Caputo Equations in the frame of fractional operators with Mittag-Leffler kernels

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 ) E1 ,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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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 ) E1 ,1,  ,a + g (t ) ba .

2 −

B ( − 1) b g (t ) E1 ,1,  ,a + f (t )dt  a 2 − B ( − 1) b = f (t ) E1 ,1,  ,b _ g (t )dt  a 2 −

g (t ) ABCa D f (t )dt =

B( − 1) B( − 1) b d2 1 f (t ) E1 ,1,  ,b − g (t ) ba − f ( t ) E − dt 2 − 2 −  a dt 2  ,1,  ,b B( − 1) f (t ) E1 ,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 ) E1 ,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 E1 ,1,  ,b − x(a) + c2 ABC Db x(a) = 0,

(10)

d1 E1 ,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 E1 ,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

x1 , x2 =  r (t ) x1 (t ) x2 (t )dt = 0, 1  2 , a

When the functions xi correspond to eigenvalues Proof. Let

1

and

2

i , i = 1,2 .

be two distinct eigenvalues of (9) corresponding to the eigenfuntions x1 and x2 ,

respectively. Then we have

D ( p(t ) ABCDb x1 (t )) + q(t ) x1 (t ) = 1r (t ) x1 (t ) ,

(19)

D ( p(t ) ABC Db x2 (t )) + q(t ) x2 (t ) = 2 r (t ) x2 (t ) .

(20)

ABC a ABC a

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Caputo Equations in the frame of fractional operators…

We multiply (19) and (20) by x2 (t ) and x1 (t ) , respectively, and subtract the results to obtain

x2 ABCa D ( p(t ) ABC Db x1 (t )) − x1 (t ) ABCa D ( p(t ) ABC Db x2 (t )) = (1 − 2 )r (t ) x1 (t ) x2 (t ).

Now, integrating from a to b, we get b

(1 − 2 )  r (t ) x1 (t ) x2 (t ) = 0 . a

1  2 , it follows that

Since



b

a

r (t ) x1 (t ) x2 (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

x1 , x2 =  r (t ) x1 (t ) x2 (t )dt = 0, 1  2 , a

When the functions xi correspond to eigenvalues

i , i = 1,2 .

Proof. The proof is similar to that of Theorem 5.

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