Journal of Appled Mathematcs and Computatonal Mechancs 06, 5(), 7-36 www.amcm.pcz.pl p-iss 99-9965 DOI: 0.75/jamcm.06..4 e-iss 353-0588 THE STURM-LIOUVILLE EIGEVALUE PROBLEM - A UMERICAL SOLUTIO USIG THE COTROL VOLUME METHOD Jarosław Sedleck, Marusz Ceselsk, Tomasz Błaszczyk Insttute of Mathematcs, Czestochowa Unversty of Technology Częstochowa, Poland Insttute of Computer and Informaton Scences Czestochowa Unversty of Technology, Częstochowa, Poland jaroslaw.sedleck@m.pcz.pl, marusz.ceselsk@cs.pcz.pl, tomasz.blaszczyk@m.pcz.pl Abstract. The soluton of the D Sturm-Louvlle problem usng the Control Volume Method s dscussed. The second order lnear dfferental equaton wth homogeneous boundary condtons s dscretzed and converted to the system of lnear algebrac equatons. The matrx assocated wth ths system s trdagonal and egenvalues of ths system are an approxmaton of the real egenvalues of the boundary value problem. The numercal results of the egenvalues for varous cases and the expermental rate of convergence are presented. Keywords: Sturm-Louvlle problem, egenvalues, numercal methods, Control Volume Method. Introducton Ths paper s concerned wth the computaton of egenvalues of regular egenvalue problems occurrng n ordnary dfferental equatons. The Sturm-Louvlle problem arses wthn n many areas of scence, engneerng and appled mathematcs. It has been studed for more than two decades. Many physcal, bologcal and chemcal processes are descrbed usng models based on the Sturm-Louvlle equatons. The Sturm-Louvlle problem appears drectly as the egenvalue problem n a one-dmensonal space. It also arses when lnear partal dfferental equatons are separable n a certan coordnate system. For a more detaled study of the nteger order Sturm-Louvlle theory, we refer the reader to [-5]. The Sturm-Louvlle problem can be solved by usng ether analytcal or numercal methods. One of the most common approaches to a numercal soluton of the consdered problem s the fnte dfference method [3, 6-8] where each dervatve s dscretzed at each grd pont wth an adequate dfference scheme. Apart from the
8 J. Sedleck, M. Ceselsk, T. Błaszczyk fnte dfference method, several analytcal ones, such as the varatonal or decomposton methods, are proposed to fnd an approxmate soluton.. Statement of the problem We consder the followng problem defned on the bounded nterval x [a, b] d dy x p( x) + q x y x = w x y x dx dx α y a + α y ' a = 0 β y b + β y ' b = 0 () () where p(x) > 0, dp/dx and q(x) are contnuous, w(x) > 0 on [a, b], α + α 0 and β + β 0. The above problem s called the regular Sturm-Louvlle Problem (SLP). The soluton to SLP conssts of a par and y, where s a constant - called an egenvalue, whle y s a nontrval (nonzero) functon - called an egenfuncton, and together they satsfy the gven SLP. For each SLP, the egenvalues form an nfnte ncreasng sequence: < < 3 < and lm k k =. For arbtrary choces of the functons p(x), q(x) and w(x) n Eq. (), the computaton of the exact values of the egenvalues for whch SLP ()-() has a nontrval egensoluton y(x) whch s very complcated or t s practcally mpossble to determne. umercal methods should, therefore, be used for computng the approxmate values of. In many cases, the mportance of a numercal approxmaton to the SLP descrbed by a dfferental egenvalue problem s to reduce the problem to that of solvng the egenvalue problem of a matrx equaton (an algebrac problem). In ths paper, we apply the Control Volume Method (also known as the Fnte Volume Method) to compute the egenvalues of the Sturm-Louvlle problem numercally. 3. Control Volume Method In ths numercal method [9] the consdered doman of SLP: x [a, b] s dvded nto control volumes Ω for =,, wth the central nodes ξ. The mesh s presented n Fgure. The auxlary nodes x = a + x, for = 0,, and x = = (b a) / have also been ntroduced. Then, the postons of central nodes are the followng: ξ = a + ( 0.5) x, =,,.
The Sturm-Louvlle egenvalue problem - a numercal soluton usng the Control Volume Method 9 Fg.. Mesh of control volumes The ntegraton of Eq. () wth respect to volume Ω leads to d dy x p( x) dv + q( x) y( x) dv = w( x) y( x) dv dx dx (3) Ω Ω Ω or wrtten n the form (assumng that Ω : [x -, x ]) x x x d dy x p( x) dx + q( x) y( x) dx = w( x) y( x) dx dx dx (4) x x x All the components n Eq. (4) can be approxmated as follows: x x q( x) y( x) dx q( ξ) y( ξ) x (5) x x w( x) y( x) dx w( ξ) y( ξ) x (6) x d dy x dy x dy x p( x) dx = p( x) + p( x) dx dx dx dx x x= x x= x y ξ y ξ y ξ+ y ξ for > for < (7) x x p( x ) + p( x) y( ξ) ya yb y( ξ ) for = for = 0.5 x 0.5 x The values of y a = y(a) and y b = y(b) are determned on the bass of approxmatons of the boundary condtons () ( ξ ) y y( ξ ) y ya b α ya + α = 0, β yb + β = 0 0.5 x 0.5 x and hence, these values are equal to (8) α β y = y ξ y = y ξ, a α α b x β + β x (9)
30 J. Sedleck, M. Ceselsk, T. Błaszczyk Substtutng (9) nto (8), the followng approxmaton of (7) has been obtaned where x x p( x ) ( ) ( ) dx y d dy x y ξ y ξ, for > p( x) dx dx x γ a ξ, for = 4α ( ) ( + ) ( ) x p x y ξ y ξ, for < + γ b y ξ, for = γ a =, b α α x (0) 4β γ = () β + β x For example, n the case of the Drchlet boundary condtons at boundares x = a (.e. α =, α = 0) and/or x = b (.e. β =, β = 0), the coeffcents γ a and γ b are equal to, and n the case of the eumann boundary condtons at x = a (α = 0, α = ) and/or x = b (β = 0, β = ), these coeffcents take the values of 0, respectvely. After substtuton of (5), (6) and (0) nto (4), the followng system of the dscrete equatons for every control volume: for Ω : ( x) ( x) γ p x a + p x + q( ξ ) y( ξ ) p x y ξ = w ξ y ξ 0 () for Ω, for =,, : ( ) = w( ξ ) y( ξ ) ( ) + p x p x p x p x y ξ + + q( ξ ) y( ξ ) y ξ x x x + (3) for Ω : ( ) ( x) ( ) ( x) p x γ p x + p x y( ξ ) + + q ξ y ξ = w ξ y ξ b (4) s obtaned. Ths system can be also wrtten n the matrx form as ( P+ Q) y=wy (5)
where P The Sturm-Louvlle egenvalue problem - a numercal soluton usng the Control Volume Method 3 ( x) ( 0) γa p x p( x) 0 0 0 + p x p( x) p( x) p( x) 0 0 + p( x) p( x ) 0 p( x ) p( x) 0 p( x) (6) p( x ) 0 0 p( x ) p( x ) + p( x ) p( x ) 0 0 0 p( x ) +γb p( x) = + and y y, y,..., y ( q( ) q q) Q = dag ξ, ξ,..., ξ (7) ( w( ) w w) W = dag ξ, ξ,..., ξ (8) T = ξ ξ ξ. Then, SLP ()-() s equvalent to the matrx egenvalue problem (5). The ordered egenvalues of (5) are denoted by, =,,. In order to evaluate egenvalues of large matrx egenvalue problem (5), the numercal methods mplemented n mathematcal software can be used. 3.. Partcular case Assumng the functons p(x) =, q(x) = 0 and w(x) = n the SLP problem, then matrces P, Q and W are reduced to the followng forms: Q = O (Zero matrx), W = I (Identty matrx) and γ a + 0 0 0 0 0 0 0 P = ( x) 0 0 0 0 0 γ b + (9)
3 J. Sedleck, M. Ceselsk, T. Błaszczyk In the case of the mxed Drchlet and/or eumann boundary condtons at both boundares, one can fnd n lterature (e.g. [4]) the explct formulas for the -th egenvalue of the SLP ()-() and they are presented n Table. For the dscrete case, the egenvalues of the matrx P can be determned n an analytcal way. The egenvalue problems of trdagonal matrces (n a smlar form as the matrx P) are consdered n [0, ]. On the bass of these results, the egenvalues of the matrx P:, =,, are adopted to the analysed problem for four mxed boundary condtons and they are also gven n Table. The Drchlet B.C. at x = a γ a = (for α =, α = 0) The eumann B.C. at x = a γ a = 0 (for α = 0, α = ) Egenvalues of Eq. () for functons p(x) =, q(x) = 0 and w(x) = and four mxed cases of boundary condtons (explct formulas for, =,, and dscrete one for, =,, ) The Drchlet B.C. at x = b γ b = (for β =, β = 0) π = b a ( ) 4 π = sn b a = ( b a) ( 0.5) π b a ( 0.5) 4 π ( ) = sn The eumann B.C. at x = b γ b = 0 (for β = 0, β = ) = ( b a) ( 0.5) π b a ( 0.5) 4 π ( ) = sn ( ) π = b a ( ) 4 ( ) π = sn ( b a) Table For more complcated cases such as non-constant functons p(x), q(x), w(x) occurrng n Eq. (), the egenvalues of system (5) should be determned n a numercal way (.e. usng mathematcal software - here the Maple s used). 3.. Error of numercal approxmaton of the egenvalues Theorem. For the Sturm-Louvlle problem ()-() there exsts a constant C such that the -th exact egenvalue and the approxmate egenvalue satsfy the followng relaton 4 4 C x = O x, x = b a /, =,..., (0) Proof: For the presented formulas n Table (n these cases, the exact and dscrete egenvalues are known), one can estmate the error of approxmaton of the dscrete egenvalues. Let us start from the Taylor seres expanson for the functon sn (x) at about pont x = 0
The Sturm-Louvlle egenvalue problem - a numercal soluton usng the Control Volume Method 33 sn ( x) n= ( ) ( n) n+ n n x = ()! ext, f we take nto account the frst two terms n the Taylor seres (), then we estmate the followng expresson as ( n) n+ n n x 4 4 x sn ( x) = x < x x x = x n=! 3 3 () For the case of the Drchlet-Drchlet boundary condtons, the error s evaluated by usng the estmaton () n the followng way: ( ) ( ) ( ) 4 sn sn ( ) π π π π = = b a b a b a 4 4 4 4 ( ) π π π ( b a) ( b a) ( b a) 4 4 < = = x = C x 4 3 (3) In the other cases (from Table ), the results are smlar. Another method of error estmaton s based on the nvestgaton of the Expermental Rate of Convergence (ERC). The total error n the estmate of the egenvalues s composed of both the error resultng from the dscretzaton of the equaton and the error of the numercal algorthm for fndng egenvalues of system (5). Here, we assume that the error s r s ( ) = = = (4) ( ) O x, x b a /,,..., where the parameters r and s are to be determned expermentally. If the -th exact egenvalue to SLP s known, then the parameters r and s can be determned usng the followng formulas for the ERC for varable values of : r= ERCr(, ) = log ( /) (5) ( ) s(, ) log ( ) + + + s= ERC = Whereas, f the exact egenvalue s unknown then, we determne the parameter r from the followng formula (6) ( ) ( /) r(, ) log ( ) ( ) r= ERC = (7)
34 J. Sedleck, M. Ceselsk, T. Błaszczyk The parameter s can be estmated usng (6) and assumng that s numercally determned for suffcently hgh value of. 4. Example of numercal smulatons In tests of verfcaton of the numercal solutons, three cases are taken nto account: Example : p(x) =, q(x) = 0, w(x) = ( + x), α =, α = 0, β =, β = 0. Example : p(x) = (x + ), q(x) = x, w(x) = exp(x), α =, α = 0, β = 0, β =. Example 3: p(x) = + sn(πx), q(x) = 0, w(x) = + sqrt(x), α =, α = 0, β = 5, β =. In all examples, the values of a = 0 and b = have been assumed. In the case of Example, the exact egenvalues are gven by ( ) = / 4 + π / ln [], whle for the remanng cases the exact egenvalues are unknown. In Tables, 4 and 5, the numercal values of the frst 8 egenvalues for dfferent values of and the calculated ((5) or (7)) for all examples are presented, respectvely. In addton, n Table 3, the calculated values of ERC s (6) for Example are shown. Egenvalues and for Example 3 4 00 0.790497 8.388896 84.97699 38.440408 Table 00 0.79838.000 8.45895.0000 85.0975.9999 38.805046.9998 400 0.7973.0000 8.4767.0000 85.099.0000 38.896.9999 800 0.79594.0000 8.4868.0000 85.895.0000 38.9907.0000 600 0.798.0067 8.490356.9995 85.9996.000 38.9476.0000 300 0.79866.9994 8.4943.0007 85.30446.0000 38.9640.0000 anal. 0.79885 8.49538 85.30596 38.9665 5 6 7 8 00 5.697 737.309749 00.600 307.7479 00 53.506.9996 739.56478.9994 005.68095.999 33.00956.9989 400 53.73969.9999 739.6839.9999 006.53680.9998 34.46964.9997 800 53.78865.0000 739.733885.0000 006.75080.9999 34.83475.9999 600 53.8057.0000 739.76760.0000 006.80430.0000 34.9603.0000 300 53.80605.0000 739.769978.0000 006.8768.0000 34.94885.0000 anal. 53.807 739.77384 006.83 34.95646
The Sturm-Louvlle egenvalue problem - a numercal soluton usng the Control Volume Method 35 ERC s for Example Table 3 ERC s (,) ERC s (,) ERC s (,3) ERC s (,4) ERC s (,5) ERC s (,6) ERC s (,7) ERC s (,8) 00 4.049 4.008 4.0037 4.007 4.0006 3.9997 3.9989 3.998 00 4.050 4.008 4.0040 4.003 4.004 4.0008 4.0004 4.0000 400 4.050 4.008 4.004 4.004 4.006 4.00 4.0007 4.0005 800 4.050 4.008 4.004 4.004 4.006 4.00 4.0008 4.0006 600 4.03 4.0073 4.0043 4.004 4.006 4.00 4.0009 4.0007 300 4.0309 4.0086 4.004 4.004 4.007 4.00 4.0008 4.0007 Egenvalues and for Example Table 4 00.70350-6.8573037-78.5053965-55.95777-00.7045757.0000 6.86856.9999 78.53837.9998 56.03905.9995 400.7048396.0000 6.869886.0000 78.546368.9999 56.069879.9999 800.7049056.0006 6.86379.000 78.548363.0000 56.077587.0000 600.7049.994 6.8633440.0000 78.5488747.9999 56.07954.0000 300.70496-6.863367-78.549006-56.079996-5 6 7 8 00 59.00064-387.68585-54.8308-7.39858-00 59.344073.999 388.4744.9988 543.56336.998 73.79747.9976 400 59.476.9998 388.6893.9997 543.67583.9996 74.430.9994 800 59.44853.9999 388.65936.9999 543.7079.9999 74.59893.9999 600 59.453735.0000 388.6709.0000 543.730508.0000 74.63867.0000 300 59.45504-388.67383-543.73655-74.6486-3 Egenvalues and for Example 3 4 Table 5 00.896997-4.0700-66.8696550-30.03860-00.8976086.9999 4.0778646.9998 66.99064.9997 30.64588.9996 400.89779365.0000 4.0798055.9999 66.9475.9999 30.99780.9999 800.89783685.9999 4.080907.0000 66.959.0000 30.08579.0000 600.89784765.9993 4.08040.9999 66.957733.0000 30.0778.0000 300.89785036-4.080444-66.959384-30.38-5 6 7 8 00 3.78678-38.96036-443.8796-588.65456-00 4.4043.9994 38.965739.999 444.65377.9990 59.85500.9998 400 4.99.9998 39.5867.9998 445.00663.9997 59.86047.9997 800 4.5343.0000 39.06405.9999 445.40.9999 59.9869.9999 600 4.57948.0000 39.844.0000 445.35338.0000 59.0664.0000 300 4.59350-39.449-445.407-59.03675 - The analyss of the results presented n Tables -5 ndcates that the rate r ( ) s close to, whle the rate s (ERC s ) s close to 4. Thus we can confrm that the relatonshp (0) s satsfed. The errors n the approxmaton of egenvalues ncrease rapdly as the ndex of the egenvalue grows. 3 4
36 J. Sedleck, M. Ceselsk, T. Błaszczyk 5. Conclusons In ths paper, the new approach based on the control volume method for fndng the egenvalues of the Sturm-Louvlle problem was dscussed. The contnuous problem descrbed by the dfferental equaton wth the adequate boundary condtons was converted to the correspondng dscrete one. The rate of convergence of the proposed numercal scheme s order. The presented results of the approxmaton of egenvalues are n close agreement wth the results obtaned n an analytcal way or n the mathematcal software feld. In the future, the presented approach can be extended to apply hgh order of accuracy dfference schemes for approxmatng egenvalues of the Sturm-Louvlle problem and can be appled to the fractonal Sturm-Louvlle problem whch s related to the correspondng fractonal Euler- -Lagrangan equaton [3]. References [] Agarwal R.P., O Regan D., An Introducton to Ordnary Dfferental Equatons, Sprnger, ew York 008. [] Atknson F.V., Dscrete and Contnuous Boundary Value Problems, Academc Press, ew York, London 964. [3] Pryce J.D., umercal Soluton of Sturm-Louvlle Problems, Oxford Unv. Press, London 993. [4] Zatsev V.F., Polyann A.D., Handbook of Exact Solutons for Ordnary Dfferental Equatons, CRC Press, ew York 995. [5] Pruess S., Estmatng the egenvalues of Sturm-Louvlle problems by approxmatng the dfferental equaton, SIAM J. umer. Anal. 973, 0, 55-68. [6] Aceto L., Ghelardon P., Maghern C., Boundary value methods as an extenson of umerov s method for Sturm-Louvlle egenvalue estmates, Appled umercal Mathematcs 009, 59 (7), 644-656. [7] Amodo P., Settann G., A matrx method for the soluton of Sturm-Louvlle problems, Journal of umercal Analyss, Industral and Appled Mathematcs (JAIAM) 0, 6 (-), -3. [8] Ascher U.M., umercal Methods for Evolutonary Dfferental Equatons, SIAM, 008. [9] Ascher U.M., Matthej R.M.M., Russell R.D., umercal Soluton of Boundary Value Problems for ODEs, Classcs n Appled Mathematcs 3, SIAM, Phladelpha 995. [0] Ellott J.F., The characterstc roots of certan real symmetrc matrces, Master s Thess, Unversty of Tennessee, 953. [] Gregory R.T., Karney D., A Collecton of Matrces for Testng Computatonal Algorthm, Wley- Interscence, 969. [] Akulenko L.D., esterov S.V., Hgh-Precson Methods n Egenvalue Problems and Ther Applcatons (seres: Dfferental and Integral Equatons and Ther Applcatons), Chapman and Hall/CRC, 004. [3] Ceselsk M., Blaszczyk T., umercal soluton of non-homogenous fractonal oscllator equaton n ntegral form, Journal of Theoretcal and Appled Mechancs 05, 53 (4), 959-968. [4] Press W.H., Teukolsky S.A., Vetterlng W.T., Flannery B.P., umercal Recpes: The Art of Scentfc Computng (3rd ed.), Cambrdge Unversty Press, ew York 007.