Invention Journal of Research Technology in Engineering & Management (IJRTEM) ISSN: 2455-3689 www.ijrtem.com Volume 2 Issue 10 ǁ October 2018 ǁ PP 39-45
Caputo Equations in the frame of fractional operators with Mittag-Leffler kernels Dong Qiang, Chengmin Hou* （Yanbian University, Jilin Yanji 133002）
ABSTRACT： In this paper, we first use fractional integral formula to study the fractional integral property of ABC and ABR. Secondly, we present integration by parts formulas for the ABC and ABR fractional derivatives with Mittag-Leffler kernels when the operators is.
3 2
(1, ) . Finally, we investigate the self-adjointness,
eigenvalues, and eigenfunction properties of the corresponding fractional Sturm-Liouville equations by using fractional integration by parts formulas.
KEYWORDS: Mittag-Leffler kernels; ABR and ABC fractional derivatives; Sturm- Liouville equations. I.
INTRODUCTION
The fractional differential plays a key role in the development of mathematics and science fields ([4], [7], [8]), especially for the discrete fractional calculus was of interest among several mathematicians and has been developing rapidly. In recent years, it is very meaningful to develop new non-local fractional derivatives ([5], [10]). The proposed kernels are non-singular such as those with Mittag-Leffler kernels. What makes those fractional derivatives with Mittag-Leffler kernels more interesting is that their corresponding fractional integrals contain Riemann-Liouville fractional integrals as a part of their structure. The advantage of develop more efficient algorithms in solving fractional dynamical systems by concentrating only on the differential equations. In this paper, we present integration by parts formulas for the ABC and ABR fractional differences with Mittag-Leffler kernels of order
3 2
(1, ) which will be used to prove the self-adjointness, eigenvalues, and eigenfunction
properties of suitable fractional difference Sturm-Liouville equations with proper boundary conditions. From [13], we know the classical Sturm-Liouville problem for a linear differen-tial equation of second order is a boundary value problem of the form:
d dx ( p(t ) ) + q(t ) x(t ) = r (t ) x(t ), t [a, b], dt dt c1 x(a) + c2 x(a) = 0, d1 x(b) + d1 x(b) = 0, where p, q, r are continuous functions on the interval [a, b] such as q (t ) 0 , −
r (t ) 0 on [a, b] . The differential equation can be written in the from: L ( x ) = r (t ) x (t ) where L( x)= D ( p(t ) D x)(t ) + q(t ) x(t ) . Parameter for which the above boundary value problem has a c
c
nontrivial solution is called an eigenvalue.
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Caputo Equations in the frame of fractional operators… II. PRELIMINARIES Definition 1. ([2]) The Mittag-Leffler function of one parameter is defined by
and the Mittag-Leffler function of two parameters , is defined by
where E ,1 ( z) = E ( z). Definition 2. ([13]) The left fractional integral of order
( a I f )(t ) =
starting at a has the following from
1 t (t − s) −1 f ( s)ds a ( )
The right fractional integral of order
( I b f )(t ) = where
0
0
ending at b has the following from
1 b ( s − t ) −1 f ( s)ds , t ( )
0 ,t [a ,b ], f are continuous on the interval [a, b] . 3 2
Definition 3. ([5]) Let f H (a, b), (1, ), = − 1 ,Then the left Caputo fractio1
nal derivative is defined by
( ABCaD f )(t ) =( ABCaD f )(t ) .
(1)
Then the left Riemann-Liouville fractional derivative is defined by
( ABRaD f )(t ) =( ABRaD f )(t ) .
(2)
In addition, the corresponding fractional integral is defined by
( ABaI f )(t ) =( a I ABaI f )(t ) , ( ABaI f )(t ) =( a I +1 f )(t ) 3 2
Let f H ( a, b), (1, ), 1
= − 1 , Then the right Caputo fractional derivative is defined by
( ABC Db f )(t ) = −( ABC Db f )(t ) . |Volume 2| Issue 10 |
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Caputo Equations in the frame of fractional operators… Then the right Riemann-Liouville fractional derivative is defined by
( ABRDb f )(t ) = −( ABRDb f )(t ) ,
(5)
the corresponding fractional integral is defined by
( ABI b f )(t ) = ( I b AB I b f )(t ) .
(6)
From [5], we have
( ABRaD ABa I f )(t ) = f (t ),
( ABRDb AB I b f )(t ) = − f (t ) .
Lemma 1. Let f (t ) are continuous functions on the interval
3 [a, b] , (1, ) , then 2
( ABaI ABCa D f )(t ) = f (t ) − f (a) − f (a)(t − a),
( ABI b ABR Db f )(t ) = f (b) − f (t ) ,
( ABI ABC Db f )(t ) = f (b) − f (t ) − f (b)(b − t ),
( ABaI ABRa D f )(t ) = f (t ) − f (a) .
Proof. Using Definition 1. Let
1 2
= + 1, (0, ) , we get
( ABaI ABRa D f )(t ) =( a I ABaI ABRa D f )(t )= a If (t ) = f (t ) − f (a). ( ABI b ABR Db f )(t ) = −( I b AB I b ABR Db f )(t ) = −( I b f )(t ) = f (b) − f (t ). ( ABaI ABCa D f )(t ) =( a I ABaI ABCa D f )(t )= a I [ f (t ) − f (a)] = f (t ) − f (a) − f (a)(t − a). ( ABI b ABC Db f )(t ) = −( I b AB I b ABC Db f )(t ) = − I b [ f (t ) − f (b)] = f (b) − f (t ) − f (b)(b − t ). Definition 4. ([3]) Let f H (a, b), a b, 1
ABC a
D f (t ) =
3 2
(1, ) , the left Caputo fractional derivative is defined by
t 1 (t − x)1− f ( x)dx, ( 2 − ) a
the left Riemann-Liouville fractional derivative is defined by ABR a
1 d2 D f (t ) = (2 − ) dt 2
t
(t − x) a
1−
f ( x)dx .
In addition, the right Caputo fractional derivative is defined by ABC
Db f (t ) = −
b 1 ( x − t )1− f ( x)dx, t ( 2 − )
the right Riemann-Liouville fractional derivative is defined by
1 d2 b ( x − t )1− f ( x)dx, 2 t (2 − ) dt where B ( ) 0, B (0) = B (1) = 1. ABR
Db f (t ) = −
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Caputo Equations in the frame of fractional operators… III.
MAIN RESULTS
From[1], recall the left generalized fractional integral operator as x
E, , w,a + ( x) = ( x − t ) −1 E, ( w( x − t ) ) (t )dt, x a .
(7)
a
Analogously, the right generalized fractional integral operator can be defined by b
E, , w,b − ( x) = (t − x) −1 E, ( w(t − x) ) (t )dt, x b .
(8)
x
Lemma 2. Let f H (a, b), a b, 1
3 2
(1, ) , using (1) and (2), then the Caputo fractional derivative with
Mittag-Leffler kernels is defined by
B ( − 1) 1 E −1 + f (t ), ,1, − ,a 2 − 2 − B ( − 1) 1 ABC Db f (t ) = − E −1 − f (t ) . ,1, − ,b 2 − 2 − ABC a
D f (t ) =
Then the Riemann-Liouville fractional derivative with Mittag-Leffler kernels is defined by ABR a
D f (t ) =
B ( − 1) d 2 1 E −1 f (t ), 2 − dt 2 ,1, − 2− ,a +
B ( − 1) d 2 1 E −1 f (t ) . 2 − dt 2 ,1, − 2− ,b − 1 B( − 1) −1 − ( −1) = = exp[ − (t − x)] , and Proof. Because (t − x) , 2 − (2 − ) 2 − B( − 1) t −1 ABC f ( x) exp[ − (t − x)]dx , so a D f (t ) = 2 − a 2 − −1 (t − x)]k [− −1 2 − due to exp[ − , so (t − x)] = 2 − k! k =0 ABR
ABC a
Db f (t ) = −
−1 k ) t k 2 − a f ( x)(t − x) dx k! −1 k (− ) t B ( − 1) 2 − = f ( x)(t − x)k dx a 2 − k =0 (k + 1) B ( − 1) 1 = E ,1, ,a + f ( x). 2 −
B ( − 1) D f (t ) = 2 − k =0
where 0, p 1, q 1,
(−
1 1 1 1 + 1 + ( p 1 and q 1 in case + = 1 + ), If p q p q b
( x) Lp (a, b), ( x) Lq (a, b) , then (t ) E1 a
b
(t )dt = (t ) E1
− + ,1, ,a 1−
a
,1,
− − ,b 1−
(t )dt .
3 −1 =− , then 2 2 − B( − 1) f (t ) ABRDb g (t )dt + f (t ) E1 ,1, ,b − g (t ) ba , 2 −
Lemma 3. Let f , g H (a, b), (1, ), 1
b
a
g (t ) ABCa D f (t )dt =
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b
a
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Caputo Equations in the frame of fractional operators…
b
a
b
g (t ) ABC Db f (t )dt = f (t ) ABRaD g (t )dt − a
Proof. Using Lemma 2, we get
b
a
=
B( − 1) f (t ) E1 ,1, ,a + g (t ) ba . 2 −
B ( − 1) b g (t ) E1 ,1, ,a + f (t )dt a 2 − B ( − 1) b = f (t ) E1 ,1, ,b _ g (t )dt a 2 −
g (t ) ABCa D f (t )dt =
B( − 1) B( − 1) b d2 1 f (t ) E1 ,1, ,b − g (t ) ba − f ( t ) E − dt 2 − 2 − a dt 2 ,1, ,b B( − 1) f (t ) E1 ,1, ,b − g (t ) ba . 2 −
b
= f (t ) ABRDb g (t )dt + a
Similarly, the proof of the second part is as follows:
b
a
b
g (t ) ABC Db f (t )dt = f (t ) ABRaD g (t )dt − a
B( − 1) f (t ) E1 ,1, ,a + g (t ) ba . 2 −
Consider the fractional Sturm-liouville problem with Caputo operator as ABC a c
D ( p(t ) ABCDb x(t )) + q(t ) x(t ) = r (t ) x(t ), t [a, b] ,
L1 x(t )=
ABC a
D ( p(t )
ABC
(9)
Db x(t )) + q(t ) x(t ) ,
3 2
where (1, ), p (t ) 0, r (t ) 0, p, q, r are real valued continuous functions on the interval
[a, b] , and
boundary conditions
c1 E1 ,1, ,b − x(a) + c2 ABC Db x(a) = 0,
(10)
d1 E1 ,1, ,b − x(b) + d 2 ABC Db x(b) = 0,
(11)
where c1 + c2 0, d1 + d 2 0 . 2
2
2
2
c
Theorem 1. The fractional Caputo operator L1 x (t ) is self-adjoint. Proof. We have
(t ) c L1 (t ) = (t ) ABCa D ( p(t ) ABC Db (t )) + q(t ) (t ) (t ) ,
(t ) L1 (t ) = (t ) c
ABC a
D ( p(t )
ABC
Db (t )) + q(t ) (t ) (t ) .
(12) (13)
Let (12) subtract to (13), we have
(t ) c L1 (t ) − (t ) c L1 (t ) = (t ) ABCa D ( p(t ) ABC Db (t )) − (t ) ABCa D ( p(t ) ABC Db (t )) . Integrating from a to b, we have b
[ (t ) a
b
c
L1 (t ) − (t ) c L1 (t )]dt b
= [ (t ) ABCa D ( p (t ) ABC Db (t ))]dt − [ (t ) ABCa D ( p (t ) ABC Db (t ))]dt a
a
b
b
a
a
= [ p(t ) ABC Db (t ) ABC Db (t )]dt − [ p(t ) ABC Db (t ) ABC Db (t )]dt = 0. |Volume 2| Issue 10 |
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Caputo Equations in the frame of fractional operators… Hence, L1 , = , L1 . That is, L1 is self-adjoint. Consider the fractional Sturm-liouville problem with Riemann-Liouville operator as c
c
ABR a
c
D ( p(t ) ABRDb x(t )) + q(t ) x(t ) = r (t ) x(t ), t [a, b] ,
(14)
L1 x(t )= ABRaD ( p(t ) ABRDb x(t )) + q(t ) x(t ) . 3 where (1, ), p (t ) 0, r (t ) 0, p, q, r are real valued continuous functions on the interval [a, b] , and 2 R
boundary conditions
c1 E1 ,1, ,b − x(a) + c2 ABR Db x(a) = 0 , d1 E ,1, ,b − x(b) + d 2 1
ABR
(15)
Db x(b) = 0 .
(16)
where c1 + c2 0, d1 + d 2 0 . 2
2
2
2
R
Theorem 2. The fractional Riemann-Liouville operator L1 x(t ) is self-adjoint. Proof. The proof is similar to that of Theorem 1. Theorem 3. The eigenvalues of the Sturm-Liouville equations with Caputo operator are real. Proof. Assume that is the eigenvalue for (9) corresponding to eigenfunction x , then x and its complex conjugate
x satisfy c
L1 x(t ) = r (t ) x(t ) ,
(17)
c
L1 x(t ) = r (t ) x(t ) .
(18)
We multiply (17) by x(t ) and (18) by
x(t ) , respectively, and subtract to obtain
( − )r (t ) x(t ) x(t ) = x(t )cL1 x(t ) − x(t )cL1 x(t ) . Integrating from a to b, and using Theorem 1, we get b
( − ) r (t ) x(t ) dt = 0 , 2
a
and r (t ) 0 , we conclude that
=.
Theorem 4. The eigenvalues of the Sturm-Liouville equations with Caputo operator are real. Proof. The proof is similar to that of Theorem 3. Theorem 5. The eigenfunctions, corresponding to distinct eigenvalues of the Sturm-Liouville equations (9) are orthogonal with respect to the weight function r on [ a , b ] that is b
x1 , x2 = r (t ) x1 (t ) x2 (t )dt = 0, 1 2 , a
When the functions xi correspond to eigenvalues Proof. Let
1
and
2
i , i = 1,2 .
be two distinct eigenvalues of (9) corresponding to the eigenfuntions x1 and x2 ,
respectively. Then we have
D ( p(t ) ABCDb x1 (t )) + q(t ) x1 (t ) = 1r (t ) x1 (t ) ,
(19)
D ( p(t ) ABC Db x2 (t )) + q(t ) x2 (t ) = 2 r (t ) x2 (t ) .
(20)
ABC a ABC a
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Caputo Equations in the frame of fractional operators… We multiply (19) and (20) by x2 (t ) and x1 (t ) , respectively, and subtract the results to obtain
x2 ABCa D ( p(t ) ABC Db x1 (t )) − x1 (t ) ABCa D ( p(t ) ABC Db x2 (t )) = (1 − 2 )r (t ) x1 (t ) x2 (t ). Now, integrating from a to b, we get b
(1 − 2 ) r (t ) x1 (t ) x2 (t ) = 0 . a
1 2 , it follows that
Since
b
a
r (t ) x1 (t ) x2 (t ) = 0 ,
which completes the proof. Theorem 6. The eigenfunctions, corresponding to distinct eigenvalues of the Sturm-Liouville equations (14) are orthogonal with respect to the weight function r on [ a , b ] that is b
x1 , x2 = r (t ) x1 (t ) x2 (t )dt = 0, 1 2 , a
When the functions xi correspond to eigenvalues
i , i = 1,2 .
Proof. The proof is similar to that of Theorem 5.
REFERENCES: [1] [2] [3] [4] [5] [6] [7] [8]
[11] [12] [13]
A.A. Kilbas, M. Saigo, R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr. Transf.Spec.F.,15No.1(2010), 31–49. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. Chuanzhi Bai, Existence and uniqueness of solutions for fractional boundary value problems with pLaplacian operator, (2018). T. Abdeljawad, D.Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ. 2017:78 (2017). T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017:130 (2017). T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016:232 (2016). R. Hilfer (Ed.), Applications of Fractional Calculus in Physics. World Scientific, Sin-gapore (2000). F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010). [9] H.L. Gray, N.F. Zhang, On a new definition of the fractional difference, Math. Comp., 50 (1988), 513–529. [10] A. Freed, K. Diethelm, Yu. Luchko, Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus. NASA’s Glenn Research Center, Ohio(2002). R. Hilfer (Ed.), Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000). R. Klages, G. Radons, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications. WileyVCH, Weinheim (2008). R. Mert, T. Abdeljawad, A. Peterson, Sturm Liouville Equations in the frame of fractional operators with Mittag-Leffler kernels and their discrete versions. Math.CA（2018）
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