Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem

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1 Applied nd Computtionl Mthemtics 5; 4(5): Pulished online Septemer, 5 ( doi:.648/.cm ISSN: (Print); ISSN: (Online) Appliction Cheyshev Polynomils for Determining the Eigenvlues of Sturm-Liouville Prolem DongYun Shen, Yong Hung * Deprtment of Mthemtics, Foshn University, Foshn, Gungdong, Chin Emil ddress: hyhy3@6.com (Yong Hung) To cite this rticle: DongYun Shen, Yong Hung. Appliction Cheyshev Polynomils for Determining the Eigenvlues of Sturm-Liouville Prolem. Applied nd Computtionl Mthemtics. Vol. 4, No. 5, 5, pp doi:.648/.cm Astrct: This pper discusses the eigenvlue prolem of second-order Sturm-Liouville eqution. We trnsform the governing differentil eqution to the Fredholm-Volterr integrl eqution pproprite end supports. By epnding the unknown function into the shifted Cheyshev polynomils, we directly get the corresponding polynomil chrcteristic equtions, where the lower nd higher-order eigenvlues cn e determined simultneously from the multi-roots. Severl emples of estimting eigenvlues re given. By comprison the ect results in open litertures, the correctness nd effectiveness of the present pproch re verified. Keywords: Sturm-Liouville Prolem, Eigenvlues, Fredholm-Volterr Integrl Eqution, Cheyshev Polynomils. Introduction Sturm-Liouville prolems ply n importnt role in severl res, such s physicl, engineering nd other scientific fields []-[]. However, it is difficult to otin ect epression of eigenvlues for such prolem different potentil functions. In the pst decdes, vrious spects of the numericl theory s well s pproimte methods for this prolem were presented to cquire the numericl results of eigenvlues. A simple symptotic correction technique ws employing y Andrew [3] to compute the eigenvlues of regulr Sturm Liouville prolems periodic or semiperiodic oundry conditions. Bsed on the oundry vlue methods, Ghelrdnoi [4] used some liner multistep methods to discretize the Sturm-Liouville prolem nd investigte the pproimtions of eigenvlues. The eigenvlues of regulr Sturm Liouville prolems periodic oundry conditions ws clculted in [5] y the finite difference scheme. In [6] nd [7], Çelik nd co-workers used the Cheyshev colloction method to investigted for the pproimte computtion of Sturm Liouville eigenvlues y trnsforming the prolems nd given oundry conditions to mtri eqution. Yun et l. [8] proposed the Cheyshev colloction method to compute the pproimte eigenvlues of regulr Sturm Liouville prolems two points nd (semi-)periodic oundry conditions. Chen nd M [9] used the Legendre Glerkin Cheyshev colloction method, which preserves the symmetry of the prolem, to compute the pproimte eigenvlues of the Sturm Liouville prolem kinds of different oundry conditions. Zhng [] cretized the Sturm Liouville prolems (SLPs) into stndrd mtri eigenvlue prolems in order to chieve high ccurcy nd high efficiency y using the mpped rycentric Cheyshev differentition mtri method. Bsed on the homotopy nlysis method, Asndy nd Shirzdi [] clculted the pproimte eigenvlues of the second nd fourth-order Sturm Liouville prolems. El-gmel nd El-hdy used the differentil qudrture method nd colloction method sinc functions for computing eigenvlues of Sturm Liouville prolems []. In this pper, we will introduce n efficient pproch to investigte the eigenvlues of Sturm-Liouville prolem. For vrious oundry conditions, we trnsform the governing eqution to the Fredholm Volterr integrl equtions. Then system of lgeric equtions will e derived sed on the Cheyshev polynomils epnsion of the unknown function. The chrcteristic vlues cn e esily determined from the eistence condition of nontrivil solution in the resulting system. Severl emples used frequently in Sturm-Liouville prolem will e used to demonstrte the ccurcy of pproimtion.. Integrl Eqution Method In the following, we consider the Liouville norml form of

2 37 DongYun Shen nd Yong Hung: Appliction Cheyshev Polynomils for Determining the Eigenvlues of Sturm-Liouville Prolem the generl Sturm-Liouville eqution: y() + λ q () y =,, () where the q is given function, y () is n unknown function stisfying certin oundry conditions, nd the prmeter λ is the eigenvlue needed to e determined. During this pper, four fmilir end conditions will e discussed: CseA : y () = y () =, () ( t) (), () ( t) () Sustituting them ck into (7), nd fter collection we get Fredholm Volterr integrl eqution s follows: y () + K( tytdt,)() + K( tytdt,)() =, (3) CseB : y () = y() =, (3) CseC : y() = y () =, (4) CseD : y () = y (), y () = y (5) K ( t, ) = t λ qt (), (4) K t, = ( t) λ qt () (5) The prolem is ctully of solving set of second-order differentil governing eqution () nd the corresponding end supports. Avoiding solving tht differentil eqution directly, we introduce n integrl eqution method to convert the prolem to Fredholm Volterr integrl equtions vrious oundry conditions. For this purpose, integrting oth sides of Eq. () respect to from to, one gets λ qt ytdt A (6) y() + () () =. Then we repet to integrte oth sides of Eq. (6) respect to from to, yielding λ y () + t qt () ytdt () = A + A, (7) where A nd A re unknown constnts tht cn e determined from the given oundry conditions. Once these two constnts A( =, ) cn e uniquely otined, we then sustitute these vlues A into Eq. (7) nd immeditely derive n integrl eqution in y () of the following form: y () + K( tytdt,)() + K( tytdt,)() =. (8) () Cse A:. y () = y () = By setting =, in (7), respectively, we cn get two liner equtions out A nd A s: A + A=, (9) A+ A= t () Solving the ove lgeric equtions, one cn otin () Cse B:. y () = y() = Applying the condition y () = in (7) leds to A+, (6) On the other hnd, we sustitute the condition y() = to (6), yielding (7) Putting the ove into Eq. (6), one gets (8) Finlly, sustituting A into Eq. (7), we otin Fredholm Volterr integrl eqution for cse B s follows: y () + K( tytdt,)() + K( tytdt,)() =, (9) K ( t, ) = t λ qt (), () K ( t, ) = λ qt (). () (3) Cse C:. y() = y () = Bering y() = in mind, setting = in (6) leds to. () Applying the condition y () = to (7), we get

3 Applied nd Computtionl Mthemtics 5; 4(5): t (3) Plugged A into Eq. (7), the finl Fredholm Volterr integrl eqution is derived s follows: y () + K( tytdt,)() + K( tytdt,)() =, (4) K ( t, ) = t λ qt (), (5) K ( t, ) = ( t ) λ qt (). (6) (4) Cse D:. y () = y (), y() = y() Setting =, in (6) nd (7), respectively, fter using the condition y () = y () nd y () = y (), we cn otin (), (7) A+ A ( + ) = t (8) Therefore, A nd A cn e otined y solving the ove lgeric equtions λ qt () ytdt (), (9) t λ qt () ytdt (). 4 (3) With these otined A, fter some simplifiction we finlly derive the Fredholm Volterr integrl eqution s follows: y () + K( tytdt,)() + K( tytdt,)() =, (3) K ( t, ) = t λ qt (), (3) t K( t,) = + λ qt () Chrcteristic Eqution of the Prolem (33) In the preceding section, for some typicl oundry conditions, we hve converted the governing differentil eqution () to the corresponding Fredholm Volterr integrl eqution (8). In the following, simple pproch will e introduced to solve the integrl equtions. It is well-known tht the first kind of Cheyshev polynomils cn e derived y the following recurrence reltions: T() ξ =, T() ξ = ξ, (34) T () ξ = ξt () ξ T () ξ, (35) i+ i i where ξ is over the intervl [, ]. After introducing + vrile sustitution = ξ +, we cn esily derive the Cheyshev polynomils over the intervl [, ] T( ) =, T( ) =, 4 T ( ) = T( ) T ( ). i+ i i (36) (37) Firstly, we epnd the unknown y () into the shifted Cheyshev polynomils in generl over the intervl [, ] s y N = ct( ),, (38) i i where c re unknown coefficients, nd i N is certin positive integer. Putting the epnsion (38) into the resulting Fredholm Volterr integrl Eq. (8) for ech cse leds to N N ct() + c K( tt,) () tdt i i i i N + c () i K tt, tdt=. i (39) Multiplying oth sides of (39) y T( ) nd then integrting respect to etween nd, one cn get system of liner lgeric equtions in c : i N ( T + K + K ) c =, =,,,.. N, (4) i i i i T = T( T ) ( d ), K i i i = K tt, () tt( dtd ), K i i = K tt, () tt( dtd ). i (4) In order to cquire nontrivil solution of the liner lgeric equtions (4), the determinnt of the coefficient mtri of the system hs to vnish, nmely: det( T + K + K ) =. (4) i i i

4 37 DongYun Shen nd Yong Hung: Appliction Cheyshev Polynomils for Determining the Eigenvlues of Sturm-Liouville Prolem Therefore, we hve derived the chrcteristic eqution in eigenvlue λ. By inspecting this otined eqution, we cn esily find tht it is ctully polynomil in eigenvlue λ tht hs multi-roots of positive solutions, which re corresponding to the lower nd higher eigenvlues. 4. Illustrtive Emples In this section, severl illustrtive emples used frequently in Sturm-Liouville prolem re presented to show the efficiency of the proposed method. We first consider the following emple y( ) = λy (). (43) where q =. This is second-order ordinry differentil eqution. A generl solution to Eq. (43) cn e immeditely otined sed on stndrd pproch for solving the ove eqution y= C cos( λ) + C sin( λ), (44) where C nd C re unknown constnts tht cn e determined from the oundry conditions. Here three oundry conditions re discussed: () y() = y() = ; () y() y = () = ; (3) y() = y() =. Sustituting the Eq. (44) to the corresponding supported ends, fter collection we cn get the ect chrcteristic vlues of (43): () () λ= ( kπ π) ; (3) λ ( kπ π) λ= kπ ; =, k Z +. In order to check the correctness nd convergence of the introduced pproch, we hve clculted the first four chrcteristic vlues of Sturm-Liouville eqution (43) y tking different N vlues in (38). The evluted results nd the ect chrcteristic vlues λ re listed in Tle for the condition y() = y() =. With N incresing from 6 to, the errors etween the numericl nd ect results drmticlly decrese. This indictes tht the numericl results hve rpid convergence N incresing. From Tle we cn find tht the otined results of N= re in ecellent greement ect ones, which re identicl to ech other up to deciml digits for the first two eigenvlues. Bsed on the present pproch, we hve clculted first four chrcteristic vlues λ for Cse nd Cse 3 N= where the results nd the reltive errors etween those re tulted in Tle. A good greement etween the present computed results nd ect results cn e oserved for the two cses from Tle. Sturm-Liouville vlue prolem rises in mny physicl, engineering nd other scientific fields. The most importnt chrcteristic vlue for such prolem is the first-order vlue, which is corresponding to the fundmentl nturl frequency or criticl uckling lods or other mteril properties in those fields. We cn find tht our results of the first-order only letting N= 8 re identicl to the ect results. Tle. Numericl nd ect results for Emple y() = y() =. k N= 6 N= 8 N= N= Ect solution Tle. Numericl nd ect chrcteristic vlues for Emple N=. y() y = () = y() = y() = k Ect solution Present Reltive errors Ect solution Present Reltive errors Net we consider the following Sturm Liouville prolem y() + cos()( y) = λy (), (45) the oundry condtion π π π π y( ) = y(), y( ) = y. By using the mpped rycentric Cheyshev differentition mtri method, Zhng [] hs clculted the chrcteristic vlues λ of sucn Sturm Liouville prolem. For N=, we employ the introduced method to determine the first four eigenvlues pproimtions. The reltive errors etween the ect solution nd our numericl results re tulted in Tle 3. Tle 3 demonstrtes tht the present results re in very good greement the ect results []. Tle 3. Numericl results of eigenvlues nd the reltive errors N=. k Ect results[] Present results ( N= ) Reltive errors As the lst emple, we discuss the following Sturm-Liouville prolem y() + ey () = λy (), (46)

5 Applied nd Computtionl Mthemtics 5; 4(5): conditions () y() = yπ = nd () y() = y( π) =. The uniderivtive Simpson method (USM)[8] nd nd Numerov s method (NM) [3] hve een used to solve this Sturm-Liouville prolem. The numericl Tle 4. Numericl results of eigenvlues nd the reltive errors N=. k y() = yπ = y() = y ( π) = results clculted y the Eq. (4), nd the reltive errors etween the present results N= nd the ect results re listed in the Tle 4 for two cses. Our results of N= coincide well the ect solutions. Ect [8] Present Reltive errors Ect [3] Present Reltive errors Conclusions This pper presented simple nd efficient method to determine the eigenvlues of the second-order Sturm-Liouville prolem. Insted of directly solving the differentil eqution, we trnsform the governing eqution to the corresponding Fredholm Volterr integrl equtions kinds of oundry conditions. By epnding the unknown functions into the shifted Cheyshev polynomils, system of liner lgeric equtions will e otined, where the lower nd higher eigenvlues cn e effectively computed from the chrcteristic polynomil equtions. Compred our results the ect solutions, the ccurcy nd effectiveness of the introduced method hve een confirmed. Acknowledgements DongYun Shen pprecites the support from the specil funds on science nd technology innovtion for Gungdong college students. References [] MA Al-Gwiz, Sturm Liouville Theory nd its Applictions, Springer-Verlg, London, 8. [] Veerle Ledou, Study of specil lgorithms for solving Sturm Liouville nd Schrödinger equtions. Ph.D Thesis, Deprtment of Applied Mthemtics nd Computer Science, Ghent University, 7. [3] Aln L Andrew, Correction of finite difference eigenvlues of periodic Sturm Liouville prolems, J. Austrl. Mth. Soc. Ser. B 3 (989) [4] Polo Ghelrdoni, Approimtions of Sturm--Liouville eigenvlues using oundry vlue methods, Appl. Numer. Mth. 3 (997) [5] DJ Condon, Corrected finite difference eigenvlues of periodic Sturm Liouville prolems, Appl. Numer. Mth. 3 (999) [6] İrhim Çelik, Approimte computtion of eigenvlues Cheyshev colloction method, Appl. Mth. Comput. 68 (5) [7] İrhim Çelik, Guzin Gokmen, Approimte solution of periodic Sturm Liouville prolems Cheyshev colloction method, Appl. Mth. Comput. 7 (5) [8] Qun Yun, Zhiqing He, Huinn Leng, An improvement for Cheyshev colloction method in solving certin Sturm Liouville prolems, Appl. Mth. Comput. 95 (8) [9] Ln Chen, He-Ping M, Approimte solution of the Sturm Liouville prolems Legendre Glerkin Cheyshev colloction method, Appl. Mth. Comput. 6 (8) [] Xuecng Zhng, Mpped rycentric Cheyshev differentition mtri method for the solution of regulr Sturm Liouville prolems, Appl. Mth. Comput. 7 () [] Seid Asndy, A new ppliction of the homotopy nlysis method: Solving the Sturm Liouville prolems, Commun. Nonliner Sci. Numer. Simult. 6 () 6. [] Mohmed El-gmel, Mhmoud Ad El-hdy, Two very ccurte nd efficient methods for computing eigenvlues of Sturm Liouville prolems, Appl. Mth. Modell. 37 (3) [3] Aln L Andrew, Twenty yers of symptotic correction for eigenvlue computtion, ANZIAM J. 4 () C96 C6.