x, y(x)) +

c1 (x − a)α−r −1

Γ (α)

,

(13)

lim Ia1−β y(x) = c0 .

(14)

x→a+

Theorem 21. Let 0 < r < α < 1 and 0 < β < 1. Let f : (a, b] × R → R be a function such that f (., y(.)) ∈ L(a, b) for any y ∈ L(a, b). If y ∈ L(a, b), then y(x) satisfies a.e. (10)–(12) if and only if y(x) satisfies a.e. (13) and (14). α,β

Proof. Let y ∈ L(a, b) satisfy a.e. (10)–(12). Since f (x, y(x)) ∈ L(a, b), then from (10) we have Dr ,a y ∈ L(a, b). Thus we can apply Lemma 17 and the formula (6) holds. By substituting the initial conditions we obtain (13). Let y ∈ L(a, b) satisfy (13) and (14). By multiplying (13) by (x − a)r and then applying Dαa to both sides we obtain (10). As for the initial condition, multiply (13) by (x − a)r and apply Ia1−α to both sides then take the limit to obtain (12).  The composition in Lemma 19 leads to the following nonlinear integral equation, c0

y(x) = Jrβ,α ,a f (x, y(x)) +

(x − a)β−1 +

Γ (β)

c1 Γ (α − r ) (x − a)α+β−r −1 . Γ (α) Γ (α + β − r )

(15)

The following theorem establishes the equivalence with this integral equation. Theorem 22. Let 0 < r < α < 1 and 0 < β < 1. Let f : (a, b] × R → R be a function such that f (., y(.)) ∈ L(a, b) for any y ∈ L(a, b). β

(a) If y ∈ L(0, T ) is such that Da y exists everywhere in (a, b] and satisfies the Cauchy-type problem (10)–(12) then y(x) satisfies the integral equation (15). (b) If y ∈ L(0, T ) satisfies the integral equation (15) then y(x) satisfies the Cauchy-type problem (10)–(12). Proof. Let y ∈ L(a, b) satisfy (10)–(12). Since f (x, y(x)) ∈ L(a, b), then from (10), we can apply Lemma 19 and the formula (7) holds. By using the initial conditions we obtain (15). α,β Let y ∈ L(a, b) satisfy the integral equation (15). Applying the operator Dr ,a to both sides and using Lemma 14 we have α,β β,α α,β Drα,β ,a y(x) = Dr ,a Jr ,a f (x, y(x)) + Dr ,a



c0

Γ (β)

(x − a)β−1 +

c1 Γ (α − r )(x − a)α+β−r −1



Γ (α)Γ (α + β − r )

= f (x, y(x)). 1−β

Applying Ia

to both sides of (15) yields

Ia1−β y(x) = Jr1,,α a f (x, y(x)) + c0 +

Γ (α − r ) Γ (α + β − r ) (x − a)α−r . Γ (α) Γ (α + β − r ) Γ (α − r + 1) c1

(16)

β

Taking the limit we obtain the initial condition (11). Applying Ia1−α (x − a)r Da to both sides of (15) and using Lemmas 4 and 6 yields Ia1−α

β

( x − a) D a y = r

Ia1−α

(x − a)

r

β

Da [Jrβ,α ,a f (x, y(x))]

+

Ia1−α

β

(x − a) Da r

c1 Γ (α − r )



Γ (α) Γ (α + β − r )

= Ia1 f (x, y(x)) + c1 . Taking the limit we obtain the initial condition (12). This completes the proof.



In the next section we use this equivalence to prove the existence and uniqueness of solutions.

( x − a)

α+β−r −1



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5. Existence and uniqueness of the solution of the Cauchy-type problem Theorem 23. Let 0 < r < α < 1, 0 < β < 1. Let f : (a, b] × R → R be a function such that f (., y(.)) ∈ L(a, b) for any y ∈ L(a, b) and the condition:

|f (x, y1 ) − f (x, y2 )| ≤ A|y1 − y2 |,

A > 0,

(17)

is satisfied for all x ∈ (a, b] and for all y1 , y2 ∈ R. Then the Cauchy-type problem (10)–(12) has a solution y ∈ L(a, b). Furthermore, this solution is unique if in addition we β require that Da y exists everywhere in (a, b]. Proof. According to Theorem 22(b), it is sufficient to show the existence of an L(a, b) solution for the equivalent integral equation (15). This equation holds in any interval (a, x1 ] ⊂ (a, b], a < x1 < b. Choose x1 such that

w := A

(b − a)α+β−r (x1 − a)α < 1. (α − r )Γ (β + 1)

We rewrite the integral equation in the form y(x) = Ty(x), where Ty(x) = v0 (x) + Jrβ,α ,a f (x, y(x)), and c0

v0 (x) =

Γ (β)

(x − a)β−1 +

c1 Γ (α − r )

Γ (α) Γ (α + β − r )

(x − a)α+β−r −1 .

It follows from Lemma 16 that if y ∈ L(a, x1 ) then Ty ∈ L(a, x1 ). Also, for any y1 , y2 in L(a, x1 ), we have

∥Ty1 − Ty2 ∥L(a,x1 ) ≤ ∥ Jrβ,α ,a { |f (x, y1 (x)) − f (x, y2 (x))| } ∥L(a,x1 ) ≤ A∥ Jrβ,α ,a { |y1 (x) − y2 (x)| } ∥L(a,x1 ) (b − a)α+β−r (x1 − a)α ∥y1 − y2 ∥L(a,x1 ) (α − r )Γ (β + 1) ≤ w∥ y1 − y2 ∥L(a,x1 ) , 0 < w < 1.

≤A

Hence by Theorem 11 there exists a unique solution y∗ ∈ L(a, x1 ) to Eq. (15) on the interval (a, x1 ]. If x1 ̸= b then we consider the interval [x1 , b]. On this interval we consider solutions y ∈ L(x1 , b) for the equation y(x) = Ty(x) := v01 (x) + Jrβ,α ,x1 f (x, y(x)),

x ∈ [x1 , b],

(18)

where

v01 (x) =

y0

Γ (β)

β−1

(x − a)

+

1



Γ (α)Γ (β)

x1



β−1

(x − t )

( t − a)

a

−r

t



α−1

( t − s)



f (s, y(s))ds

dt .

a

Now we select x2 ∈ (x1 , b] such that

w2 := A

(b − a)α+β−r (x2 − x1 )α < 1. (α − r )Γ (β + 1)

Since the solution is uniquely defined on the interval (a, x1 ], we can consider v01 (x) to be a known function. For y1 , y2 ∈ L(x1 , x2 ), it follows from the Lipschitz condition and 16 that

∥Ty1 − Ty2 ∥L(x1 ,x2 ) ≤ ∥ Jrβ,α ,x1 { |f (x, y1 (x)) − f (x, y2 (x))| } ∥L(x1 ,x2 ) ≤ A∥ Jrβ,α ,x1 { |y1 (x) − y2 (x)| } ∥L(x1 ,x2 ) (b − a)α+β−r (x2 − x1 )α ∥y1 (x) − y2 (x)∥L(x1 ,x2 ) (α − r )Γ (β + 1) ≤ w2 ∥ |y1 (x) − y2 (x)|∥L(x1 ,x2 ) , 0 < w2 < 1. ≤A

β,α

Since 0 < w2 < 1, T is a contraction. Since f (x, y(x)) ∈ L(x1 , x2 ) for any y ∈ L(x1 , x2 ), then Jr ,x1 f ∈ L(x1 , x2 ). Moreover, clearly v01 (x) is in L(x1 , x2 ). Thus the right-hand side of (18) is in L(x1 , x2 ). Therefore T maps L(x1 , x2 ) into itself. By

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Theorem 11, there exists a unique solution y∗1 ∈ L(x1 , x2 ) to the equation on the interval [x1 , x2 ]. Therefore if y∗ (x) =



y∗0 (x), y∗1 (x),

a < x ≤ x1 , x1 < x ≤ x2 ,

then y∗ is the unique solution of (15) in L(a, x2 ) on the interval (a, x2 ]. If x2 ̸= b, we repeat the process as necessary, say M − 2 times, to obtain the unique solutions y∗k ∈ L(xk , xk+1 ), k = 2, 3, . . . , M, where a = x0 < x1 < · · · < xM = b, such that

wk+1 = A

(b − a)α+β−r (xk+1 − xk )α < 1. (α − r )Γ (β + 1)

As a result we have the unique solution y∗ ∈ L(a, b) of (15) given by y∗ (x) = y∗k (x),

x ∈ (xk , xk+1 ], k = 0, 1, . . . , M − 1.

(19)

β

This solution is also a solution for (10)–(12). If Da y exists everywhere in (a, b] then the uniqueness follows from Theorem 22(a).  Example Consider the problem

Drα,β y(x) = f (x, y(x)),

x > 0, 0 < r < α < 1, 0 < β < 1,

1−β lim I0 y x→0+

(x) = c0 ,   β lim I01−α xr D0 y (x) = c1 ,

x→0+

with f (x, y(x)) = tan

−1

y(x) +



1, 0,

0 ≤ x ≤ 1, x > 1.

Then clearly f satisfies the conditions of Theorem 23. Thus the problem has a solution y ∈ L(0, b) for any b > 0. Acknowledgment The author is grateful for the support provided by King Fahd University of Petroleum & Minerals. References [1] D. Baleanu, Z.B. Güvenç, J.T. Machado (Eds.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer, 2010. [2] R. Caponetto, G. Dongola, L. Fortuna, I. Petráš, Fractional Order Systems: Modeling and Control Applications, in: World Scientific Series on Nonlinear Science, vol. 72, World Scientific, 2010. [3] R. Hilfer, Threefold introduction to fractional derivatives, in: [5], pp. 17–73. [4] R. Hilfer, Y. Luchko, Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann–Liouville fractional derivatives, Fractional Calculus & Applied Analysis 12 (2009) 299–318. [5] R. Klages, G. Radons, I. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim, 2008. [6] C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-order systems and controls, in: Advances in Industrial Control, Springer, 2010. [7] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, 2010. [8] B.L.S.P. Rao, Statistical Inference for Fractional Diffusion Processes, Wiley, 2010. [9] T. Sandev, Ž. Tomovski, General time fractional wave equation for a vibrating string, Journal of Physics A: Mathematical and Theoretical 43 (2010) 055204. [10] H.M. Srivastava, Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Applied Mathematics and Computation 211 (2009) 198–210. [11] Živorad Tomovski, R. Hilfer, H. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms and Special Functions 21 (2010) 797–814. [12] E. Gerolymatou, I. Vardoulakis, R. Hilfer, Modelling infiltration by means of a nonlinear fractional diffusion model, Journal of Physics D: Applied Physics 39 (2006) 4104–4110. [13] R. Hilfer, L. Anton, Fractional master equations and fractal time random walks, Physical Review E 51 (1995) R848–R851. [14] F. Mainardi, R. Gorenflo, Time-fractional derivatives in relaxation processes: a tutorial survey, Fractional Calculus & Applied Analysis 10 (2007) 269–308. [15] T. Wenchang, P. Wenxiao, X. Mingyu, A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, International Journal of Non-Linear Mechanics 38 (2003) 645–650. [16] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chemical Physics 284 (2002) 399–408. [17] B. Ahmad, J.J. Nieto, Riemann–Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory 13 (2012) 329–336. [18] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, 1993. [19] I. Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999. [20] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993, (Engl. Trans. from the Russian 1987).

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