Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 915830 7 pageshttpdxdoiorg1011552013915830

Research ArticleFundamental Spectral Theory of Fractional SingularSturm-Liouville Operator

Erdal Bas

Department of Mathematics Faculty of Science Firat University 23119 Elazig Turkey

Correspondence should be addressed to Erdal Bas erdalmatyahoocom

Received 29 May 2013 Accepted 19 July 2013

Academic Editor Kehe Zhu

Copyright copy 2013 Erdal Bas This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouvilleproblem We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal respectively Furthermore weprove new approximations about the topic

1 Introduction

Sturm-Liouville problem was first developed in a number ofpapers that were published by these authors in 1836 and 1837Charles-Francois Sturm (1803ndash1855) Professor of Mechanicsat the Sorbonne had been interested since about 1833 inthe problem of heat flow in bars so he was well aware ofeigenvalue-type problems He worked closely with his friendJoseph Liouville (1809ndash1882) Professor ofMathematics at theCollege de France on the general properties of second-orderdifferential equations Liouville also made many contribu-tions to the general field of analysis see [1]

A Sturm-Liouville boundary value problem consists of asecond order linear ordinary differential equation

minus (119901119910

1015840)

1015840

+ 119902119910 = 120582119908119910 (119886 119887)

(1)

and boundary conditions Here (119886 119887) is a bounded orunbounded open interval of the real line 119877 The coefficients119901 119902 119908 (119886 119887) into 119877 120582 isin C the complex fieldSpectral analysis finds applications in many diverse fieldsMathematical techniques could be developed into a moresuitable and significant course by presenting them withinthe more general Sturm-Liouville theory in 119871

2 The Sturm-

Liouville problems are important in many areas of scienceengineering and mathematics It is known that the spectralcharacteristics are spectra spectral functions scattering datanorming constants etc According to the theory 119886 linearsecond-order differential operator which is self-adjoint has

an orthogonal sequence of eigenfunctions in 119871

2 Spectral

properties of Sturm-Liouville operators are often deriveddirectly or indirectly as a consequence of an established linkbetween large distance asymptotic behavior of solutions ofthe associated differential equation and spectral propertiesof the corresponding differential operator Sturm-Liouvilleproblems are divided into regular and singular types Dif-ferential equations such as Bessel hydrogen atom HermitteJakobi and Legendre equations can be transformed intoSturm-Liouville equations There are many studies on theseissues [2ndash7] We also discuss the radial part of Schrodingerrsquosequation for the Bessel equation

Fractional calculus is ldquothe theory of derivatives andintegrals of any arbitrary real or complex order whichunify and generalize the notions of integer-order differen-tiation and 119899-fold integrationrdquo [6ndash13] In recent years theconcept of fractional calculus originated from Leibniz hasachieved increasing interest during the last two decades Inparticular the last decade has scientific papers concerningfractional quantummechanics It has been proved that manysystems in different fields of science and engineering canbe modeled more accurately using fractional derivatives[8ndash17] Fractional calculus has increasing importance forthe last years because fractional calculus has been appliedto almost every field of science They are viscoelasticityelectrical engineering electrochemistry biology biophysicsand bioengineering signal and image processing mechanicsmechatronics physics and control theory We note that

2 Journal of Function Spaces and Applications

ordinary derivatives in a traditional Sturm-Liouville problemare replaced with fractional derivatives and the resultingproblems are solved using some numerical methods [18ndash23]Furthermore Klimek and Argawal [24] define a fractionalSturm-Liouville operator introduce a regular fractionalSturm-Liouville problem and investigate the properties ofthe eigenfunctions and the eigenvalues of the operator In thispaper our purpose is to introduce singular fractional Sturm-Liouville problem having Bessel type and prove spectralproperties of spectral data for the operator

Let us give the boundary value problem for Besselequation and necessary data as follows

2 Preliminaries

Now consider the following Bessel equation

119889

2119910

119889119909

2+ (120582 minus

V2 minus 14

119909

2)119910 = 0 (2)

where 120582 and V are real numbers The Bessel equation forhaving the analogous singularity is given in [5]

Definition 1 (see [10]) Let 0 lt 120572 le 1The left-sided and right-sided Riemann-Liouville integrals of order 120572 respectively aregiven by the formulas

(119868

120572

119886+119891) (119909) =

1

Γ (120572)

int

119909

119886

(119909 minus 119904)

120572minus1119891 (119904) 119889119904 119909 gt 119886

(119868

120572

119887minus119891) (119909) =

1

Γ (120572)

int

119887

119909

(119904 minus 119909)

120572minus1119891 (119904) 119889119904 119909 lt 119887

(3)

where Γ denotes the gamma function

Definition 2 (see [10]) Let 0 lt 120572 le 1 The left-sidedand right-sided Riemann-Liouville derivatives of order 120572respectively are defined as follows

(119863

120572

119886+119891) (119909) = 119863 (119868

1minus120572

119886+119891) (119909) 119909 gt 119886

(119863

120572

119887minus119891) (119909) = minus119863 (119868

1minus120572

119887minus119891) (119909) 119909 lt 119887

(4)

Analogous formulas yield the left-sided and right-sidedCaputo derivatives of order 120572

(

119862

119863

120572

119886+119891) (119909) = (119868

1minus120572

119886+119863119891) (119909) 119909 gt 119886 0 lt 120572 le 1

(

119862

119863

120572

119887minus119891) (119909)

= (119868

1minus120572

119887minus(minus119863)119891) (119909) 119909 lt 119887 0 lt 120572 le 1

(5)

Definition 3 (see [14]) The general function119901Ψ

119902(119911) is

defined for 119911 isin C 119886119897 119887

119895isin C and120572

119897 120573

119895isin R (119897 = 1 119901 119895 =

1 119902) by the series

119901Ψ

119902(119911) =

119901Ψ

119902[

(119886

1 120572

1)

1119901

(119887

1 120573

1)

1119902

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911] =

infin

sum

119896=0

prod

119901

119897=1Γ (119886

119897+ 120572

119897119896)

prod

119902

119895=1Γ (119887

119895+ 120573

119895119896)

119911

119896

119896

(6)

This general Wright function was investigated by Fox whopresented its asymptotic expansion for large values of theargument 119911 under the condition

119902

sum

119895=1

120573

119895minus

119901

sum

119897=1

120572

119897gt 1 (7)

If these conditions are satisfied the series in (6) is convergentfor any 119911 isin C

Theorem 4 (see [14]) Let 119886119897 119887

119895isin C and 120572

119897 120573

119895isin R (119897 =

1 119901 119895 = 1 119902) and let

Δ =

119902

sum

119895=1

120573

119895minus

119901

sum

119897=1

120572

119897

120575 =

119901

prod

119897=1

1003816

1003816

1003816

1003816

120572

119897

1003816

1003816

1003816

1003816

minus120572119897

119902

prod

119895=1

1003816

1003816

1003816

1003816

1003816

120573

119895

1003816

1003816

1003816

1003816

1003816

120573119895

120583 =

119902

sum

119895=1

119887

119895minus

119901

sum

119897=1

119886

119897+

119901 minus 119902

2

(8)

(i) If Δ gt minus1 then the series in (6) is absolutely convergentfor all 119911 isin C

(ii) If Δ = minus1 then the series in (6) is absolutely convergentfor |119911| lt 120575 and for |119911| = 120575 andR(120583) gt 12

Property 1 The fractional differential operators definedin(4)-(5) satisfy the following identities

(i)

int

119887

119886

119891 (119909)119863

120572

119887minus119892 (119909) 119889119909

= int

119887

119886

119892 (119909)

119862

119863

120572

119886+119891 (119909) 119889119909 minus 119891 (119909) 119868

1minus120572

119887minus119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119887

119886

(9)

(ii)

int

119887

119886

119891 (119909)119863

120572

119887minus119892 (119909)

119862

119863

120572

119886+119896 (119909) 119889119909

= int

119887

119886

119892 (119909)

119862

119863

120572

119886+119891 (119909)

119862

119863

120572

119886+119896 (119909) 119889119909

minus119891 (119909) 119868

1minus120572

119887minus119892 (119909)

119862

119863

120572

119886+119896 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119887

119886

(10)

(iii)

int

119887

119886

119891 (119909)119863

120572

119886+119892 (119909) 119889119909

= int

119887

119886

119892 (119909)

119862

119863

120572

119887minus119891 (119909) 119889119909 + 119891 (119909) 119868

1minus120572

119886+119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119887

119886

(11)

Journal of Function Spaces and Applications 3

Property 2 (see [24]) Assume that 120572 isin (0 1) 120573 gt 120572 and119891 isin 119862[119886 119887] Then the relations

119863

120572

119886+119868

120572

119886+119891 (119909) = 119891 (119909)

119863

120572

119887minus119868

120572

119887minus119891 (119909) = 119891 (119909)

119863

120572

119886+119868

120573

119886+119891 (119909) = 119868

120573minus120572

119886+119891 (119909)

119863

120572

119887minus119868

120573

119887minus119891 (119909) = 119868

120573minus120572

119887minus119891 (119909)

119862

119863

120572

119886+119868

120572

119886+119891 (119909) = 119891 (119909)

119862

119863

120572

119887minus119868

120572

119887minus119891 (119909) = 119891 (119909)

(12)

hold for any 119909 isin [119886 119887] Furthermore the integral operatorsdefined in (3) satisfy the following semigroup properties

119868

120572

119886+119868

120573

119886+= 119868

120572+120573

119886+ 119868

120572

119887minus119868

120573

119887minus= 119868

120572+120573

119887minus

(13)

Now let us take up a singular fractional boundaryproblem for Bessel operator and give some spectral results

3 Main Results

31 A Singular Fractional Sturm-Liouville Problem for BesselOperator Fractional Sturm-Liouville problem for Besseloperator denotes the differential part containing the left- andright-sided derivatives Let us use the form of the integrationby parts formulas (10) (11) for this new approximationProperties of eigenfunctions and eigenvalues in the theoryof classical Sturm-Liouville problems are related to theintegration by parts formula for the first-order derivativesIn the corresponding fractional version we note that bothleft and right derivatives appear and the essential pairs arethe left Riemann-Liouville derivative with the right Caputoderivative and the right Riemann-Liouville derivative withthe left Caputo one Spectral properties of Sturm-Liouvilleoperators are often derived directly or indirectly as aconsequence of an established link between large distanceasymptotic behavior of solutions of the associated differentialequation and spectral properties of the corresponding Besseloperator

Definition 5 Let 120572 isin (0 1) Fractional Bessel operator iswritten as

L120572[119861]

= 119863

120572

1minus119901(119909)

119862119863

120572

0++ (119902 (119909) minus

V2 minus 14

119909

2) (14)

Considering the fractional Bessel equation

L120572[119861]

119910

120582(119909) + 120582119908

120572(119909) 119910

120582(119909) = 0 (15)

where 119901(119909) = 0 119908

120572(119909) gt 0 for all 119909 isin (0 1] 119908

120572(119909) is weight

function and 119901 119902 are real valued continuous functions ininterval (0 1]

The boundary conditions for the operator L are thefollowing

119910

120582(0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901 (1)

119862

119863

120572

0+119910 (1) = 0

(16)

where 11988921+ 119889

2

2= 0 The fractional boundary-value problem

(15)-(16) is fractional Sturm-Liouville problem for Besseloperator

Theorem 6 Fractional Bessel operator is self-adjoint on (0 1]

Proof Let us consider the following equation

⟨L120572[119861]

120593 120601⟩ = int

1

0

L120572[119861]

120593 (119909) sdot 120601 (119909) 119889119909

= int

1

0

120601 (119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909)

]

]

]

119889119909

= int

1

0

120601 (119909)119863

120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

(17)

By means of equality (10) and boundary conditions (16) weobtain the identity

⟨L120572[119861]

120593 120601⟩ = int

1

0

119901 (119909)

119862

119863

120572

0+120601 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

minus120601 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+120601 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

(18)

4 Journal of Function Spaces and Applications

On the other hand by performing similar operations we find

⟨120593L120572[119861]

120601⟩ = int

1

0

119901(119909)

119862

119863

120572

0+120593(119909)

119862

119863

120572

0+120601 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120601 (119909) 120593 (119909) 119889119909

(19)

The right-hand sides of (18) and (19) are equal hence wemaysee that the left sides are equal that is

⟨L120572[119861]

120593 120601⟩ = ⟨120593L120572[119861]

120601⟩ (20)

Theorem 7 The eigenvalues of fractional Bessel operator (15)-(16) are real

Proof Let us observe that the following relation results fromequality (10)

int

1

0

119891 (119909)L120572[119861]

119892 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+119891(119909)

119862

119863

120572

0+119892 (119909) 119889119909

minus119891 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)119892 (119909) 119891 (119909) 119889119909

(21)

Suppose that 120582 is the eigenvalue for (15)-(16) correspondingto eigenfunction 119910 the following equalities satisfy 119910 and itscomplex conjugate 119910

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0 (22)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(23)

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0

(24)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(25)

where 11988921+119889

2

2= 0 Wemultiply (22) by function 119910 and (24) by

function 119910 respectively and subtract

(120582 minus 120582)119908

120572(119909) 119910 (119909) 119910 (119909)

= 119910 (119909)L120572[119861]

119910 (119909) minus 119910 (119909)L120572[119861]

119910 (119909)

(26)

Now we integrate over interval (0 1] and applying relation(21) and we note that the right-hand side of the integratedequality contains only boundary terms

(120582 minus 120582)int

1

0

119908

120572(119909) 119910 (119909) 119910 (119909) 119889119909

= int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909 minus int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909

= int

1

0

119910 (119909)

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

minus int

1

0

119910

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+(119909) 119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

= minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

(27)

By virtue of the boundary conditions (23) (25) we find

(120582 minus 120582)int

1

0

119908

120572(119909)

1003816

1003816

1003816

1003816

119910 (119909)

1003816

1003816

1003816

1003816

2

119889119909 = 0

(28)

Because 119910 is a nontrivial solution and 119908120572(119909) gt 0 it is easily

seen that 120582 = 120582 The eigenvalues are real

Theorem 8 The eigenfunctions corresponding with distincteigenvalues of fractional Bessel operator (15)-(16) are orthog-onal weight function 119908

120572on (0 1] that is

int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0 120582

1= 120582

2

(29)

Journal of Function Spaces and Applications 5

Proof We have by assumptions fractional Sturm-Liouvilleoperator for Bessel type fulfilled by two different eigenvalues(120582

1 120582

2) and the respective eigenfunctions (119910

1205821 119910

1205822)

L120572[119861]

119910

1205821(119909) + 120582

1119908

120572(119909) 119910

1205821(119909) = 0 (30)

119910

1205821(119909) = 0

119889

1119910

1205821(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205821(1) = 0

(31)

L120572[119861]

119910

1205822(119909) + 120582

2119908

120572(119909) 119910

1205822(119909) = 0 (32)

119910

1205822(119909) = 0

119889

1119910

1205822(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205822(1) = 0

(33)

We multiply (30) by function 119910

1205822and (32) by function 119910

1205821

respectively and subtract

(120582

1minus 120582

2) 119908

120572(119909) 119910

1205821119910

1205822= 119910

1205821L120572[119861]

119910

1205822minus 119910

1205822L120572[119861]

119910

1205821

(34)

Integrating over interval (0 1] and applying relation (21)we note that the right-hand side of the integrated equalitycontains only boundary terms

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909

=int

1

0

119910

1205821(119909)L

120572[119861]119910

1205822(119909) 119889119909

minus int

1

0

119910

1205822(119909)L

120572[119861]119910

1205821(119909) 119889119909

= int

1

0

119910

1205821(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205822(119909)

]

]

]

119889119909

minus int

1

0

119910

1205822(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205821(119909)

]

]

]

119889119909

= minus119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038161

+119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038160

+119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038161

minus119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038160

(35)

Using the boundary conditions (31) (33) we obtain that

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0

(36)

where 1205821

= 120582

2 Then the eigenfunctions are orthogonal of

this operator

Remark 9 Let us now give certain auxiliary functionsBecause we use the functions the first of them is as follows

119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572)

= (1 minus 0)

120572minus1

(119909 minus 0)

120572

1Ψ

2[

(1 1)

(120572 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(37)

where1Ψ

2is the Fox-Wright function [14]

1Ψ

2[

(119886

1 120572

1)

(119887

1120573

1) (119887

2120573

2)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911]

=

infin

sum

119896=0

Γ (119886

1+ 120572

1119896)

Γ (119887

1+ 120573

1119896) Γ (119887

2+ 120573

2119896)

119911

119896

119896

(38)

The properties of the function are determined by the param-eters

Δ = 120573

1+ 120573

2minus 120572

1= minus1

120575 =

1003816

1003816

1003816

1003816

120572

1

1003816

1003816

1003816

1003816

minus1205721 10038161003816

1003816

1003816

120573

1

1003816

1003816

1003816

1003816

1205731 10038161003816

1003816

1003816

120573

2

1003816

1003816

1003816

1003816

1205732= 1

120583 = 119887

1+ 119887

2minus 120572

1+

1 minus 2

2

= 2120572 minus

1

2

(39)

Considering Theorem 4 we note that this function is con-tinuous in (0 1] when order 120572 gt 12 that is 120583 gt 12 For0 lt 120572 le 12 it is discontinuous at end 119909 = 1 The explicitlycalculated function allows to estimate the second componentof stationary function 120601

0of the differential part of Sturm-

Liouville operator

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120601

0(119909) = 0

(40)

which looks as follows

120601

0(119909) = 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(41)

The next function is the following integral

120593 (119909) = 119868

120572

0+119868

120572

1minus1 = 119868

120572

0+

(1 minus 119909)

120572

Γ (120572 + 1)

= (1 minus 0)

120572

(119909 minus 0)

120572

times

1Ψ

2[

(1 1)

(120572 + 1 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(42)

Again using Theorem 4 and calculating parameters accord-ing to (39)

Δ = minus1 120575 = 1 120583 = 2120572 +

1

2

(43)

6 Journal of Function Spaces and Applications

Finally

120572 gt 0 997904rArr 120583 gt

1

2

(44)

and the obtained Fox-Wright function (42) is continuous ininterval (0 1] for any positive order 120572

Theorem 10 Let 120572 gt 12 119909 isin (0 1] and define

119884

120582(119910) = (119902 (119909) minus

V2 minus 14

119909

2)119910

120582(119909) + 120582119908

120572119910

120582(119909)

Δ = 119889

2+ 119889

1120595 (120572 0 1)

(45)

Assume that Δ = 0Then (15)-(16) are equivalent to the integralequation

119910

120582(119909)

= minus119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) + 119860 (119909) (119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910))

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(46)

where the coefficient 119860(119909) is

119860 (119909) =

119889

1

Δ

120595 (120572 0 119909)(47)

and functions 120595 are defined in (41)

Proof By means of composition rules (15) can be rewrittenas follows

119863

120572

1minus119901 (119909)

119862

119863

120572

0+[119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)] = 0 (48)

The last equality suggests that is a stationary function of frac-tional singular Sturm-Liouville problem for Bessel operator119863

120572

1minus119901(119909)

119862

119863

120572

0+which according to (41) can be found as

120601

0= 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(49)

Equation (15) in the form of

119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) = 120585

1+ 120585

2120595 (120572 0 119909) (50)

proves we should connect coefficients 120585119895values 119889

119895 119895 = 1 2

determining the boundary conditions (16)Let us note that the following formula results from

composition rules (11) and (50)

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909) = minus119868

1

1minus119884

120582(119910) + 120585

2

(51)

For continuous function 119910120582 we obtain the following values as

the ends

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=0= minusint

120587

0

119884

120582(119910) + 120585

2

119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=1= 120585

2

(52)

respectively for 119910120582 Using (50) we find

119910

120582(0) = 120601

0(0) = 120585

1

119910

120582(1) = 120601

0(1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

= 120585

1+ 120585

2120595 (120572 0 1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(53)

The following set of linear equations for coefficients 120585119895results

from (52)ndash(54)

120585

1= 0

119889

1120585

1+ 120585

2(119889

2+ 119889

1120595 (120572 0 1)) = 119889

1119865

(54)

where 119865 = 119868

120572

0+(1119901(119909))119868

120572

1minus119884

120582(119910)|

119909=1

Since Δ = 0 the solution for coefficients 120585119895(119895 = 1 2) is

unique

120585

1= 0

120585

2=

119889

1119865

Δ

(55)

Substituting the previous solution into (50) we recover theequivalent integral equation (46)

Furthermore we give notation such as

119898

119901= min119909isin[01]

1003816

1003816

1003816

1003816

119901 (119909)

1003816

1003816

1003816

1003816

119860 = 119860 (119909) 119872

120593=

1003817

1003817

1003817

1003817

120593 (119909)

1003817

1003817

1003817

1003817

(56)

The proof is completed

4 Conclusion

In the paper we have extended the scope of some spectralproperties of singular fractional Sturm-Liouville problemWe pointed that its eigenvalues related to the Bessel operatorwith the certain boundary conditions are real and its eigen-functions corresponding to distinct eigenvalues are orthogo-nal Furthermore we showed that fractional Bessel operatoris self-adjoint Spectral properties of Sturm-Liouville theoryare applied to the fractional theory Our results are importantin point of the fractional Sturm-Liouville theory

Acknowledgments

The author sincerely thanks the editor and the reviewer fortheir valuable suggestions and comments

References

[1] R S Johnson An Introduction To Sturm-Liouville Theory Uni-versity of Newcastle 2006

[2] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

ordinary derivatives in a traditional Sturm-Liouville problemare replaced with fractional derivatives and the resultingproblems are solved using some numerical methods [18ndash23]Furthermore Klimek and Argawal [24] define a fractionalSturm-Liouville operator introduce a regular fractionalSturm-Liouville problem and investigate the properties ofthe eigenfunctions and the eigenvalues of the operator In thispaper our purpose is to introduce singular fractional Sturm-Liouville problem having Bessel type and prove spectralproperties of spectral data for the operator

Let us give the boundary value problem for Besselequation and necessary data as follows

2 Preliminaries

Now consider the following Bessel equation

119889

2119910

119889119909

2+ (120582 minus

V2 minus 14

119909

2)119910 = 0 (2)

where 120582 and V are real numbers The Bessel equation forhaving the analogous singularity is given in [5]

Definition 1 (see [10]) Let 0 lt 120572 le 1The left-sided and right-sided Riemann-Liouville integrals of order 120572 respectively aregiven by the formulas

(119868

120572

119886+119891) (119909) =

1

Γ (120572)

int

119909

119886

(119909 minus 119904)

120572minus1119891 (119904) 119889119904 119909 gt 119886

(119868

120572

119887minus119891) (119909) =

1

Γ (120572)

int

119887

119909

(119904 minus 119909)

120572minus1119891 (119904) 119889119904 119909 lt 119887

(3)

where Γ denotes the gamma function

Definition 2 (see [10]) Let 0 lt 120572 le 1 The left-sidedand right-sided Riemann-Liouville derivatives of order 120572respectively are defined as follows

(119863

120572

119886+119891) (119909) = 119863 (119868

1minus120572

119886+119891) (119909) 119909 gt 119886

(119863

120572

119887minus119891) (119909) = minus119863 (119868

1minus120572

119887minus119891) (119909) 119909 lt 119887

(4)

Analogous formulas yield the left-sided and right-sidedCaputo derivatives of order 120572

(

119862

119863

120572

119886+119891) (119909) = (119868

1minus120572

119886+119863119891) (119909) 119909 gt 119886 0 lt 120572 le 1

(

119862

119863

120572

119887minus119891) (119909)

= (119868

1minus120572

119887minus(minus119863)119891) (119909) 119909 lt 119887 0 lt 120572 le 1

(5)

Definition 3 (see [14]) The general function119901Ψ

119902(119911) is

defined for 119911 isin C 119886119897 119887

119895isin C and120572

119897 120573

119895isin R (119897 = 1 119901 119895 =

1 119902) by the series

119901Ψ

119902(119911) =

119901Ψ

119902[

(119886

1 120572

1)

1119901

(119887

1 120573

1)

1119902

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911] =

infin

sum

119896=0

prod

119901

119897=1Γ (119886

119897+ 120572

119897119896)

prod

119902

119895=1Γ (119887

119895+ 120573

119895119896)

119911

119896

119896

(6)

This general Wright function was investigated by Fox whopresented its asymptotic expansion for large values of theargument 119911 under the condition

119902

sum

119895=1

120573

119895minus

119901

sum

119897=1

120572

119897gt 1 (7)

If these conditions are satisfied the series in (6) is convergentfor any 119911 isin C

Theorem 4 (see [14]) Let 119886119897 119887

119895isin C and 120572

119897 120573

119895isin R (119897 =

1 119901 119895 = 1 119902) and let

Δ =

119902

sum

119895=1

120573

119895minus

119901

sum

119897=1

120572

119897

120575 =

119901

prod

119897=1

1003816

1003816

1003816

1003816

120572

119897

1003816

1003816

1003816

1003816

minus120572119897

119902

prod

119895=1

1003816

1003816

1003816

1003816

1003816

120573

119895

1003816

1003816

1003816

1003816

1003816

120573119895

120583 =

119902

sum

119895=1

119887

119895minus

119901

sum

119897=1

119886

119897+

119901 minus 119902

2

(8)

(i) If Δ gt minus1 then the series in (6) is absolutely convergentfor all 119911 isin C

(ii) If Δ = minus1 then the series in (6) is absolutely convergentfor |119911| lt 120575 and for |119911| = 120575 andR(120583) gt 12

Property 1 The fractional differential operators definedin(4)-(5) satisfy the following identities

(i)

int

119887

119886

119891 (119909)119863

120572

119887minus119892 (119909) 119889119909

= int

119887

119886

119892 (119909)

119862

119863

120572

119886+119891 (119909) 119889119909 minus 119891 (119909) 119868

1minus120572

119887minus119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119887

119886

(9)

(ii)

int

119887

119886

119891 (119909)119863

120572

119887minus119892 (119909)

119862

119863

120572

119886+119896 (119909) 119889119909

= int

119887

119886

119892 (119909)

119862

119863

120572

119886+119891 (119909)

119862

119863

120572

119886+119896 (119909) 119889119909

minus119891 (119909) 119868

1minus120572

119887minus119892 (119909)

119862

119863

120572

119886+119896 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119887

119886

(10)

(iii)

int

119887

119886

119891 (119909)119863

120572

119886+119892 (119909) 119889119909

= int

119887

119886

119892 (119909)

119862

119863

120572

119887minus119891 (119909) 119889119909 + 119891 (119909) 119868

1minus120572

119886+119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119887

119886

(11)

Journal of Function Spaces and Applications 3

Property 2 (see [24]) Assume that 120572 isin (0 1) 120573 gt 120572 and119891 isin 119862[119886 119887] Then the relations

119863

120572

119886+119868

120572

119886+119891 (119909) = 119891 (119909)

119863

120572

119887minus119868

120572

119887minus119891 (119909) = 119891 (119909)

119863

120572

119886+119868

120573

119886+119891 (119909) = 119868

120573minus120572

119886+119891 (119909)

119863

120572

119887minus119868

120573

119887minus119891 (119909) = 119868

120573minus120572

119887minus119891 (119909)

119862

119863

120572

119886+119868

120572

119886+119891 (119909) = 119891 (119909)

119862

119863

120572

119887minus119868

120572

119887minus119891 (119909) = 119891 (119909)

(12)

hold for any 119909 isin [119886 119887] Furthermore the integral operatorsdefined in (3) satisfy the following semigroup properties

119868

120572

119886+119868

120573

119886+= 119868

120572+120573

119886+ 119868

120572

119887minus119868

120573

119887minus= 119868

120572+120573

119887minus

(13)

Now let us take up a singular fractional boundaryproblem for Bessel operator and give some spectral results

3 Main Results

31 A Singular Fractional Sturm-Liouville Problem for BesselOperator Fractional Sturm-Liouville problem for Besseloperator denotes the differential part containing the left- andright-sided derivatives Let us use the form of the integrationby parts formulas (10) (11) for this new approximationProperties of eigenfunctions and eigenvalues in the theoryof classical Sturm-Liouville problems are related to theintegration by parts formula for the first-order derivativesIn the corresponding fractional version we note that bothleft and right derivatives appear and the essential pairs arethe left Riemann-Liouville derivative with the right Caputoderivative and the right Riemann-Liouville derivative withthe left Caputo one Spectral properties of Sturm-Liouvilleoperators are often derived directly or indirectly as aconsequence of an established link between large distanceasymptotic behavior of solutions of the associated differentialequation and spectral properties of the corresponding Besseloperator

Definition 5 Let 120572 isin (0 1) Fractional Bessel operator iswritten as

L120572[119861]

= 119863

120572

1minus119901(119909)

119862119863

120572

0++ (119902 (119909) minus

V2 minus 14

119909

2) (14)

Considering the fractional Bessel equation

L120572[119861]

119910

120582(119909) + 120582119908

120572(119909) 119910

120582(119909) = 0 (15)

where 119901(119909) = 0 119908

120572(119909) gt 0 for all 119909 isin (0 1] 119908

120572(119909) is weight

function and 119901 119902 are real valued continuous functions ininterval (0 1]

The boundary conditions for the operator L are thefollowing

119910

120582(0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901 (1)

119862

119863

120572

0+119910 (1) = 0

(16)

where 11988921+ 119889

2

2= 0 The fractional boundary-value problem

(15)-(16) is fractional Sturm-Liouville problem for Besseloperator

Theorem 6 Fractional Bessel operator is self-adjoint on (0 1]

Proof Let us consider the following equation

⟨L120572[119861]

120593 120601⟩ = int

1

0

L120572[119861]

120593 (119909) sdot 120601 (119909) 119889119909

= int

1

0

120601 (119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909)

]

]

]

119889119909

= int

1

0

120601 (119909)119863

120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

(17)

By means of equality (10) and boundary conditions (16) weobtain the identity

⟨L120572[119861]

120593 120601⟩ = int

1

0

119901 (119909)

119862

119863

120572

0+120601 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

minus120601 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+120601 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

(18)

4 Journal of Function Spaces and Applications

On the other hand by performing similar operations we find

⟨120593L120572[119861]

120601⟩ = int

1

0

119901(119909)

119862

119863

120572

0+120593(119909)

119862

119863

120572

0+120601 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120601 (119909) 120593 (119909) 119889119909

(19)

The right-hand sides of (18) and (19) are equal hence wemaysee that the left sides are equal that is

⟨L120572[119861]

120593 120601⟩ = ⟨120593L120572[119861]

120601⟩ (20)

Theorem 7 The eigenvalues of fractional Bessel operator (15)-(16) are real

Proof Let us observe that the following relation results fromequality (10)

int

1

0

119891 (119909)L120572[119861]

119892 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+119891(119909)

119862

119863

120572

0+119892 (119909) 119889119909

minus119891 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)119892 (119909) 119891 (119909) 119889119909

(21)

Suppose that 120582 is the eigenvalue for (15)-(16) correspondingto eigenfunction 119910 the following equalities satisfy 119910 and itscomplex conjugate 119910

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0 (22)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(23)

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0

(24)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(25)

where 11988921+119889

2

2= 0 Wemultiply (22) by function 119910 and (24) by

function 119910 respectively and subtract

(120582 minus 120582)119908

120572(119909) 119910 (119909) 119910 (119909)

= 119910 (119909)L120572[119861]

119910 (119909) minus 119910 (119909)L120572[119861]

119910 (119909)

(26)

Now we integrate over interval (0 1] and applying relation(21) and we note that the right-hand side of the integratedequality contains only boundary terms

(120582 minus 120582)int

1

0

119908

120572(119909) 119910 (119909) 119910 (119909) 119889119909

= int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909 minus int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909

= int

1

0

119910 (119909)

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

minus int

1

0

119910

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+(119909) 119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

= minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

(27)

By virtue of the boundary conditions (23) (25) we find

(120582 minus 120582)int

1

0

119908

120572(119909)

1003816

1003816

1003816

1003816

119910 (119909)

1003816

1003816

1003816

1003816

2

119889119909 = 0

(28)

Because 119910 is a nontrivial solution and 119908120572(119909) gt 0 it is easily

seen that 120582 = 120582 The eigenvalues are real

Theorem 8 The eigenfunctions corresponding with distincteigenvalues of fractional Bessel operator (15)-(16) are orthog-onal weight function 119908

120572on (0 1] that is

int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0 120582

1= 120582

2

(29)

Journal of Function Spaces and Applications 5

Proof We have by assumptions fractional Sturm-Liouvilleoperator for Bessel type fulfilled by two different eigenvalues(120582

1 120582

2) and the respective eigenfunctions (119910

1205821 119910

1205822)

L120572[119861]

119910

1205821(119909) + 120582

1119908

120572(119909) 119910

1205821(119909) = 0 (30)

119910

1205821(119909) = 0

119889

1119910

1205821(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205821(1) = 0

(31)

L120572[119861]

119910

1205822(119909) + 120582

2119908

120572(119909) 119910

1205822(119909) = 0 (32)

119910

1205822(119909) = 0

119889

1119910

1205822(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205822(1) = 0

(33)

We multiply (30) by function 119910

1205822and (32) by function 119910

1205821

respectively and subtract

(120582

1minus 120582

2) 119908

120572(119909) 119910

1205821119910

1205822= 119910

1205821L120572[119861]

119910

1205822minus 119910

1205822L120572[119861]

119910

1205821

(34)

Integrating over interval (0 1] and applying relation (21)we note that the right-hand side of the integrated equalitycontains only boundary terms

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909

=int

1

0

119910

1205821(119909)L

120572[119861]119910

1205822(119909) 119889119909

minus int

1

0

119910

1205822(119909)L

120572[119861]119910

1205821(119909) 119889119909

= int

1

0

119910

1205821(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205822(119909)

]

]

]

119889119909

minus int

1

0

119910

1205822(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205821(119909)

]

]

]

119889119909

= minus119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038161

+119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038160

+119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038161

minus119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038160

(35)

Using the boundary conditions (31) (33) we obtain that

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0

(36)

where 1205821

= 120582

2 Then the eigenfunctions are orthogonal of

this operator

Remark 9 Let us now give certain auxiliary functionsBecause we use the functions the first of them is as follows

119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572)

= (1 minus 0)

120572minus1

(119909 minus 0)

120572

1Ψ

2[

(1 1)

(120572 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(37)

where1Ψ

2is the Fox-Wright function [14]

1Ψ

2[

(119886

1 120572

1)

(119887

1120573

1) (119887

2120573

2)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911]

=

infin

sum

119896=0

Γ (119886

1+ 120572

1119896)

Γ (119887

1+ 120573

1119896) Γ (119887

2+ 120573

2119896)

119911

119896

119896

(38)

The properties of the function are determined by the param-eters

Δ = 120573

1+ 120573

2minus 120572

1= minus1

120575 =

1003816

1003816

1003816

1003816

120572

1

1003816

1003816

1003816

1003816

minus1205721 10038161003816

1003816

1003816

120573

1

1003816

1003816

1003816

1003816

1205731 10038161003816

1003816

1003816

120573

2

1003816

1003816

1003816

1003816

1205732= 1

120583 = 119887

1+ 119887

2minus 120572

1+

1 minus 2

2

= 2120572 minus

1

2

(39)

Considering Theorem 4 we note that this function is con-tinuous in (0 1] when order 120572 gt 12 that is 120583 gt 12 For0 lt 120572 le 12 it is discontinuous at end 119909 = 1 The explicitlycalculated function allows to estimate the second componentof stationary function 120601

0of the differential part of Sturm-

Liouville operator

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120601

0(119909) = 0

(40)

which looks as follows

120601

0(119909) = 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(41)

The next function is the following integral

120593 (119909) = 119868

120572

0+119868

120572

1minus1 = 119868

120572

0+

(1 minus 119909)

120572

Γ (120572 + 1)

= (1 minus 0)

120572

(119909 minus 0)

120572

times

1Ψ

2[

(1 1)

(120572 + 1 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(42)

Again using Theorem 4 and calculating parameters accord-ing to (39)

Δ = minus1 120575 = 1 120583 = 2120572 +

1

2

(43)

6 Journal of Function Spaces and Applications

Finally

120572 gt 0 997904rArr 120583 gt

1

2

(44)

and the obtained Fox-Wright function (42) is continuous ininterval (0 1] for any positive order 120572

Theorem 10 Let 120572 gt 12 119909 isin (0 1] and define

119884

120582(119910) = (119902 (119909) minus

V2 minus 14

119909

2)119910

120582(119909) + 120582119908

120572119910

120582(119909)

Δ = 119889

2+ 119889

1120595 (120572 0 1)

(45)

Assume that Δ = 0Then (15)-(16) are equivalent to the integralequation

119910

120582(119909)

= minus119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) + 119860 (119909) (119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910))

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(46)

where the coefficient 119860(119909) is

119860 (119909) =

119889

1

Δ

120595 (120572 0 119909)(47)

and functions 120595 are defined in (41)

Proof By means of composition rules (15) can be rewrittenas follows

119863

120572

1minus119901 (119909)

119862

119863

120572

0+[119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)] = 0 (48)

The last equality suggests that is a stationary function of frac-tional singular Sturm-Liouville problem for Bessel operator119863

120572

1minus119901(119909)

119862

119863

120572

0+which according to (41) can be found as

120601

0= 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(49)

Equation (15) in the form of

119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) = 120585

1+ 120585

2120595 (120572 0 119909) (50)

proves we should connect coefficients 120585119895values 119889

119895 119895 = 1 2

determining the boundary conditions (16)Let us note that the following formula results from

composition rules (11) and (50)

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909) = minus119868

1

1minus119884

120582(119910) + 120585

2

(51)

For continuous function 119910120582 we obtain the following values as

the ends

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=0= minusint

120587

0

119884

120582(119910) + 120585

2

119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=1= 120585

2

(52)

respectively for 119910120582 Using (50) we find

119910

120582(0) = 120601

0(0) = 120585

1

119910

120582(1) = 120601

0(1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

= 120585

1+ 120585

2120595 (120572 0 1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(53)

The following set of linear equations for coefficients 120585119895results

from (52)ndash(54)

120585

1= 0

119889

1120585

1+ 120585

2(119889

2+ 119889

1120595 (120572 0 1)) = 119889

1119865

(54)

where 119865 = 119868

120572

0+(1119901(119909))119868

120572

1minus119884

120582(119910)|

119909=1

Since Δ = 0 the solution for coefficients 120585119895(119895 = 1 2) is

unique

120585

1= 0

120585

2=

119889

1119865

Δ

(55)

Substituting the previous solution into (50) we recover theequivalent integral equation (46)

Furthermore we give notation such as

119898

119901= min119909isin[01]

1003816

1003816

1003816

1003816

119901 (119909)

1003816

1003816

1003816

1003816

119860 = 119860 (119909) 119872

120593=

1003817

1003817

1003817

1003817

120593 (119909)

1003817

1003817

1003817

1003817

(56)

The proof is completed

4 Conclusion

In the paper we have extended the scope of some spectralproperties of singular fractional Sturm-Liouville problemWe pointed that its eigenvalues related to the Bessel operatorwith the certain boundary conditions are real and its eigen-functions corresponding to distinct eigenvalues are orthogo-nal Furthermore we showed that fractional Bessel operatoris self-adjoint Spectral properties of Sturm-Liouville theoryare applied to the fractional theory Our results are importantin point of the fractional Sturm-Liouville theory

Acknowledgments

The author sincerely thanks the editor and the reviewer fortheir valuable suggestions and comments

References

[1] R S Johnson An Introduction To Sturm-Liouville Theory Uni-versity of Newcastle 2006

[2] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

Property 2 (see [24]) Assume that 120572 isin (0 1) 120573 gt 120572 and119891 isin 119862[119886 119887] Then the relations

119863

120572

119886+119868

120572

119886+119891 (119909) = 119891 (119909)

119863

120572

119887minus119868

120572

119887minus119891 (119909) = 119891 (119909)

119863

120572

119886+119868

120573

119886+119891 (119909) = 119868

120573minus120572

119886+119891 (119909)

119863

120572

119887minus119868

120573

119887minus119891 (119909) = 119868

120573minus120572

119887minus119891 (119909)

119862

119863

120572

119886+119868

120572

119886+119891 (119909) = 119891 (119909)

119862

119863

120572

119887minus119868

120572

119887minus119891 (119909) = 119891 (119909)

(12)

hold for any 119909 isin [119886 119887] Furthermore the integral operatorsdefined in (3) satisfy the following semigroup properties

119868

120572

119886+119868

120573

119886+= 119868

120572+120573

119886+ 119868

120572

119887minus119868

120573

119887minus= 119868

120572+120573

119887minus

(13)

Now let us take up a singular fractional boundaryproblem for Bessel operator and give some spectral results

3 Main Results

31 A Singular Fractional Sturm-Liouville Problem for BesselOperator Fractional Sturm-Liouville problem for Besseloperator denotes the differential part containing the left- andright-sided derivatives Let us use the form of the integrationby parts formulas (10) (11) for this new approximationProperties of eigenfunctions and eigenvalues in the theoryof classical Sturm-Liouville problems are related to theintegration by parts formula for the first-order derivativesIn the corresponding fractional version we note that bothleft and right derivatives appear and the essential pairs arethe left Riemann-Liouville derivative with the right Caputoderivative and the right Riemann-Liouville derivative withthe left Caputo one Spectral properties of Sturm-Liouvilleoperators are often derived directly or indirectly as aconsequence of an established link between large distanceasymptotic behavior of solutions of the associated differentialequation and spectral properties of the corresponding Besseloperator

Definition 5 Let 120572 isin (0 1) Fractional Bessel operator iswritten as

L120572[119861]

= 119863

120572

1minus119901(119909)

119862119863

120572

0++ (119902 (119909) minus

V2 minus 14

119909

2) (14)

Considering the fractional Bessel equation

L120572[119861]

119910

120582(119909) + 120582119908

120572(119909) 119910

120582(119909) = 0 (15)

where 119901(119909) = 0 119908

120572(119909) gt 0 for all 119909 isin (0 1] 119908

120572(119909) is weight

function and 119901 119902 are real valued continuous functions ininterval (0 1]

The boundary conditions for the operator L are thefollowing

119910

120582(0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901 (1)

119862

119863

120572

0+119910 (1) = 0

(16)

where 11988921+ 119889

2

2= 0 The fractional boundary-value problem

(15)-(16) is fractional Sturm-Liouville problem for Besseloperator

Theorem 6 Fractional Bessel operator is self-adjoint on (0 1]

Proof Let us consider the following equation

⟨L120572[119861]

120593 120601⟩ = int

1

0

L120572[119861]

120593 (119909) sdot 120601 (119909) 119889119909

= int

1

0

120601 (119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909)

]

]

]

119889119909

= int

1

0

120601 (119909)119863

120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

(17)

By means of equality (10) and boundary conditions (16) weobtain the identity

⟨L120572[119861]

120593 120601⟩ = int

1

0

119901 (119909)

119862

119863

120572

0+120601 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

minus120601 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+120593 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+120601 (119909)

119862

119863

120572

0+120593 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120593 (119909) 120601 (119909) 119889119909

(18)

4 Journal of Function Spaces and Applications

On the other hand by performing similar operations we find

⟨120593L120572[119861]

120601⟩ = int

1

0

119901(119909)

119862

119863

120572

0+120593(119909)

119862

119863

120572

0+120601 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120601 (119909) 120593 (119909) 119889119909

(19)

The right-hand sides of (18) and (19) are equal hence wemaysee that the left sides are equal that is

⟨L120572[119861]

120593 120601⟩ = ⟨120593L120572[119861]

120601⟩ (20)

Theorem 7 The eigenvalues of fractional Bessel operator (15)-(16) are real

Proof Let us observe that the following relation results fromequality (10)

int

1

0

119891 (119909)L120572[119861]

119892 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+119891(119909)

119862

119863

120572

0+119892 (119909) 119889119909

minus119891 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)119892 (119909) 119891 (119909) 119889119909

(21)

Suppose that 120582 is the eigenvalue for (15)-(16) correspondingto eigenfunction 119910 the following equalities satisfy 119910 and itscomplex conjugate 119910

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0 (22)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(23)

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0

(24)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(25)

where 11988921+119889

2

2= 0 Wemultiply (22) by function 119910 and (24) by

function 119910 respectively and subtract

(120582 minus 120582)119908

120572(119909) 119910 (119909) 119910 (119909)

= 119910 (119909)L120572[119861]

119910 (119909) minus 119910 (119909)L120572[119861]

119910 (119909)

(26)

Now we integrate over interval (0 1] and applying relation(21) and we note that the right-hand side of the integratedequality contains only boundary terms

(120582 minus 120582)int

1

0

119908

120572(119909) 119910 (119909) 119910 (119909) 119889119909

= int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909 minus int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909

= int

1

0

119910 (119909)

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

minus int

1

0

119910

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+(119909) 119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

= minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

(27)

By virtue of the boundary conditions (23) (25) we find

(120582 minus 120582)int

1

0

119908

120572(119909)

1003816

1003816

1003816

1003816

119910 (119909)

1003816

1003816

1003816

1003816

2

119889119909 = 0

(28)

Because 119910 is a nontrivial solution and 119908120572(119909) gt 0 it is easily

seen that 120582 = 120582 The eigenvalues are real

Theorem 8 The eigenfunctions corresponding with distincteigenvalues of fractional Bessel operator (15)-(16) are orthog-onal weight function 119908

120572on (0 1] that is

int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0 120582

1= 120582

2

(29)

Journal of Function Spaces and Applications 5

Proof We have by assumptions fractional Sturm-Liouvilleoperator for Bessel type fulfilled by two different eigenvalues(120582

1 120582

2) and the respective eigenfunctions (119910

1205821 119910

1205822)

L120572[119861]

119910

1205821(119909) + 120582

1119908

120572(119909) 119910

1205821(119909) = 0 (30)

119910

1205821(119909) = 0

119889

1119910

1205821(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205821(1) = 0

(31)

L120572[119861]

119910

1205822(119909) + 120582

2119908

120572(119909) 119910

1205822(119909) = 0 (32)

119910

1205822(119909) = 0

119889

1119910

1205822(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205822(1) = 0

(33)

We multiply (30) by function 119910

1205822and (32) by function 119910

1205821

respectively and subtract

(120582

1minus 120582

2) 119908

120572(119909) 119910

1205821119910

1205822= 119910

1205821L120572[119861]

119910

1205822minus 119910

1205822L120572[119861]

119910

1205821

(34)

Integrating over interval (0 1] and applying relation (21)we note that the right-hand side of the integrated equalitycontains only boundary terms

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909

=int

1

0

119910

1205821(119909)L

120572[119861]119910

1205822(119909) 119889119909

minus int

1

0

119910

1205822(119909)L

120572[119861]119910

1205821(119909) 119889119909

= int

1

0

119910

1205821(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205822(119909)

]

]

]

119889119909

minus int

1

0

119910

1205822(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205821(119909)

]

]

]

119889119909

= minus119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038161

+119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038160

+119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038161

minus119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038160

(35)

Using the boundary conditions (31) (33) we obtain that

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0

(36)

where 1205821

= 120582

2 Then the eigenfunctions are orthogonal of

this operator

Remark 9 Let us now give certain auxiliary functionsBecause we use the functions the first of them is as follows

119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572)

= (1 minus 0)

120572minus1

(119909 minus 0)

120572

1Ψ

2[

(1 1)

(120572 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(37)

where1Ψ

2is the Fox-Wright function [14]

1Ψ

2[

(119886

1 120572

1)

(119887

1120573

1) (119887

2120573

2)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911]

=

infin

sum

119896=0

Γ (119886

1+ 120572

1119896)

Γ (119887

1+ 120573

1119896) Γ (119887

2+ 120573

2119896)

119911

119896

119896

(38)

The properties of the function are determined by the param-eters

Δ = 120573

1+ 120573

2minus 120572

1= minus1

120575 =

1003816

1003816

1003816

1003816

120572

1

1003816

1003816

1003816

1003816

minus1205721 10038161003816

1003816

1003816

120573

1

1003816

1003816

1003816

1003816

1205731 10038161003816

1003816

1003816

120573

2

1003816

1003816

1003816

1003816

1205732= 1

120583 = 119887

1+ 119887

2minus 120572

1+

1 minus 2

2

= 2120572 minus

1

2

(39)

Considering Theorem 4 we note that this function is con-tinuous in (0 1] when order 120572 gt 12 that is 120583 gt 12 For0 lt 120572 le 12 it is discontinuous at end 119909 = 1 The explicitlycalculated function allows to estimate the second componentof stationary function 120601

0of the differential part of Sturm-

Liouville operator

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120601

0(119909) = 0

(40)

which looks as follows

120601

0(119909) = 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(41)

The next function is the following integral

120593 (119909) = 119868

120572

0+119868

120572

1minus1 = 119868

120572

0+

(1 minus 119909)

120572

Γ (120572 + 1)

= (1 minus 0)

120572

(119909 minus 0)

120572

times

1Ψ

2[

(1 1)

(120572 + 1 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(42)

Again using Theorem 4 and calculating parameters accord-ing to (39)

Δ = minus1 120575 = 1 120583 = 2120572 +

1

2

(43)

6 Journal of Function Spaces and Applications

Finally

120572 gt 0 997904rArr 120583 gt

1

2

(44)

and the obtained Fox-Wright function (42) is continuous ininterval (0 1] for any positive order 120572

Theorem 10 Let 120572 gt 12 119909 isin (0 1] and define

119884

120582(119910) = (119902 (119909) minus

V2 minus 14

119909

2)119910

120582(119909) + 120582119908

120572119910

120582(119909)

Δ = 119889

2+ 119889

1120595 (120572 0 1)

(45)

Assume that Δ = 0Then (15)-(16) are equivalent to the integralequation

119910

120582(119909)

= minus119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) + 119860 (119909) (119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910))

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(46)

where the coefficient 119860(119909) is

119860 (119909) =

119889

1

Δ

120595 (120572 0 119909)(47)

and functions 120595 are defined in (41)

Proof By means of composition rules (15) can be rewrittenas follows

119863

120572

1minus119901 (119909)

119862

119863

120572

0+[119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)] = 0 (48)

The last equality suggests that is a stationary function of frac-tional singular Sturm-Liouville problem for Bessel operator119863

120572

1minus119901(119909)

119862

119863

120572

0+which according to (41) can be found as

120601

0= 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(49)

Equation (15) in the form of

119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) = 120585

1+ 120585

2120595 (120572 0 119909) (50)

proves we should connect coefficients 120585119895values 119889

119895 119895 = 1 2

determining the boundary conditions (16)Let us note that the following formula results from

composition rules (11) and (50)

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909) = minus119868

1

1minus119884

120582(119910) + 120585

2

(51)

For continuous function 119910120582 we obtain the following values as

the ends

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=0= minusint

120587

0

119884

120582(119910) + 120585

2

119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=1= 120585

2

(52)

respectively for 119910120582 Using (50) we find

119910

120582(0) = 120601

0(0) = 120585

1

119910

120582(1) = 120601

0(1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

= 120585

1+ 120585

2120595 (120572 0 1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(53)

The following set of linear equations for coefficients 120585119895results

from (52)ndash(54)

120585

1= 0

119889

1120585

1+ 120585

2(119889

2+ 119889

1120595 (120572 0 1)) = 119889

1119865

(54)

where 119865 = 119868

120572

0+(1119901(119909))119868

120572

1minus119884

120582(119910)|

119909=1

Since Δ = 0 the solution for coefficients 120585119895(119895 = 1 2) is

unique

120585

1= 0

120585

2=

119889

1119865

Δ

(55)

Substituting the previous solution into (50) we recover theequivalent integral equation (46)

Furthermore we give notation such as

119898

119901= min119909isin[01]

1003816

1003816

1003816

1003816

119901 (119909)

1003816

1003816

1003816

1003816

119860 = 119860 (119909) 119872

120593=

1003817

1003817

1003817

1003817

120593 (119909)

1003817

1003817

1003817

1003817

(56)

The proof is completed

4 Conclusion

In the paper we have extended the scope of some spectralproperties of singular fractional Sturm-Liouville problemWe pointed that its eigenvalues related to the Bessel operatorwith the certain boundary conditions are real and its eigen-functions corresponding to distinct eigenvalues are orthogo-nal Furthermore we showed that fractional Bessel operatoris self-adjoint Spectral properties of Sturm-Liouville theoryare applied to the fractional theory Our results are importantin point of the fractional Sturm-Liouville theory

Acknowledgments

The author sincerely thanks the editor and the reviewer fortheir valuable suggestions and comments

References

[1] R S Johnson An Introduction To Sturm-Liouville Theory Uni-versity of Newcastle 2006

[2] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

On the other hand by performing similar operations we find

⟨120593L120572[119861]

120601⟩ = int

1

0

119901(119909)

119862

119863

120572

0+120593(119909)

119862

119863

120572

0+120601 (119909) 119889119909

+

119889

1

119889

2

120593 (1) 120601 (1)

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)120601 (119909) 120593 (119909) 119889119909

(19)

The right-hand sides of (18) and (19) are equal hence wemaysee that the left sides are equal that is

⟨L120572[119861]

120593 120601⟩ = ⟨120593L120572[119861]

120601⟩ (20)

Theorem 7 The eigenvalues of fractional Bessel operator (15)-(16) are real

Proof Let us observe that the following relation results fromequality (10)

int

1

0

119891 (119909)L120572[119861]

119892 (119909) 119889119909

= int

1

0

119901 (119909)

119862

119863

120572

0+119891(119909)

119862

119863

120572

0+119892 (119909) 119889119909

minus119891 (119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119892 (119909)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1

0

+ int

1

0

(119902 (119909) minus

V2 minus 14

119909

2)119892 (119909) 119891 (119909) 119889119909

(21)

Suppose that 120582 is the eigenvalue for (15)-(16) correspondingto eigenfunction 119910 the following equalities satisfy 119910 and itscomplex conjugate 119910

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0 (22)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(23)

L120572[119861]

119910 (119909) + 120582119908

120572(119909) 119910 (119909) = 0

(24)

119910 (0) = 0

119889

1119910 (1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910 (1) = 0

(25)

where 11988921+119889

2

2= 0 Wemultiply (22) by function 119910 and (24) by

function 119910 respectively and subtract

(120582 minus 120582)119908

120572(119909) 119910 (119909) 119910 (119909)

= 119910 (119909)L120572[119861]

119910 (119909) minus 119910 (119909)L120572[119861]

119910 (119909)

(26)

Now we integrate over interval (0 1] and applying relation(21) and we note that the right-hand side of the integratedequality contains only boundary terms

(120582 minus 120582)int

1

0

119908

120572(119909) 119910 (119909) 119910 (119909) 119889119909

= int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909 minus int

1

0

119910 (119909)L120572[119861]

119910 (119909) 119889119909

= int

1

0

119910 (119909)

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

minus int

1

0

119910

[

[

[

119863

120572

1minus119901(119909)

119862

119863

120572

0+(119909) 119910 (119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910 (119909)

]

]

]

119889119909

= minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

+119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038161

minus119910 (119909) 119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910 (119909)

1003816

1003816

1003816

1003816

10038160

(27)

By virtue of the boundary conditions (23) (25) we find

(120582 minus 120582)int

1

0

119908

120572(119909)

1003816

1003816

1003816

1003816

119910 (119909)

1003816

1003816

1003816

1003816

2

119889119909 = 0

(28)

Because 119910 is a nontrivial solution and 119908120572(119909) gt 0 it is easily

seen that 120582 = 120582 The eigenvalues are real

Theorem 8 The eigenfunctions corresponding with distincteigenvalues of fractional Bessel operator (15)-(16) are orthog-onal weight function 119908

120572on (0 1] that is

int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0 120582

1= 120582

2

(29)

Journal of Function Spaces and Applications 5

Proof We have by assumptions fractional Sturm-Liouvilleoperator for Bessel type fulfilled by two different eigenvalues(120582

1 120582

2) and the respective eigenfunctions (119910

1205821 119910

1205822)

L120572[119861]

119910

1205821(119909) + 120582

1119908

120572(119909) 119910

1205821(119909) = 0 (30)

119910

1205821(119909) = 0

119889

1119910

1205821(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205821(1) = 0

(31)

L120572[119861]

119910

1205822(119909) + 120582

2119908

120572(119909) 119910

1205822(119909) = 0 (32)

119910

1205822(119909) = 0

119889

1119910

1205822(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205822(1) = 0

(33)

We multiply (30) by function 119910

1205822and (32) by function 119910

1205821

respectively and subtract

(120582

1minus 120582

2) 119908

120572(119909) 119910

1205821119910

1205822= 119910

1205821L120572[119861]

119910

1205822minus 119910

1205822L120572[119861]

119910

1205821

(34)

Integrating over interval (0 1] and applying relation (21)we note that the right-hand side of the integrated equalitycontains only boundary terms

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909

=int

1

0

119910

1205821(119909)L

120572[119861]119910

1205822(119909) 119889119909

minus int

1

0

119910

1205822(119909)L

120572[119861]119910

1205821(119909) 119889119909

= int

1

0

119910

1205821(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205822(119909)

]

]

]

119889119909

minus int

1

0

119910

1205822(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205821(119909)

]

]

]

119889119909

= minus119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038161

+119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038160

+119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038161

minus119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038160

(35)

Using the boundary conditions (31) (33) we obtain that

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0

(36)

where 1205821

= 120582

2 Then the eigenfunctions are orthogonal of

this operator

Remark 9 Let us now give certain auxiliary functionsBecause we use the functions the first of them is as follows

119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572)

= (1 minus 0)

120572minus1

(119909 minus 0)

120572

1Ψ

2[

(1 1)

(120572 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(37)

where1Ψ

2is the Fox-Wright function [14]

1Ψ

2[

(119886

1 120572

1)

(119887

1120573

1) (119887

2120573

2)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911]

=

infin

sum

119896=0

Γ (119886

1+ 120572

1119896)

Γ (119887

1+ 120573

1119896) Γ (119887

2+ 120573

2119896)

119911

119896

119896

(38)

The properties of the function are determined by the param-eters

Δ = 120573

1+ 120573

2minus 120572

1= minus1

120575 =

1003816

1003816

1003816

1003816

120572

1

1003816

1003816

1003816

1003816

minus1205721 10038161003816

1003816

1003816

120573

1

1003816

1003816

1003816

1003816

1205731 10038161003816

1003816

1003816

120573

2

1003816

1003816

1003816

1003816

1205732= 1

120583 = 119887

1+ 119887

2minus 120572

1+

1 minus 2

2

= 2120572 minus

1

2

(39)

Considering Theorem 4 we note that this function is con-tinuous in (0 1] when order 120572 gt 12 that is 120583 gt 12 For0 lt 120572 le 12 it is discontinuous at end 119909 = 1 The explicitlycalculated function allows to estimate the second componentof stationary function 120601

0of the differential part of Sturm-

Liouville operator

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120601

0(119909) = 0

(40)

which looks as follows

120601

0(119909) = 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(41)

The next function is the following integral

120593 (119909) = 119868

120572

0+119868

120572

1minus1 = 119868

120572

0+

(1 minus 119909)

120572

Γ (120572 + 1)

= (1 minus 0)

120572

(119909 minus 0)

120572

times

1Ψ

2[

(1 1)

(120572 + 1 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(42)

Again using Theorem 4 and calculating parameters accord-ing to (39)

Δ = minus1 120575 = 1 120583 = 2120572 +

1

2

(43)

6 Journal of Function Spaces and Applications

Finally

120572 gt 0 997904rArr 120583 gt

1

2

(44)

and the obtained Fox-Wright function (42) is continuous ininterval (0 1] for any positive order 120572

Theorem 10 Let 120572 gt 12 119909 isin (0 1] and define

119884

120582(119910) = (119902 (119909) minus

V2 minus 14

119909

2)119910

120582(119909) + 120582119908

120572119910

120582(119909)

Δ = 119889

2+ 119889

1120595 (120572 0 1)

(45)

Assume that Δ = 0Then (15)-(16) are equivalent to the integralequation

119910

120582(119909)

= minus119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) + 119860 (119909) (119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910))

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(46)

where the coefficient 119860(119909) is

119860 (119909) =

119889

1

Δ

120595 (120572 0 119909)(47)

and functions 120595 are defined in (41)

Proof By means of composition rules (15) can be rewrittenas follows

119863

120572

1minus119901 (119909)

119862

119863

120572

0+[119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)] = 0 (48)

The last equality suggests that is a stationary function of frac-tional singular Sturm-Liouville problem for Bessel operator119863

120572

1minus119901(119909)

119862

119863

120572

0+which according to (41) can be found as

120601

0= 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(49)

Equation (15) in the form of

119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) = 120585

1+ 120585

2120595 (120572 0 119909) (50)

proves we should connect coefficients 120585119895values 119889

119895 119895 = 1 2

determining the boundary conditions (16)Let us note that the following formula results from

composition rules (11) and (50)

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909) = minus119868

1

1minus119884

120582(119910) + 120585

2

(51)

For continuous function 119910120582 we obtain the following values as

the ends

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=0= minusint

120587

0

119884

120582(119910) + 120585

2

119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=1= 120585

2

(52)

respectively for 119910120582 Using (50) we find

119910

120582(0) = 120601

0(0) = 120585

1

119910

120582(1) = 120601

0(1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

= 120585

1+ 120585

2120595 (120572 0 1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(53)

The following set of linear equations for coefficients 120585119895results

from (52)ndash(54)

120585

1= 0

119889

1120585

1+ 120585

2(119889

2+ 119889

1120595 (120572 0 1)) = 119889

1119865

(54)

where 119865 = 119868

120572

0+(1119901(119909))119868

120572

1minus119884

120582(119910)|

119909=1

Since Δ = 0 the solution for coefficients 120585119895(119895 = 1 2) is

unique

120585

1= 0

120585

2=

119889

1119865

Δ

(55)

Substituting the previous solution into (50) we recover theequivalent integral equation (46)

Furthermore we give notation such as

119898

119901= min119909isin[01]

1003816

1003816

1003816

1003816

119901 (119909)

1003816

1003816

1003816

1003816

119860 = 119860 (119909) 119872

120593=

1003817

1003817

1003817

1003817

120593 (119909)

1003817

1003817

1003817

1003817

(56)

The proof is completed

4 Conclusion

In the paper we have extended the scope of some spectralproperties of singular fractional Sturm-Liouville problemWe pointed that its eigenvalues related to the Bessel operatorwith the certain boundary conditions are real and its eigen-functions corresponding to distinct eigenvalues are orthogo-nal Furthermore we showed that fractional Bessel operatoris self-adjoint Spectral properties of Sturm-Liouville theoryare applied to the fractional theory Our results are importantin point of the fractional Sturm-Liouville theory

Acknowledgments

The author sincerely thanks the editor and the reviewer fortheir valuable suggestions and comments

References

[1] R S Johnson An Introduction To Sturm-Liouville Theory Uni-versity of Newcastle 2006

[2] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

Proof We have by assumptions fractional Sturm-Liouvilleoperator for Bessel type fulfilled by two different eigenvalues(120582

1 120582

2) and the respective eigenfunctions (119910

1205821 119910

1205822)

L120572[119861]

119910

1205821(119909) + 120582

1119908

120572(119909) 119910

1205821(119909) = 0 (30)

119910

1205821(119909) = 0

119889

1119910

1205821(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205821(1) = 0

(31)

L120572[119861]

119910

1205822(119909) + 120582

2119908

120572(119909) 119910

1205822(119909) = 0 (32)

119910

1205822(119909) = 0

119889

1119910

1205822(1) + 119889

2119868

1minus120572

1minus119901(1)

119862

119863

120572

0+119910

1205822(1) = 0

(33)

We multiply (30) by function 119910

1205822and (32) by function 119910

1205821

respectively and subtract

(120582

1minus 120582

2) 119908

120572(119909) 119910

1205821119910

1205822= 119910

1205821L120572[119861]

119910

1205822minus 119910

1205822L120572[119861]

119910

1205821

(34)

Integrating over interval (0 1] and applying relation (21)we note that the right-hand side of the integrated equalitycontains only boundary terms

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909

=int

1

0

119910

1205821(119909)L

120572[119861]119910

1205822(119909) 119889119909

minus int

1

0

119910

1205822(119909)L

120572[119861]119910

1205821(119909) 119889119909

= int

1

0

119910

1205821(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205822(119909)

]

]

]

119889119909

minus int

1

0

119910

1205822(119909)

[

[

[

119863

120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

+(119902 (119909) minus

V2 minus 14

119909

2)119910

1205821(119909)

]

]

]

119889119909

= minus119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038161

+119910

1205821(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205822(119909)

1003816

1003816

1003816

1003816

10038160

+119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038161

minus119910

1205822(119909) 119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

1205821(119909)

1003816

1003816

1003816

1003816

10038160

(35)

Using the boundary conditions (31) (33) we obtain that

(120582

1minus 120582

2) int

1

0

119908

120572(119909) 119910

1205821(119909) 119910

1205822(119909) 119889119909 = 0

(36)

where 1205821

= 120582

2 Then the eigenfunctions are orthogonal of

this operator

Remark 9 Let us now give certain auxiliary functionsBecause we use the functions the first of them is as follows

119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572)

= (1 minus 0)

120572minus1

(119909 minus 0)

120572

1Ψ

2[

(1 1)

(120572 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(37)

where1Ψ

2is the Fox-Wright function [14]

1Ψ

2[

(119886

1 120572

1)

(119887

1120573

1) (119887

2120573

2)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911]

=

infin

sum

119896=0

Γ (119886

1+ 120572

1119896)

Γ (119887

1+ 120573

1119896) Γ (119887

2+ 120573

2119896)

119911

119896

119896

(38)

The properties of the function are determined by the param-eters

Δ = 120573

1+ 120573

2minus 120572

1= minus1

120575 =

1003816

1003816

1003816

1003816

120572

1

1003816

1003816

1003816

1003816

minus1205721 10038161003816

1003816

1003816

120573

1

1003816

1003816

1003816

1003816

1205731 10038161003816

1003816

1003816

120573

2

1003816

1003816

1003816

1003816

1205732= 1

120583 = 119887

1+ 119887

2minus 120572

1+

1 minus 2

2

= 2120572 minus

1

2

(39)

Considering Theorem 4 we note that this function is con-tinuous in (0 1] when order 120572 gt 12 that is 120583 gt 12 For0 lt 120572 le 12 it is discontinuous at end 119909 = 1 The explicitlycalculated function allows to estimate the second componentof stationary function 120601

0of the differential part of Sturm-

Liouville operator

119863

120572

1minus119901 (119909)

119862

119863

120572

0+120601

0(119909) = 0

(40)

which looks as follows

120601

0(119909) = 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(41)

The next function is the following integral

120593 (119909) = 119868

120572

0+119868

120572

1minus1 = 119868

120572

0+

(1 minus 119909)

120572

Γ (120572 + 1)

= (1 minus 0)

120572

(119909 minus 0)

120572

times

1Ψ

2[

(1 1)

(120572 + 1 minus1) (120572 + 1 1)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus

119909 minus 0

1 minus 0

]

(42)

Again using Theorem 4 and calculating parameters accord-ing to (39)

Δ = minus1 120575 = 1 120583 = 2120572 +

1

2

(43)

6 Journal of Function Spaces and Applications

Finally

120572 gt 0 997904rArr 120583 gt

1

2

(44)

and the obtained Fox-Wright function (42) is continuous ininterval (0 1] for any positive order 120572

Theorem 10 Let 120572 gt 12 119909 isin (0 1] and define

119884

120582(119910) = (119902 (119909) minus

V2 minus 14

119909

2)119910

120582(119909) + 120582119908

120572119910

120582(119909)

Δ = 119889

2+ 119889

1120595 (120572 0 1)

(45)

Assume that Δ = 0Then (15)-(16) are equivalent to the integralequation

119910

120582(119909)

= minus119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) + 119860 (119909) (119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910))

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(46)

where the coefficient 119860(119909) is

119860 (119909) =

119889

1

Δ

120595 (120572 0 119909)(47)

and functions 120595 are defined in (41)

Proof By means of composition rules (15) can be rewrittenas follows

119863

120572

1minus119901 (119909)

119862

119863

120572

0+[119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)] = 0 (48)

The last equality suggests that is a stationary function of frac-tional singular Sturm-Liouville problem for Bessel operator119863

120572

1minus119901(119909)

119862

119863

120572

0+which according to (41) can be found as

120601

0= 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(49)

Equation (15) in the form of

119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) = 120585

1+ 120585

2120595 (120572 0 119909) (50)

proves we should connect coefficients 120585119895values 119889

119895 119895 = 1 2

determining the boundary conditions (16)Let us note that the following formula results from

composition rules (11) and (50)

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909) = minus119868

1

1minus119884

120582(119910) + 120585

2

(51)

For continuous function 119910120582 we obtain the following values as

the ends

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=0= minusint

120587

0

119884

120582(119910) + 120585

2

119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=1= 120585

2

(52)

respectively for 119910120582 Using (50) we find

119910

120582(0) = 120601

0(0) = 120585

1

119910

120582(1) = 120601

0(1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

= 120585

1+ 120585

2120595 (120572 0 1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(53)

The following set of linear equations for coefficients 120585119895results

from (52)ndash(54)

120585

1= 0

119889

1120585

1+ 120585

2(119889

2+ 119889

1120595 (120572 0 1)) = 119889

1119865

(54)

where 119865 = 119868

120572

0+(1119901(119909))119868

120572

1minus119884

120582(119910)|

119909=1

Since Δ = 0 the solution for coefficients 120585119895(119895 = 1 2) is

unique

120585

1= 0

120585

2=

119889

1119865

Δ

(55)

Substituting the previous solution into (50) we recover theequivalent integral equation (46)

Furthermore we give notation such as

119898

119901= min119909isin[01]

1003816

1003816

1003816

1003816

119901 (119909)

1003816

1003816

1003816

1003816

119860 = 119860 (119909) 119872

120593=

1003817

1003817

1003817

1003817

120593 (119909)

1003817

1003817

1003817

1003817

(56)

The proof is completed

4 Conclusion

In the paper we have extended the scope of some spectralproperties of singular fractional Sturm-Liouville problemWe pointed that its eigenvalues related to the Bessel operatorwith the certain boundary conditions are real and its eigen-functions corresponding to distinct eigenvalues are orthogo-nal Furthermore we showed that fractional Bessel operatoris self-adjoint Spectral properties of Sturm-Liouville theoryare applied to the fractional theory Our results are importantin point of the fractional Sturm-Liouville theory

Acknowledgments

The author sincerely thanks the editor and the reviewer fortheir valuable suggestions and comments

References

[1] R S Johnson An Introduction To Sturm-Liouville Theory Uni-versity of Newcastle 2006

[2] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces and Applications

Finally

120572 gt 0 997904rArr 120583 gt

1

2

(44)

and the obtained Fox-Wright function (42) is continuous ininterval (0 1] for any positive order 120572

Theorem 10 Let 120572 gt 12 119909 isin (0 1] and define

119884

120582(119910) = (119902 (119909) minus

V2 minus 14

119909

2)119910

120582(119909) + 120582119908

120572119910

120582(119909)

Δ = 119889

2+ 119889

1120595 (120572 0 1)

(45)

Assume that Δ = 0Then (15)-(16) are equivalent to the integralequation

119910

120582(119909)

= minus119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) + 119860 (119909) (119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910))

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(46)

where the coefficient 119860(119909) is

119860 (119909) =

119889

1

Δ

120595 (120572 0 119909)(47)

and functions 120595 are defined in (41)

Proof By means of composition rules (15) can be rewrittenas follows

119863

120572

1minus119901 (119909)

119862

119863

120572

0+[119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)] = 0 (48)

The last equality suggests that is a stationary function of frac-tional singular Sturm-Liouville problem for Bessel operator119863

120572

1minus119901(119909)

119862

119863

120572

0+which according to (41) can be found as

120601

0= 120585

1+ 120585

2119868

120572

0+

(1 minus 119909)

120572minus1

Γ (120572) 119901 (119909)

= 120585

1+ 120585

2120595 (120572 0 119909)

(49)

Equation (15) in the form of

119910

120582(119909) + 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910) = 120585

1+ 120585

2120595 (120572 0 119909) (50)

proves we should connect coefficients 120585119895values 119889

119895 119895 = 1 2

determining the boundary conditions (16)Let us note that the following formula results from

composition rules (11) and (50)

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909) = minus119868

1

1minus119884

120582(119910) + 120585

2

(51)

For continuous function 119910120582 we obtain the following values as

the ends

119868

1minus120572

1minus119901 (119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=0= minusint

120587

0

119884

120582(119910) + 120585

2

119868

1minus120572

1minus119901(119909)

119862

119863

120572

0+119910

120582(119909)

1003816

1003816

1003816

1003816

1003816119909=1= 120585

2

(52)

respectively for 119910120582 Using (50) we find

119910

120582(0) = 120601

0(0) = 120585

1

119910

120582(1) = 120601

0(1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

= 120585

1+ 120585

2120595 (120572 0 1) minus 119868

120572

0+

1

119901 (119909)

119868

120572

1minus119884

120582(119910)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=1

(53)

The following set of linear equations for coefficients 120585119895results

from (52)ndash(54)

120585

1= 0

119889

1120585

1+ 120585

2(119889

2+ 119889

1120595 (120572 0 1)) = 119889

1119865

(54)

where 119865 = 119868

120572

0+(1119901(119909))119868

120572

1minus119884

120582(119910)|

119909=1

Since Δ = 0 the solution for coefficients 120585119895(119895 = 1 2) is

unique

120585

1= 0

120585

2=

119889

1119865

Δ

(55)

Substituting the previous solution into (50) we recover theequivalent integral equation (46)

Furthermore we give notation such as

119898

119901= min119909isin[01]

1003816

1003816

1003816

1003816

119901 (119909)

1003816

1003816

1003816

1003816

119860 = 119860 (119909) 119872

120593=

1003817

1003817

1003817

1003817

120593 (119909)

1003817

1003817

1003817

1003817

(56)

The proof is completed

4 Conclusion

In the paper we have extended the scope of some spectralproperties of singular fractional Sturm-Liouville problemWe pointed that its eigenvalues related to the Bessel operatorwith the certain boundary conditions are real and its eigen-functions corresponding to distinct eigenvalues are orthogo-nal Furthermore we showed that fractional Bessel operatoris self-adjoint Spectral properties of Sturm-Liouville theoryare applied to the fractional theory Our results are importantin point of the fractional Sturm-Liouville theory

Acknowledgments

The author sincerely thanks the editor and the reviewer fortheir valuable suggestions and comments

References

[1] R S Johnson An Introduction To Sturm-Liouville Theory Uni-versity of Newcastle 2006

[2] A Zettl Sturm-Liouville Theory vol 121 of MathematicalSurveys and Monographs American Mathematical SocietyProvidence RI USA 2005

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 7

[3] W O Amrein A M Hinz and D B Pearson Eds Sturm-Liouville Theory Past and Present Birkhauser Basel Switzer-land 2005

[4] E S Panakhov and R Yilmazer ldquoA Hochstadt-Liebermantheorem for the hydrogen atom equationrdquo Applied and Com-putational Mathematics vol 11 no 1 pp 74ndash80 2012

[5] B M Levitan and I S Sargsjan Introduction to SpectralTheory Self adjoint Ordinary Differential Operators AmericanMathematical Society Providence RI USA 1975

[6] J Qi and S Chen ldquoEigenvalue problems of themodel fromnon-local continuum mechanicsrdquo Journal of Mathematical Physicsvol 52 no 7 Article ID 073516 2011

[7] E S Panakhov and M Sat ldquoReconstruction of potentialfunction for Sturm-Liouville operator with Coulomb potentialrdquoBoundary Value Problems vol 2013 article 49 2013

[8] A Carpinteri and F Mainardi Eds Fractals and FractionalCalculus in Continum Mechanics Telos Springer 1998

[9] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[11] R Hilfer Ed Applications of Fractional Calculus in PhysicsWorld Scientific Singapore 2000

[12] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Philadelphia Pa USA 1993

[13] K S Miller and B Ross An Introduction to the FractionalCalculus and Fractional Differential Equations John Wiley ampSons New York NY USA 1993

[14] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Amsterdam The Netherlands 2006

[15] R Yilmazer and E Bas ldquoFractional solutions of confluenthypergeometric equationrdquo Journal of the Chungcheong Mathe-matical Society vol 25 no 2 pp 149ndash157 2012

[16] X Jiang and H Qi ldquoThermal wave model of bioheat transferwithmodified Riemann-Liouville fractional derivativerdquo Journalof Physics A vol 45 no 48 Article ID 485101 2012

[17] X Jiang and M Xu ldquoThe time fractional heat conductionequation in the general orthogonal curvilinear coordinate andthe cylindrical coordinate systemsrdquo Physica A vol 389 no 17pp 3368ndash3374 2010

[18] E Nakai and G Sadasue ldquoMartingale Morrey-Campanatospaces and fractional integralsrdquo Journal of Function Spaces andApplications vol 2012 Article ID 673929 29 pages 2012

[19] Y Wang L Liu and Y Wu ldquoExistence and uniqueness of apositive solution to singular fractional differential equationsrdquoBoundary Value Problems vol 2012 article 81 2012

[20] D Baleanu and O G Mustafa ldquoOn the existence interval forthe initial value problem of a fractional differential equationrdquoHacettepe Journal of Mathematics and Statistics vol 40 no 4pp 581ndash587 2011

[21] M Klimek On Solutions of Linear Fractional DifferentialEquations of a Variational Type The Publishing Office ofCzestochowa University of Technology Czestochowa Poland2009

[22] Q M Al-Mdallal ldquoAn efficient method for solving fractionalSturm-Liouville problemsrdquoChaos Solitons and Fractals vol 40no 1 pp 183ndash189 2009

[23] V S Erturk ldquoComputing eigenelements of Sturm-Liouvilleproblems of fractional order via fractional differential trans-form methodrdquo Mathematical amp Computational Applicationsvol 16 no 3 pp 712ndash720 2011

[24] M Klimek and O P Argawal ldquoOn a regular fractional Sturm-Liouville problem with derivatives of order in (0 1)rdquo in Pro-ceedings of the 13th International CarpathianControl ConferenceMay 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of