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
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
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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
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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
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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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
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