Eigenvalue Computation of Regular 4th Order Sturm-Liouville Problems
Abstract: In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.
Cite this paper: Alalyani, A. (2019) Eigenvalue Computation of Regular 4th Order Sturm-Liouville Problems. Applied Mathematics, 10, 784-803. doi: 10.4236/am.2019.109056.
References

[1]   Alalyani, A. (2019) Numerical Methods for Eigenvalue Computation of Fourth-Order Self-Adjoint Ordinary Differential Operators. PhD Dissertation, Florida Institute of Technology, Melbourne.

[2]   Everitt, W.N. (1957) The Sturm-Liouville Problem for Fourth Order Differential Equations. The Quarterly Journal of Mathematics, 8, 146-160.
https://doi.org/10.1093/qmath/8.1.146

[3]   Fulton, C. (1989) The Bessel-Squared Equation in the Lim-2, Lim-3, Im-4 Cases. The Quarterly Journal of Mathematics, 40, 423-456.
https://doi.org/10.1093/qmath/40.4.423

[4]   Bailey, P., Gordon, M. and Shampine, L. (1978) Automatic Solution of the Sturm-Liouville Problem. ACM Transactions on Mathematical Software, 4, 193-208.
https://doi.org/10.1145/355791.355792

[5]   Bailey, P.B., Everitt, W.N. and Zettl, A. (2001) The SLEIGN2 Sturm-Liouville Code. ACM Transactions on Mathematical Software, 21, 143-192.
https://doi.org/10.1145/383738.383739

[6]   Pruess, S. and Fulton, C. (1993) Mathematical Software for Sturm-Liouville Problems. ACM Transactions on Mathematical Software, 19, 360-376.
https://doi.org/10.1145/155743.155791

[7]   Fulton, C. and Pruess, S. (1998) The Computation of Spectral Density Functions for Singular Sturm-Liouville Problems Involving Simple Continuous Spectra. ACM Transactions on Mathematical Software, 34, 107-129.
https://doi.org/10.1145/285861.285867

[8]   Pryce, J.D. (1993) Numerical Solution of Sturm-Liouville Problems. Oxford University Press, Oxford.

[9]   Pryce, J.D. and Marletta, M. (1992) Automatic Solution of Sturm-Liouville Problems Using the Pruess Method. Journal of Computational and Applied Mathematics, 39, 57-78.
https://doi.org/10.1016/0377-0427(92)90222-J

[10]   Pryce, J. (1986) Error Control of Phase-Function Shooting Methods for Sturm-Liouville Problems. IMA Journal of Numerical Analysis, 6, 103-123.
https://doi.org/10.1093/imanum/6.1.103

[11]   Ledoux, V., Van Daele, M. and Vanden Berghe, G. (2004) CP Methods of Higher Order for Sturm-Liouville and Schrödinger Equations. Computer Physics Communications, 162, 153-165.
https://doi.org/10.1016/j.cpc.2004.07.001

[12]   Ledoux, V., Van Daele, M. and Vanden Berghe, G. (2005) MATSLISE: A Matlab Package for the Numerical Solution of Sturm-Liouville and Schrödinger Equations. ACM Transactions on Mathematical Software, 31, 532-554.
https://doi.org/10.1145/1114268.1114273

[13]   Ixaru, L. (2000) CP Methods for the Schrödinger Equation. Journal of Computational and Applied Mathematics, 125, 347-357.
https://doi.org/10.1016/S0377-0427(00)00478-7

[14]   Greenberg, L. and Marletta, M. (1997) Algorithm 775: The Code SLEUTH for Solving Fourth-Order Sturm-Liouville Problems. ACM Transactions on Mathematical Software, 23, 453-493.
https://doi.org/10.1145/279232.279231

[15]   Greenberg, L. and Marletta, M. (1995) Oscillation Theory and Numerical Solution of Fourth-Order Sturm-Liouville Problems. IMA Journal of Numerical Analysis, 15, 319-356.
https://doi.org/10.1093/imanum/15.3.319

[16]   Greenberg, L. (1991) An Oscillation Method for Fourth-Order, Self-Adjoint, Two-Point Boundary Value Problems with Nonlinear Eigenvalues. SIAM Journal on Mathematical Analysis, 22, 1021-1042.
https://doi.org/10.1137/0522067

[17]   Chanane, B. (2010) Accurate Solutions of Fourth Order Sturm-Liouville Problems. Journal of Computational and Applied Mathematics, 234, 3064-3071.
https://doi.org/10.1016/j.cam.2010.04.023

[18]   Chanane, B. (1998) Eigenvalues of Fourth Order Sturm-Liouville Problems Using Fliess Series. Journal of Computational and Applied Mathematics, 96, 91-97.
https://doi.org/10.1016/S0377-0427(98)00086-7

[19]   Chanane, B. (2002) Fliess Series Approach to the Computation of the Eigenvalues of Fourth-Order Sturm-Liouville Problems. Applied Mathematics Letters, 15, 459-563.
https://doi.org/10.1016/S0893-9659(01)00159-8

[20]   Chanane, B. (1998) Eigenvalues of Sturm-Liouville Problems Using Fliess Series. Applicable Analysis, 69, 233-238.
https://doi.org/10.1080/00036819808840659

[21]   Mihaila, B. and Mihaila, I. (2002) Numerical Approximations Using Chebyshev Polynomial Expansions: El-gendi’s Method Revisited. Journal of Physics A: Mathematical and General, 35, 731-746.
https://doi.org/10.1088/0305-4470/35/3/317

[22]   El-Gamel, M. and Sameeh, M. (2012) An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems. Applied Mathematics, 3, 920-925.
https://doi.org/10.4236/am.2012.38137

[23]   El-Gamel, M. and Abd El-hady, M. (2013) Two Very Accurate and Efficient Methods for Computing Eigenvalues of Sturm-Liouville Problems. Applied Mathematical Modeling, 37, 5039-5046.
https://doi.org/10.1016/j.apm.2012.10.019

[24]   Fox, L. (1962) Chebyshev Methods for Ordinary Differential Equations. The Computer Journal, 4, 318-331.
https://doi.org/10.1093/comjnl/4.4.318

[25]   Saleh Taher, A.H., Malek, A. and Momeni-Masuleh, S.H. (2013) Chebyshev Differential Matrices for Efficient Computation of the Eigenvalues of Fourth-Order Sturm-Liouville Problems. Applied Mathematical Modeling, 37, 4634-4642.
https://doi.org/10.1016/j.apm.2012.09.062

[26]   Ycel, U. (2015) Numerical Approximations of Sturm-Liouville Eigenvalues Using Chebyshev Polynomial Expansions Method. Cogent Mathematics, 2, Article ID: 1045223.
https://doi.org/10.1080/23311835.2015.1045223

[27]   Ycel, U. and Boubaker, K. (2012) Differential Quadrature Method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for Efficient Computation of the Eigenvalues of Fourth-Order Sturm-Liouville Problems. Applied Mathematical Modelling, 36, 158-167.
https://doi.org/10.1016/j.apm.2011.05.030

[28]   Khmelnytskaya, K., Kravchenko, V. and Baldenebro-Obeso, J. (2012) Spectral Parameter Power Series for Fourth-Order Sturm-Liouville Problems. Applied Mathematics and Computation, 219, 3610-3524.
https://doi.org/10.1016/j.amc.2012.09.055

[29]   Kravchenko, V. and Porter, R. (2010) Spectral Parameter Power Series for Sturm-Liouville Problems. Mathematical Methods in the Applied Sciences, 33, 459-468.
https://doi.org/10.1002/mma.1205

[30]   Kravchenko, V. and Porter, R. (2015) Eigenvalue Problems, Spectral Parameter Power Series, and Modern Applications. Mathematical Methods in the Applied Sciences, 38, 1945-1969.
https://doi.org/10.1002/mma.3213

[31]   Fulton, C. and Krall, A. (1983) Self-Adjoint 4th Order Boundary Value Problem in the Lim-4 Case. Proceedings of Symposium on Ordinary Differential Equations and Operators, Dundee, Lecture Notes in Mathematics 1032, 240-256.
https://doi.org/10.1007/BFb0076800

[32]   Atkinson, F. (1964) Discrete and Continuous Boundary Problems. Academic Press, New York.
https://doi.org/10.1063/1.3051875

[33]   Iserles, A., Marthinsen, A. and Nrsett, S.P. (1999) On the Implementation of the Method of Magnus Series for Linear Differential Equations. BIT Numerical Mathematics, 39, 281-304.
https://doi.org/10.1023/A:1022393913721

[34]   Greenberg, L. and Marletta, M. (1997) The Validity of Richardson Extrapolation in Coefficient Approximation Methods for Fourth Order and Higher Order Self-Adjoint ODE Eigenvalue Problems. Technical Report, Department of Mathematics and Computer Science, University of Leicester, Leicester.

[35]   Greenberg, L. (1991) A Prüfer Method for Calculating Eigenvalues of Self-Adjoint Systems of Ordinary Differential Equations: Parts 1 and 2. Technical Report TR91-24, University of Maryland at College Park, College Park.

[36]   Gallier, J. (2005) Manifolds, Lie Groups, Lie Algebras, Riemannian Manifolds, with Applications to Computer Vision and Robotics [Lecture Notes]. Department of Computer and Information Science, University of Pennsylvania, Philadelphia.

[37]   Magnus, W. (1954) On the Exponential Solution of Differential Equations for a Linear Operator. Communications on Pure and Applied Mathematics, 7, 649-673.
https://doi.org/10.1002/cpa.3160070404

[38]   Iserles, A. and Nrsett, S.P. (1999) On the Solution of Linear Differential Equations in Lie Groups. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 357, 983-1019.
https://doi.org/10.1098/rsta.1999.0362

[39]   Stoer, J. and Bulirsch, R. (1993) Introduction to Numerical Analysis. Second Edition, Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4757-2272-7

[40]   Iserles, A., Munthe-Kaas, H.Z., Norsett, S.P. and Zanna, A. (2005) Lie-Group Methods. Acta Numerica, 9, 1-148.
https://doi.org/10.1017/S0962492900002154

Top