Fractional and operational calculus with generalized fractional ... - ICP

Integral Transforms and Special Functions Vol. 21, No. 11, November 2010, 797–814

Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions

Downloaded By: [Srivastava, Hari M.] At: 18:19 27 October 2010

Živorad Tomovskia , Rudolf Hilferb and H.M. Srivastavac * a Institute of Mathematics, Faculty of Natural Sciences and Mathematics, St. Cyril and Methodius University, MK-91000 Skopje, Republic of Macedonia; b Institute for Computational Physics (ICP), Universität Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany; c Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

(Received 09 November 2009 )

Dedicated to Professor Rudolf Gorenflo on the Occasion of his Eightieth Birth Anniversary In this paper, we study a certain family of generalized Riemann–Liouville fractional derivative operators α,β Da± of order α and type β, which were introduced and investigated in several earlier works [R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000; R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130; R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), pp. 399–408; R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I.M. Sokolov, eds., Wiley-VCH Verlag, Weinheim, 2008, pp. 17–73; R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. R848–R851; R. Hilfer,Y. Luchko, and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318; F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal. 10 (2007), pp. 269–308; T. Sandev and Ž. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A Math. Theor. 43 (2010), 055204; H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210]. In particular, we derive various compositional properties, which are associated with Mittag–Leffler functions and Hardy-type inequalities for the generalized fractional α,β derivative operator Da± . Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given. Keywords: Riemann–Liouville fractional derivative operator; generalized Mittag–Leffler function; Hardy-type inequalities; Laplace transform method; Volterra differintegral equations; fractional differential equations; fractional kinetic equations; Lebesgue integrable functions; Fox–Wright hypergeometric functions 2000 Mathematics Subject Classification: Primary: 26A33, 33C20, 33E12; Secondary: 47B38, 47G10

*Corresponding author. Email: [email protected]

ISSN 1065-2469 print/ISSN 1476-8291 online © 2010 Taylor & Francis DOI: 10.1080/10652461003675737 http://www.informaworld.com

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Introduction, Definitions and Preliminaries

Applications of fractional calculus require fractional derivatives of different kinds [5–9,11,16,20,24]. Differentiation and integration of fractional order are traditionally defined by p the right-sided Riemann–Liouville fractional integral operator Ia+ and the left-sided Riemann– p Liouville fractional integral operator Ia− , and the corresponding Riemann–Liouville fractional p p derivative operators Da+ and Da− , as follows [3, Chapter 13,14, pp. 69–70, 19]:


! µ " Ia+ f (x) =

and

! µ " Ia− f (x) =

1 # (µ) 1 # (µ)

# #

x a a x

f (t) dt (x − t)1−µ f (t) dt (t − x)1−µ

$ % ! µ " d n ! n−µ " Da± f (x) = ± Ia± f (x) dx

!

! " x > a; " (µ) > 0 ,

!

x < a; " (µ) > 0

"

" " (µ) ! 0; n = [" (µ)] + 1 ,

(1.1) (1.2)

(1.3)

where the function f is locally integrable, " (µ) denotes the real part of the complex number µ ∈ C and [" (µ)] means the greatest integer in " (µ). Recently, a remarkably large family of generalized Riemann–Liouville fractional derivatives of order α (0 < α < 1) and type β (0 " β " 1) was introduced as follows [5–7,9,11,20]. α,β

Definition 1 The right-sided fractional derivative Da+ and the left-sided fractional derivative α,β Da− of order α (0 < α < 1) and type β (0 " β " 1) with respect to x are defined by $ & ' & '% α,β β(1−α) d (1−β)(1−α) Da± f (x) = ±Ia± Ia± f (x) , dx

(1.4)

whenever the second member of (1.4) exists. This generalization (1.4) yields the classical Riemann–Liouville fractional derivative operator when β = 0. Moreover, for β = 1, it gives the fractional derivative operator introduced by Liouville [15, p. 10], which is often attributed to Caputo now-a-days and which should more appropriately be referred to as the Liouville–Caputo fractional derivative. Several authors [16, 24] called the general operators in (1.4) the Hilfer α,β fractional derivative operators. Applications of Da± are given in [7]. α,β

The purpose of this paper is to investigate the generalized fractional derivative operators Da± of order α and type β. In particular, several fractional differential equations arising in applications are solved. Using the formulas (1.1) and (1.2) in conjunction with (1.3) when n = 1, the fractional derivative α,β operator Da± can be rewritten in the form: &

$ ' & '% α,β β(1−α) α+β−αβ Da± f (x) = ± Ia± Da± f (x) .

(1.5)

The difference between fractional derivatives of different types becomes apparent from their Laplace transformations. For example, it is found for 0 < α < 1 that [5, 6, 24] (& & ' ) ' α,β (1−β)(1−α) L D0+ f (x) (s) = s α L [f (x)] (s) − s β(α−1) I0+ f (0+) (1.6) (0 < α < 1),

Integral Transforms and Special Functions

where

799

' & (1−β)(1−α) f (0+) I0+

is the Riemann–Liouville fractional integral of order (1 − β) (1 − α) evaluated in the limit as t → 0+, it being understood (as usual) that [2, Chapters 4 and 5] # ∞ L [f (x)] (s) := e−sx f (x) dx =: F (s), (1.7) 0

provided that the defining integral in (1.7) exists. The familiar Mittag–Leffler functions Eµ (z) and Eµ,ν (z) are defined by the following series:


Eµ (z) :=

∞ * n=0

zn =: Eµ,1 (z) # (µn + 1)

and Eµ,ν (z) :=

∞ * n=0

zn # (µn + ν)

! " z ∈ C; "(µ) > 0

! " z, ν ∈ C; "(µ) > 0 ,

(1.8)

(1.9)

respectively. These functions are natural extensions of the exponential, hyperbolic and trigonometric functions, since ! " ! " E1 (z) = ez , E2 z2 = cosh z, E2 −z2 = cos z, E1,2 (z) =

ez − 1 z

and

E2,2 (z2 ) =

sinh z . z

For a detailed account of the various properties, generalizations and applications of the Mittag– Leffler functions, the reader may refer to the recent works by, for example, Gorenflo et al. [4] and Kilbas et al. [12–14, Chapter 1]. The Mittag–Leffler function (1.1) and some of its various generalizations have only recently been calculated numerically in the whole complex plane [10,22]. By means of the series representation, a generalization of the Mittag–Leffler function Eµ,ν (z) of (1.2) was introduced by Prabhakar [18] as follows: λ Eµ,ν (z) =

∞ * n=0

zn (λ)n # (µn + ν) n!

! " z, ν, λ ∈ C; " (µ) > 0 ,

(1.10)

where (and throughout this investigation) (λ)ν denotes the familiar Pochhammer symbol or the shifted factorial, since (1)n = n!

(n ∈ N0 := N ∪ {0}; N := {1, 2, 3, . . .}),

defined (for λ, ν ∈ C and in terms of the familiar Gamma function) by + # (λ + ν) 1 (ν = 0; λ ∈ C \ {0}) = (λ)ν := # (λ) λ (λ + 1) · · · (λ + n − 1) (ν = n ∈ N; λ ∈ C) . Clearly, we have the following special cases: 1 Eµ,ν (z) = Eµ,ν (z)

and

1 Eµ,1 (z) = Eµ (z) .

Indeed, as already observed earlier by Srivastava and Saxena [23, p. 201, Equation (1.6)], the λ generalized Mittag–Leffler function Eµ,ν (z) itself is actually a very specialized case of a rather

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extensively investigated function p &q as indicated below [14, p. 45, Equation (1.9.1)]:   (λ, 1) ; 1 λ  Eµ,ν z. (z) = 1 &1 # (λ) (ν, µ) ;

Here, and in what follows, p &q denotes the Wright (or, more appropriately, the Fox–Wright) generalization of the hypergeometric p Fq function, which is defined as follows [14, p. 56 et seq.]: ! "   ! " ∞ (a1 , A1 ) , . . . , ap , Ap ; * # (a1 + A1 k) · · · # ap + Ap k zk   ! " p &q " z := ! # (b1 + B1 k) · · · # bq + Bq k k! k=0 (b1 , B1 ) , . . . , bq , Bq ;   $* % q p * "(Aj ) > 0 (j = 1, . . . , p) ; "(Bj ) > 0 (j = 1, . . . , q) ; 1 + " Bj − Aj ! 0  , j =1

j =1

in which we have assumed, in general, that aj , Aj ∈ C

(j = 1, . . . , p)

bj , Bj ∈ C

and

(j = 1, . . . , q)

and that the equality holds true only for suitably bounded values of |z|. Special functions of the Mittag–Leffler and the Fox–Wright types are known to play an important role in the theory of fractional and operational calculus and their applications in the basic processes of evolution, relaxation, diffusion, oscillation and wave propagation. Just as we have remarked above, the Mittag–Leffler type functions have only recently been calculated numerically in the whole complex plane [10,22]. λ The following Laplace transform formula for the generalized Mittag–Leffler function Eµ,ν (z) was given by Prabhakar [18]: 4 5 λ L x ν−1 Eµ,ν (ωx µ ) (s) =

s λµ−ν (s µ − ω)λ 6ω6 & ' 6 6 λ, µ, ω ∈ C; " (ν) > 0; " (s) > 0; 6 µ 6 < 1 . s

Prabhaker [18] also introduced the following fractional integral operator: # x ! λ " ! " λ Eµ,ν,ω;a+ ϕ (x) = ω (x − t)µ ϕ (t) dt (x − t)ν−1 Eµ,ν (x > a)

(1.11)

(1.12)

a

in the space L(a, b) of Lebesgue integrable functions on a finite closed interval [a, b] (b > a) of the real line R given by 7 8 # b |f (x)| dx < ∞ , L(a, b) = f : 'f '1 = (1.13) a

it being tacitly assumed (throughout the present investigation) that, in situations such as those occurring in (1.12) and in conjunction with the usages of the definitions in (1.3), (1.4) and (1.5), a in all such function spaces as, for example, the function space L(a, b) coincides precisely with the lower terminal a in the integrals involved in the definitions (1.3), (1.4) and (1.5). The fractional integral operator (1.12) was investigated earlier by Kilbas et al. [13] and its λ generalization involving a family of more general Mittag–Leffler type functions than Eµ,ν (z) was studied recently by Srivastava and Tomovski [24].


2.

801

Properties of the generalized fractional derivative operator and relationships with the Mittag–Leffler functions


In this section, we derive several continuity properties of the generalized fractional derivative α,β operator Da+ . Each of the following results (Lemma 1 as well as Theorems 1 and 2) are easily derivable by suitably specializing the corresponding general results proven recently by Srivastava and Tomovski [24]. Lemma 1 [24] The following fractional derivative formula holds true: & 5' # (ν) α,β 4 Da+ (t − a)ν−1 (x) = (x − a)ν−α−1 # (ν − α) ! " x > a; 0 < α < 1; 0 " β " 1; " (ν) > 0 .

Theorem 1 [24] The following relationship holds true: & 4 55' 4 5 α,β 4 λ λ Da+ (t − a)ν−1 Eµ,ν ω (t − a)µ ω (x − a)µ (x) = (x − a)ν−α−1 Eµ,ν−α ! " x > a; 0 < α < 1; 0 " β " 1; λ, ω ∈ C; " (µ) > 0; " (ν) > 0 .

(2.1)

(2.2)

Theorem 2 [24] The following relationship holds true for any Lebesgue integrable function ϕ ∈ L(a, b): " α,β ! λ λ Da+ Eµ,ν,ω;a+ ϕ = Eµ,ν−α,ω;a+ ϕ (2.3) ! " x > a (a = a); 0 < α < 1; 0 " β " 1; λ, ω ∈ C; " (µ) > 0; " (ν) > 0 . In addition to the space L(a, b) given by (1.13), we shall need the weighted Lp -space ! " Xcp (a, b) c ∈ R; 1 " p " ∞ ,

which consists of those complex-valued Lebesgue integrable functions f on (a, b) for which 'f 'Xcp < ∞, with 'f 'Xcp = In particular, when c =

$#

a

b

6c 6 6t f (t)6p dt t

%1/p

! " 1"p a) in R, by 9 : α,p α Wa+ (a, b) = f : f ∈ Lp (a, b) and Da+ f ∈ Lp (a, b) (0 < α " 1) ,

α where Da+ f denotes the fractional derivative of f of order α (0 < α " 1). Alternatively, in Theorems 3 and 4, we can make use of a suitable p-variant of the space Lαa+ (a, b) which was defined, for "(α) > 0, by Kilbas et al. [14, p. 144, Equation (3.2.1)] as follows 9 ! ": α α La+ (a, b) = f : f ∈ L(a, b) and Da+ f ∈ L(a, b) "(α) > 0 .

See also the notational convention mentioned in connection with (1.13).

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Ž. Tomovski et al. α,β

Theorem 3 For 0 < α < 1 and 0 < β < 1, the operator Da+ is bounded in the space α+β−αβ,1 Wa+ (a, b) and < = ; ; ; ; (b − a)β(1−α) ; α,β ; ; α+β−αβ ; A := . (2.4) ;Da+ ; " A ;Da+ ; 1 1 β (1 − α) # [β (1 − α)] Proof Using a known result [19, Equation (2.72)], we get ; ; ; & '; ; α,β ; ; β(1−α) ; α+β−αβ Da+ ϕ ; ;Da+ ϕ ; = ;Ia+ 1

1


"

; ; (b − a) ; α+β−αβ ; ϕ; . ;Da+ 1 β (1 − α) # [β (1 − α)] β(1−α)

#

α The weighted Hardy-type inequality for the integral operator Ia+ is stated as the following lemma. α Lemma 2 [14, p. 82] If 1 < p < ∞ and α > 0, then the integral operator I0+ is bounded from p p L (0, ∞) into X1/p−α (0, ∞) as follows:

$#

0

∞

6! α " 6p x −αp 6 I0+ f (x)6 dx "

%1/p

! " $# ∞ %1/p # 1/p( |f (x)|p dx # (α + 1/p) 0

$

% 1 1 + ( =1 . p p

(2.5)

α,β

Applying the last two inequalities to the fractional derivative operator Da+ , we get $#

∞

0

"

x

−αp

6& ' 6p %1/p 6 α,β 6 6 D0+ f (x)6 dx

! " # 1/p (

! " # β (1 − α) + 1/p

$#

∞

0

6& ' 6p %1/p 6 α+β−αβ 6 f (x)6 dx 6 D0+

$

% 1 1 + =1 . p p(

(2.6)

Hence, we arrive at the following result.

Theorem 4 If 1 < p < ∞, 0 < α < 1 and 0 " β " 1, then the fractional derivative operator α+β−αβ,p p α,β D0+ is bounded from W0+ (0, ∞) into X1/p−α (0, ∞). 3.

Fractional differential equations with constant and variable coefficients

The eigenfunctions of the Riemann–Liouville fractional derivatives are defined as the solutions of the following fractional differential equation: ! α " D0+ f (x) = λf (x) , (3.1) where λ is the eigenvalue. The solution of (3.1) is given by

f (x) = x 1−α Eα,α (λx α ) .

(3.2)


803 α,β

More generally, the eigenvalue equation for the fractional derivative D0+ of order α and type β reads as follows: ' & α,β (3.3) D0+ f (x) = λf (x) and its solution is given by [6, Equation (124)]

f (x) = x (1−β)(1−α) Eα,α+β(1−α) (λx α ) ,

(3.4)

which, in the special case when β = 0, corresponds to (3.2). A second important special case of (3.3) occurs when β = 1: ' & α,1 f (x) = λf (x) . (3.5) D0+ Downloaded By: [Srivastava, Hari M.] At: 18:19 27 October 2010

In this case, the eigenfunction is given by

f (x) = Eα (λx α ) .

(3.6)

We now divide this section into the following five subsections. 3.1. A General Family of Fractional Differential Equations In this section, we assume that 0 < α1 " α2 < 1, 0 " β1 " 1, 0 " β2 " 1

and

a, b, c ∈ R,

and consider the following fractional differential equation: & ' & ' α ,β α ,β a D0+1 1 y (x) + b D0+2 2 y (x) + cy (x) = f (x)

in the space of Lebesgue integrable functions y ∈ L(0, ∞) with the initial conditions: & ' (1−β )(1−αi ) I0+ i y (0+) = ci (i = 1, 2),

(3.7)

(3.8)

where, without loss of generality, we assume that

(1 − β1 )(1 − α1 ) " (1 − β2 )(1 − α2 ). If c1 < ∞, then c2 = 0

unless

(1 − β1 )(1 − α1 ) = (1 − β2 )(1 − α2 ).

An equation of the form (3.7) was introduced in [7] for dielectric relaxation in glasses. Although the Laplace transformed relaxation function and the corresponding dielectric susceptibility were found, its general solution was not given in [7]. Here we proceed to find its general solution. Theorem 5 The fractional differential equation (3.7) with the initial conditions (3.8) has its solution in the space L(0, ∞) given by > ∞ & c ' 1 * & a 'm y (x) = − ac1 x (α2 −α1 )m+α2 +β1 (1−α1 )−1 Eαm+1 − x α2 2 ,(α2 −α1 )m+α2 +β1 (1−α1 ) b m=0 b b & c ' − x α2 + bc2 x (α2 −α1 )m+α2 +β2 (1−α2 )−1 Eαm+1 2 ,(α2 −α1 )m+α2 +β2 (1−α2 ) b & '& c '? + Em+1 (3.9) − x α2 . α2 ,(α2 −α1 )m+α2 ,− bc ;0+ f b

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Proof Our demonstration of Theorem 5 is based upon the Laplace transform method. Indeed, if we make use of the notational convention depicted in (1.7) and the Laplace transform formula (1.6), by applying the operator L to both the sides of (3.7), it is easily seen that Y (s) = ac1

s β1 (α1 −1) s β2 (α2 −1) F (s) + bc + α . 2 α α α α 1 2 1 2 1 as + bs + c as + bs + c as + bs α2 + c

Furthermore, since


s βi (αi −1) 1 = α α 1 2 as + bs + c b

$

s βi (αi −1) s α2 + bc

%

 

1+

a b

1 &

s α1 s α2 + bc



' =

∞ 1 * & a 'm s α1 m+βi αi −βi − ! "m+1 b m=0 b s α2 + bc

> * ∞ & a 'm 1 =L x (α2 −α1 )m+α2 +βi (1−αi )−1 (−1)m b m=0 b & c '? − x α2 (s) · Eαm+1 2 ,(α2 −α1 )m+α2 +βi (1−αi ) b

(i = 1, 2)

and

as α1

< = > * ∞ ∞ & F (s) s α1 m a 'm 1 1 * & a 'm − − = F = L (p) ! " m+1 + bs α2 + c b m=0 b b m=0 b s α2 + bc $ %? & c ' α2 · x (α2 −α1 )m+α2 −1 Eαm+1 − ∗ f (s) x (x) 2 ,(α2 −α1 )m+α2 b @ ∞ A & a 'm & '& c ' 1* =L Em+1 − x α2 (s) (−1)m α2 ,(α2 −α1 )m+α2 ,− bc ;0+ f b m=0 b b

in terms of the Laplace convolution, by applying the inverse Laplace transform, we get the solution (3.9) asserted by Theorem 5. # 3.2. An Application of Theorem 5 The next problem is to solve the fractional differential equation (3.7) in the space of Lebesgue integrable functions y ∈ L(0, ∞) when α1 = α2 = α

β1 * = β2 ,

and

that is, the following fractional differential equation: & ' & ' α,β α,β a D0+ 1 y (x) + b D0+ 2 y (x) + cy (x) = f (x) under the initial conditions given by & ' (1−β )(1−α) I0+ i y (0+) = ci

(i = 1, 2).

We now assume, without loss of generality, that β2 " β1 . If c1 < ∞, then c2 = 0

unless

β1 = β2 .

(3.10)

(3.11)


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Corollary 1 The fractional differential equation (3.10) under the initial conditions (3.11) has its solution in the space L (0, ∞) given by


y (x) =

$

% $ % ac1 c x β1 +α(1−β1 )−1 Eα,β1 +α(1−β1 ) − xα a+b a+b $ $ % % bc2 c β2 +α(1−β2 )−1 α + x Eα,β2 +α(1−β2 ) − x a+b a+b $ %& ' 1 1 c + f Eα,1,− (x) . a+b ;0+ a+b

(3.12)

Proof Our proof of Corollary 1 is much akin to that of Theorem 5. We choose to omit the details involved. # 3.3. A Fractional Differential Equation Related to the Process of Dielectric Relaxation Let 0 < α1 " α2 " α3 < 1

and

0 " βi " 1

(i = 1, 2, 3; a, b, c, e ∈ R).

Consider the following fractional differential equation: & ' & ' & ' α ,β α ,β α ,β a D0+1 1 y (x) + b D0+2 2 y (x) + c D0+3 3 y (x) + ey (x) = f (x)

(3.13)

in the space of Lebesgue integrable functions y ∈ L(0, ∞) with the initial conditions given by & ' (1−β )(1−αi ) I0+ i y (0+) = ci

(i = 1, 2, 3).

(3.14)

Without loss of generality, we assume that (1 − β1 )(1 − α1 ) " (1 − β2 )(1 − α2 ) " (1 − β3 )(1 − α3 ). If c1 < ∞, then c2 = 0

unless

(1 − β1 )(1 − α1 ) = (1 − β2 )(1 − α2 )

and c3 = 0

unless

(1 − β1 )(1 − α1 ) = (1 − β2 )(1 − α2 ) = (1 − β3 )(1 − α3 ).

Hilfer [7] observed that a particular case of the fractional differential equation (3.13) when α1 = 1, βi = 1

(i = 1, 2, 3), e = 1

and

f (x) = 0

describes the process of dielectric relaxation in glycerol over 12 decades in frequency.


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Theorem 6 The fractional differential equation (3.13) with the initial conditions (3.14) has its solution in the space L (0, ∞) given by ∞ m $ % * (−1)m * m k m−k (α3 −α2 )m+(α2 −α1 )k+α3 −1 a b x y (x) = cm+1 k=0 k m=0 > & e ' · ac1 x β1 (1−α1 ) Eα3 ,(α3 −α2 )m+(α2 −α1 )k+α3 +β1 (1−α1 ) − x α3 c & e ' + bc2 x β2 (1−α2 ) Eα3 ,(α3 −α2 )m+(α2 −α1 )k+α3 +β2 (1−α2 ) − x α3 c & e '? + cc3 x β3 (1−α3 ) Eα3 ,(α3 −α2 )m+(α2 −α1 )k+α3 +β3 (1−α3 ) − x α3 c ∞ m $ % m * & '& e ' * m k m−k m+1 (−1) + a E (3.15) b f − x α3 . ,(α −α −α α )m+(α )k+α 3 3 2 2 1 3 m+1 k c c m=0 k=0 Proof Making use of the above-demonstrated technique based upon the Laplace and the inverse Laplace transformations once again, it is not difficult to deduce the solution (3.15) just as we did in our proof of Theorem 5. # 3.4. An Interesting Consequence of Theorem 6 Let 0 0 , (4.1) (x − t)ν−1 f (t) dt = g (x) # (ν) 0

where a ∈ C and g ∈ L (0, b) (b > 0). Here, in this section, we consider the following general class of differintegral equations of the Volterra type involving the generalized fractional derivative operators: # x ! α,µ " a (4.2) D0+ f (x) + (x − t)ν−1 f (t) dt = g (x) # (ν) 0 ! " 0 < α < 1; 0 " µ " 1; " (ν) > 0

in the space of Lebesgue integrable functions f ∈ L (0, ∞) with the initial condition given by & ' (1−µ)(1−α) I0+ f (0+) = c. (4.3)

Theorem 8 The fractional differintegral equation (4.2) with the initial condition (4.3) has its solution in the space L (0, ∞) given by " ! " ! 1 f (x) = cx α−µ(α−1)−1 Eα+ν,α−µ(α−1) −ax α+ν + Eα+ν,α,−a;0+ g (x) , (4.4)

where c is an arbitrary constant.


809

Proof By applying the Laplace transform operator L to both sides of (4.2) and using the formula (1.6), we readily get s µ(α−1)+ν sν F (s) = c α+ν + α+ν G (s) , s +a s +a which, in view of the Laplace transform formula (1.11) and Laplace convolution theorem, yields 4 ! "5 F (s) = cL x α−µ(α−1)−1 Eα+ν,α−µ(α−1) −ax α+ν (s) 4! ! "" 5 + L x α−1 Eα+ν,α −ax α+ν ∗ g (x) (s) .

The solution (4.4) asserted by Theorem 8 would now follow by appealing to the inverse Laplace transform to each member of this last equation. #


Next, we consider some illustrative examples of the solution (4.4). Example 1

If we put g (x) = x µ−1

in Theorem 8 and apply the special case of the following integral formula when γ = 1 [8], then # x ! " γ γ ω (x − t)ρ t µ−1 dt = # (µ) x α+µ−1 Eρ,α+µ (ωx ρ ) , (4.5) (x − t)α−1 Eρ,α 0

we can deduce a particular case of the solution (4.4) given by Corollary 3.

Corollary 3 The following fractional differintegral equation: # x ! α,µ " a D0+ f (x) + (x − t)ν−1 f (t) dt = x µ−1 # (ν) 0 ! " 0 < α < 1; 0 " µ " 1; " (ν) > 0

with the initial condition (4.3) has its solution in the space L (0, ∞) given by 4 ! " ! "5 f (x) = x α−µ(α−1)−1 cEα+ν,α−µ(α−1) −ax α+ν + # (µ) x αµ Eα+ν,α+µ −ax α+ν . Example 2

(4.6)

(4.7)

If, in Theorem 8, we put

! " g (x) = x µ−1 Eα+ν,µ −ax α+ν

and apply the special case of the following integral formula when γ = σ = 1 [24], then # x ! " γ +σ γ σ ω (x − t)ρ t ν−1 Eρ,ν (x − t)µ−1 Eρ,µ (ωt ρ ) dt = x µ+ν−1 Eρ,µ+ν (ωx ρ ) ,

(4.8)

0

and we get another particular case of the solution (4.4) given by Corollary 4.

Corollary 4 The following fractional differintegral equation: # x ! α,µ " ! " a D0+ f (x) + (x − t)ν−1 f (t) dt = x µ−1 Eα+ν,µ −ax α+ν # (ν) 0 ! " 0 < α < 1; 0 " µ " 1; " (ν) > 0

with the initial condition (4.3) has its solution in the space L (0, ∞) given by 4 ! " ! "5 2 f (x) = x α−µ(α−1)−1 cEα+ν,α−µ(α−1) −ax α+ν + x αµ Eα+ν,α+µ −ax α+ν ,


(4.9)

(4.10)

810

Example 3

Ž. Tomovski et al.

If we put

! " g (x) = x β+ν−1 Eα+ν,β+ν −bx α+ν

and apply the following integral formula [23], then # x ! " ! " (x − t)α−1 Eα+ν,α −a (x − t)α+ν t β+ν−1 Eα+ν,β+ν −bt α+ν dt 0

! " ! " Eα+ν,β −bx α+ν − Eα+ν,β −ax α+ν β−1 = x a−b

(a *= b) ,

(4.11)


and we get yet another particular case of the solution (4.4) given by Corollary 5. Corollary 5 The following fractional differintegral equation: # x ! α,µ " ! " a D0+ f (x) + (x − t)ν−1 f (t) dt = x β+ν−1 Eα+ν,β+ν −bx α+ν # (ν) 0 ! " 0 < α < 1; 0 " µ " 1; " (ν) > 0 with the initial condition (4.3) has its solution in the space L (0, ∞) given by ! " f (x) = cx α−µ(α−1)−1 Eα+ν,α−µ(α−1) −ax α+ν ! " ! " Eα+ν,β −bx α+ν − Eα+ν,β −ax α+ν β−1 + x (a * = b) , a−b

(4.12)

(4.13)


4.2. A Cauchy-Type Problem for Fractional Differential Equations Kilbas et al. [10] established the explicit solution of the Cauchy-type problem for the following fractional differential equation: & ' ! α " γ Da+ f (x) = λ Eρ,α,ν;a+ f (x) + g (x) (4.14) (a ∈ C; g ∈ L [0, b])

in the space of Lebesgue integrable functions f ∈ L (0, ∞) with the initial conditions given by ! α " ! " Da+ f (a+) = bk bk ∈ C (k = 1, · · · , n) . (4.15) Let us consider the following more general Volterra-type fractional differintegral equation: & ' ! α,µ " γ D0+ f (x) = λ Eρ,α,ν;0+ f (x) + g (x) (4.16)

in the space of Lebesgue integrable functions f ∈ L (0, ∞) with initial condition given by & ' (1−µ)(1−α) I0+ f (0+) = c. (4.17)

Theorem 9 The fractional differintegral equation (4.16) with the initial condition (4.17) has its solution in the space L (0, ∞) given by f (x) = c

∞ *

γk

λk x 2αk+α+µ−µα−1 Eρ,2αk+α+µ−µα (νx ρ )

k=0

+ where c is an arbitrary constant.

∞ * k=0

& ' γk λk Eρ,2αk+α,ν;0+ g (x) ,

(4.18)


811

Proof Applying Laplace transforms on both sides of (4.16), we get F (s) = c

s µ(α−1) G (s) ( ργ −α ) + ( ργ −α ) . s s α − λ (s ρ −ν)γ s α − λ (ssρ −ν)γ

(4.19)

On the other hand, by virtue of (1.11), it is not difficult to see that @∞ A * s µ(α−1) k 2αk+α+µ−µα−1 γ k ρ ( ργ −α ) = L λ x Eρ,2αk+α+µ−µα (νx ) (s) s α − λ (ssρ −ν)γ k=0


and G (s) ( ργ −α ) = L s α − λ (ssρ −ν)γ

@
0 , (5.1)

where N (t) denotes the number density of a given species at time t, N0 = N (0) is the number density of that species at time t = 0, c is a constant and (for convenience) f ∈ L(0, ∞), it being tacitly assumed that f (0) = 1 in order to satisfy the initial condition N (0) = N0 . By applying the Laplace transform operator L to each member of (5.1), we readily obtain