( ) + ( ) = ( ), [, ]

Transcription

1 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 S79 AN APPLICATION OF COMPARISON CRITERIA TO FRACTIONAL SPECTRAL PROBLEM HAVING COULOMB POTENTIAL y Erdl BAS * nd Fund METIN TURK Deprtment of Mthemtics, Fculty of Science, Firt University, Elzig, Turkey Deprtment of Mthemtics, Fculty of Science, Brtın University, Brtın, Turkey Originl scientific pper Introduction In this study, the zeros of eigenfunctions of spectrl theory re considered in frctionl Sturm-Liouville prolem. The st nd nd comprison theorems for frctionl Sturm-Liouville eqution with oundry condition nd their proofs re given. In this wy, our new pproximtion will contriute to construct frctionl Sturm-Liouville theory. Also, its n ppliction is given in cse of Coulom potentil nd the results re presented y symolic grph. Key words: therml conduction, ngulr momentum, Coulom potentil, energy eqution, Sturm-Liouville, Schroedinger eqution Chrles Frncois Sturm ws counted s the founder of Sturm-Liouville theory in the 830s. Doutless Sturm s the most importnt contriution is to introduce nd center on Sturm-Liouville opertors. Only, his most remrkle results re the oscilltion nd comprison theorems. The fundmentl theory of Sturm-Liouville differentil equtions, is deeply influenced y the development of quntum mechnics. Sturm tkes into consider the PDE for the diffusion of het. The improvement of such diffusion process in time derives system of trnsformtions whose effect on the zeros of function hs een much studied. Sturm-Liouville eqution is in fct -D Schroedinger eqution, which is PDE ut in the event of sphericlly symmetric potentils such s the Coulom potentil, it is reduced to the ODE, one for ech pir of ngulr momentum quntum numers. The importnce of mthemtics rises from the study of prolems in the rel world nd lot of physicl pplictions like therml conduction, virting of strings, intercting of tomic prticles, quntum mechnics, mss-spring system, etc. []. The concept of Sturm-Liouville prolems plys n importnt role in mthemtics nd physics. Sturm-Liouville oundry vlue prolem is given: d dy ( ) + ( ) = ( ), [, ] d p r d q r y λ w r y r () r r Ay( ) + Ay ( ) = 0, A + A > 0 () By ( ) + By ( ) = 0, B + B > 0 (3) * Corresponding uthor, e-mil: erdlmt@yhoo.com

2 S80 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 where pr (), wr () > 0nd p (), r wr, () nd qr () re continuous functions over the finite intervl [,]. Lter on, mny studies hve een mde on this suject. Regulr-singulr Sturm-Liouville prolems re defined. Inverse prolems hve een proved y using vrious spectrl dts. There re numerous studies out Sturm-Liouville prolem, [-6]. Together with coming up L Hopitl s frctionl clculus ide in 695, mny dvnced pplictions out this suject in vrious res hve een given for yers. It hs een demonstrted tht mny systems in different fields of physics, chemistry, iology, economics, control theory, opticl systems, engineering cn e modelled more ccurtely y mens of frctionl derivtives. Definitions nd theory of frctionl derivtives nd integrl, frctionl differentil equtions, frctionl PDE cn e found in [7-]. Mjority of the mthemticl theory to the study of frctionl clculus ws improved in the 0 th century [-6]. For lst centuries the theory of frctionl derivtives hs een improved minly s pure theoreticl fields of mthemtics useful only for mthemticins. And especilly in the recent yers, importnt studies hve een imed t comining frctionl derivtive nd integrl suject with Sturm-Liouville suject hve een otined, [7, 8]. Fundmentl spectrl theory for regulr-singulr Sturm-Liouville prolems is given. Furthermore the spectrl theory for frctionl Sturm-Liouville prolems hving Bessel nd Coulom type singulrity is investigted, in 03 [7, 9-4]. Now, let s mention out Coulom potentil; motion of electrons moving under the Coulom potentil is of significnce in quntum theory. We use to find energy levels for hydrogen tom nd single vlence electron toms. For hydrogen tom, the Coulom potentil is given y U = e / s, s is the rdius of the nucleus nd e is electronic chrge. As regrds, using the time-dependent Schroedinger eqution: ω h = ω ih + U ( r, y, z) ω, d d d = ω r y z t m r where ω is the wve function, h the Plnck constnt, nd m the mss of electron. In this eqution, pplying Fourier trnsform: = iλt ω e ω dt π It shll trnsform energy eqution dependent on the cse: h + U = E m ω ω ω Whence energy eqution in the field with the Coulom potentil trnsforms: h e ω + E + ω = 0 m s Replcing hydrogen tom to other potentil re, the energy eqution is: h e ω + E+ + qryz (,, ) ω = 0 m s In consequence of some trnsformtions, we get Sturm-Liouville eqution with Coulom potentil: 3 R

3 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 S8 A y + + q( r) y = λ y (4) r where λ is prmeter corresponding to the energy [5]. In this study, the zeros of eigenfunctions of spectrl theory re investigted, first nd second comprison theorems for frctionl Sturm-Liouville prolems with their proofs re given. This note will provide n importnt contriution to develop frctionl Sturm-Liouville theory. In the following section, we get definitions nd properties of frctionl clculus [9-,, 6]. Preliminries Definition. Let 0 < <. The left-sided nd right-sided Riemnn-Liouville integrls of order re given y the formuls, respectively: r I + f r = ( r s) f ( s) d, s r > Γ (5) ( )( ) ( )( ) ( ) ( ) I f r = ( s r) f ( s) d, s r < Γ (6) r where Γ denotes the gmm function. Definition. Let 0 < <. The left-sided nd right-sided Riemnn-Liouville derivtives of order re defined s, respectively: ( D+ )( ) D ( + )( ) ( D )( ) D( )( ) f r = I f r r > f r = I f r r < Similr formuls re given for the left- nd right-sided Cputo derivtives of order : ( D + )( ) ( + D )( ) ( D )( ) ( D) ( ) f r = I f r r > f r = I f r r < Property 3. Let 0 < <, the frctionl differentil opertors defined in eqs. (5)-(0) stisfy the following identities: ( ) D ( ) d = ()D ( ) d ( ) ( ) f r g r r g r f r r f r I g r + ( ) D ( ) d = ()D ( ) d + ( ) ( ) f r g r r g r f r r f r I g r + + ( ) D ()D ( ) d = ()D ()D ( ) d ( ) ()D ( ) f r gr k r r gr f r k r r f r I gr k r Now, we prove fundmentl theorems of Sturm ccording to the frctionl Sturm-Liouville prolem, lso giving some pplictions. Min results Let us consider the following two frctionl Sturm-Liouville oundry vlue prolems: (7) (8) (9) (0) () () (3)

4 S8 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 D pr ()D + ur () + grur ()() = 0 D pr ()D + vr () + hrvr ()() = 0 where pr ( ) > 0, r [, ] nd pr (), gr, () nd hr () re rel vlued continuous functions in intervl [, ] nd (0,). st comprison theorem. Suppose tht we re given two frctionl Sturm-Liouville eqs. (4) nd (5). If gr () < hr () over the entire intervl [,, ] then etween every two zeros of ny non-trivil solution of the first eqution there is t lest one zero of every solution of the second eqution. Proof. Let us multiply eq. (4) y function vr () nd eq. (5) y function ur () nd sutrcting, we find: eq. (6): vr ()D pr ()D ur () + grurvr ()()() ur ()D pr ()D vr () hrvrur ()()() = [ ] vr ()D pr ()D ur () ur ()D pr ()D vr () = hr () gr () vrur ()() + + Let r nd r re two consecutive zeros of u. Integrting from r to r, we cn write the r r r + + = [ ] vr ()D pr ()D ur ()d r ur ()D pr ()D vr ()d r hr () gr () vrur ()()dr Applying property eq. (3) in lst eqution, we otin: r r pr vr ur r vri pr ur () D + ()D + ()d () ()D + () pr ur vr r+ uri pr vr = () D + ()D + ()d () ()D + () = vri () pr ()D ur () + vri () pr ()D ur () + + r + + ri ) ()D () r pr + vr = r + uri () pr ()D vr () u( = [() hr gr ()]()()d vrur r Let us ssume tht v does not equl to zero nywhere in the intervl (, r ). Without loss of generlity, we cn ssume tht ur () > 0nd vr () > 0in the intervl (, r ) nd ur (), vr () re incresing functions. Therefore, the right hnd side of the equlity is positive. But left hnd side of the equlity is negtive for sufficiently lrge vlue of vr () t r point. Then we rrive t contrdiction nd this completes the proof. nd comprison theorem. Let ur () e the solution of eq. (4) stisfying the initil conditions: u ( ) =, D + ur ( ) = 0 r= nd vr () is the solution of eq. (5) stisfying the sme initil conditions. Assume vr () > 0, ur () > 0nd ur (), vr () re incresing functions. Moreover, suppose tht gr () < hr () over the entire intervl [,]. r r (4) (5) (6)

5 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 S83 If ur () hs m zeros in the intervl < r, then vr () hs not less thn m zeros in the sme intervl nd the k th zero of vr () is less thn the k th zero of ur. () Proof. Let r denote the zero of ur () closest to the point. Now let us prove tht vr () hs t lest one zero in the intervl ( r, ). Assume the contrry, without loss of generlity, we my suppose tht vr () > 0nd ur () > 0in the intervl ( r, ), since ur ( ) = 0. Integrting the identity of eq. (6) from to r nd pply reltion of eq. (3) we otin: vri () pr ()D ur () + vri () pr ()D ur () + r + () ()D + () [ () ()] ()()d uri pr vr = hr gr vrur r Since ssumption vr () > 0, ur () > 0nd ur (), vr () re incresing functions in the intervl ( r, ), the left side in the previous equlity is negtive. But the right side is positive. This cse is contrdiction. Hence, the theorem proves. To mke these comprison theorems clerer, we give the following exmple which includes only Riemnn-Liouville frctionl derivtive. Appliction. Let us consider two frctionl Sturm-Liouville equtions hving Coulom potentil function: nd D y ( r) + λ + yr ( ) = 0, r (0,) (7) r / y (0) = 0, y () = 0 D y ( r) + λ + yr ( ) = 0, r (0,) r / y (0) = 0, y () = 0 We estimte some zeros of eigenfunctions of frctionl Sturm-Liouville prolems of eqs. (7) nd (8), y using Adomins decomposition method [7]. λ.6609, λ 3.550, λ , nd µ.3705, µ.85, µ , respectively. We cn see lternting of the zeros in fig. s symolic. Appliction. Similry, the following two prolems hold: r + = / D y( r) λ yr ( ) 0, r (0,) y (0) = 0, y () = 0 r + = 4 / D y( r) λ yr ( ) 0, r (0,) y (0) = 0, y () = 0 By performing necessry opertions nd computtions, some zeros of eigenfunctions of frctionl Sturm-Liouville prolems of eqs. (9) nd (0) re τ , τ , τ , τ4 43.5, τ nd η , η , η3 4.60, η , η (8) (9) (0)

6 S84 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 0 λ λ λ 3 λ 4 λ 5 μ μ μ 3 μ 4 μ 5 Conclusion As known Sturm comprison theorems re crucil in clssicl spectrl theory. Becuse, these theorems ply n importnt role in the proof of oscilltion theorem which is one of the most importnt theorems in spectrl theory. In this regrd, to move these importnt results to frctionl clculus, we give the proof of clssicl Sturm comprison theorems in frctionl cse under certin conditions. This study will e n importnt step for the proof of the oscilltion theorem in frctionl cse. Acknowledgment References Figure. The lternting zeros of eigenfunctions (symolic grph) This pper includes prt of Ph.D thesis dt of Fund Metin Turk. [] Akno, T. T., Fkinlede, O. A., Numericl Computtion of Sturm-Liouville Prolem with Roin Boundry Condition, Interntionl Journl of Mthemticl nd Computtionl Sciences, 9 (05),, pp [] Johnson, R. S., An introduction to Sturm-Liouville theory, University of Newcstle, Newcstle, UK, 006 [3] Zettl, A., Sturm-Liouville Theory, Mthemticl Surveys nd Monogrphs, Americn Mthemticl Society, Providence, R. I., USA, 005 [4] Amrein, W. O., et l., Sturm-Liouville Theory: Pst nd Present, Birkhuser, Bsel, Switzerlnd, 005 [5] Levitn, B. M., Srgsjn, I. S., Introduction to Spectrl Theory: Self djoint Ordinry Differentil Opertors, Americn Mth. Soc. Providence, R. I., USA, 975 [6] Pnkhov, E. S., On the Determintion the Differentil Opertor Peculrity in Zero, VINITI, (980), pp. -6 [7] Hilfer, R., Applictions of Frctionl Clculus in Physics, World Scientific, Singpore, 000 [8] Crpinteri, A., Minrdi, F., Frctls nd Frctionl Clculus in Continum Mechnics (Ed. A., Crpinteri, F. Minrdi), Telos: Springer-Verlg, Berlin, 998 [9] Podluny, I., Frctionl Differentil Equtions, Acdemic Press, Sn Diego, Cl., USA, 999 [0] Smko, S. G., et l., Frctionl Integrls nd Derivtives: Theory nd Applictions, Gordon nd Brech, Phildelphi, Penn., USA, 993 [] Miller, K. S., Ross, B., An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions, John Wiley nd Sons, New York, USA, 993 [] Blenu, D., et l., Asymptotic Integrtion of (+Alph)-Order Frctionl Differentil Equtions, Computers & Mthemtics with Applictions, 6 (0), 3, pp [3] Aolhssn, R., et l., Conditionl Optimiztion Prolems: Frctionl Order Cse, Journl of Optimiztion Theory nd Appl., 56 (03),, pp [4] Blenu, D., Guo-Cheng, W., New Applictions of the Vritionl Itertion Method-From Differentil Equtions to Q-Frctionl Difference Equtions, Advnces in Difference Eq., (03), Dec., pp. -6 [5] Sid Grce, R., et l., On the Oscilltion of Frctionl Differentil Equtions, Frc. Clc. App. Anl., 5 (0),, pp. -3

7 Bs, E., et l.: An Appliction of Comprison Criteri to Frctionl Spectrl Prolem... THERMAL SCIENCE: Yer 08, Vol., Suppl., pp. S79-S85 S85 [6] Klimek, M., On Solutions of Liner Frctionl Differentil Equtions of Vritionl Type, The Pulishing Office of Czestochow University of Technology, Czestochow, Polnd, 009 [7] Al-Mdlll, Q. M., An Efficient Method for Solving Frctionl Sturm-Liouville Prolems, Chos, Solitons & Frctls, 40 (009),, pp [8] Klimek, M., Argwl, O. P., Frctionl Sturm-Liouville Prolem, Computers nd Mthemtics with Applictions, 66 (03), 5, pp [9] Bs, E., Fundmentl Spectrl Theory of Frctionl Singulr Sturm-Liouville Opertor, Journl of Function Spces nd Applictions, 03 (03), ID [0] Erturk, V. S., Computing Eigenelements of Sturm-Liouville Prolems of Frctionl Order vi Frctionl Differentil Trnsform Method, Mthemticl nd Computtionl Applictions, 6 (0), 3, pp [] Klimek, M., Argwl, O. P., On Regulr Frctionl Sturm-Liouville Prolem with Derivtives of Order in (0,), Proceeding, 3 th Int. Cont. Conf., Ttry, Slovki, 0 [] Bs, E., Metin, F., Frctionl Singulr Sturm-Liouville Opertor for Coulom Potentil, Advnces in Differences Equtions, (03), On-line first, [3] Bs, E., Metin, F., Spectrl Anlysis for Frctionl Hydrogen Atom Eqution, Advnces in Pure Mthemtics, 5 (05), Nov., pp [4] Bs, E., Ozrsln, R., Sturm-Liouville Prolem vi Coulom Type in Difference Equtions, Filomt 3 (07), 4, pp [5] Blohincev, D. I., Foundtions of Quntum Mechnics, GITTL, Moscow, 949 [6] Kils, A. A., et l., Theory nd Applictions of Frctionl Differentil Equtions, Elsevier, Amsterdm, The Netherlnds, 006 Pper sumitted: June, 07 Pper revised: Novemer 0, 07 Pper ccepted: Novemer 0, Society of Therml Engineers of Seri Pulished y the Vinč Institute of Nucler Sciences, Belgrde, Seri. This is n open ccess rticle distriuted under the CC BY-NC-ND 4.0 terms nd conditions