Applications of Mathematics, first online, pp. 1-18
The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula
Chein-Shan Liu, Botong Li
Received January 10, 2021. Published online April 8, 2024.
Abstract: The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an $n$-dimensional matrix eigenvalue problem is derived with a special matrix ${\bf A}:=[a_{ij}]$, that is, $a_{ij}=0$ if
$i+ j$ is odd.
Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function $q(x)$ in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method.
Keywords: symmetric Sturm-Liouville problem; inverse potential problem; special matrix eigenvalue problem; product formula; fictitious time integration method
References: [1] A. L. Andrew: Asymptotic correction of Numerov's eigenvalue estimates with natural boundary conditions. J. Comput. Appl. Math. 125 (2000), 359-366. DOI 10.1016/S0377-0427(00)00479-9 | MR 1803202 | Zbl 0970.65086
[2] G. Borg: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Acta Math. 78 (1946), 1-96. (In German.) DOI 10.1007/BF02421600 | MR 0015185 | Zbl 0063.00523
[3] I. Çelik: Approximate calculation of eigenvalues with the method of weighted residuals-collocation method. Appl. Math. Comput. 160 (2005), 401-410. DOI 10.1016/j.amc.2003.11.011 | MR 2102818 | Zbl 1064.65073
[4] I. Çelik: Approximate computation of eigenvalues with Chebyshev collocation method. Appl. Math. Comput. 168 (2005), 125-134. DOI 10.1016/j.amc.2004.08.024 | MR 2170019 | Zbl 1082.65555
[5] M. Dehghan: An efficient method to approximate eigenfunctions and high-index eigenvalues of regular Sturm-Liouville problems. Appl. Math. Comput. 279 (2016), 249-257. DOI 10.1016/j.amc.2016.01.026 | MR 3458019 | Zbl 1410.65276
[6] P. Ghelardoni: Approximations of Sturm-Liouville eigenvalues using boundary value methods. Appl. Numer. Math. 23 (1997), 311-325. DOI 10.1016/S0168-9274(96)00073-6 | MR 1445127 | Zbl 0877.65056
[7] P. Ghelardoni, C. Magherini: BVMs for computing Sturm-Liouville symmetric potentials. Appl. Math. Comput. 217 (2010), 3032-3045. DOI 10.1016/j.amc.2010.08.036 | MR 2733748 | Zbl 1204.65092
[8] S. H. Gould: Variational Methods for Eigenvalue Problems: An Introduction to the Methods of Rayleigh, Ritz, Weinstein, and Aronszajn. Dover, New York (1995). DOI 10.3138/9781487596002 | MR 1350533 | Zbl 0077.09603
[9] O. H. Hald: The inverse Sturm-Liouville problem and the Rayleigh-Ritz method. Math. Comput. 32 (1978), 687-705. DOI 10.1090/S0025-5718-1978-0501963-2 | MR 0501963 | Zbl 0432.65050
[10] O. H. Hald: The inverse Sturm-Liouville problem with symmetric potentials. Acta Math. 141 (1978), 263-291. DOI 10.1007/BF02545749 | MR 0505878 | Zbl 0431.34013
[11] D. Hinton, P. W. Schaefer (Eds.): Spectral Theory & Computational Methods of Sturm-Liouville Problems. Lecture Notes in Pure and Applied Mathematics 191. Marcel Dekker, New York (1997). MR 1460546 | Zbl 0866.00046
[12] M. Kobayashi: Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials. Comput. Math. Appl. 18 (1989), 357-364. DOI 10.1016/0898-1221(89)90220-4 | MR 0999264 | Zbl 0682.65054
[13] C.-S. Liu: A Lie-group shooting method for computing eigenvalues and eigenfunctions of Sturm-Liouville problems. CMES, Comput. Model. Eng. Sci. 26 (2008), 157-168. DOI 10.3970/cmes.2008.026.157 | MR 2426635 | Zbl 1232.65110
[14] C.-S. Liu: Analytic solutions of the eigenvalues of Mathieu's equation. J. Math. Research 12 (2020), Article ID p1, 11 pages. DOI 10.5539/jmr.v12n1p1
[15] C.-S. Liu: Accurate eigenvalues for the Sturm-Liouville problems, involving generalized and periodic ones. J. Math. Res. 14 (2022), Article ID p1, 19 pages. DOI 10.5539/jmr.v14n4p1
[16] C.-S. Liu, S. N. Atluri: A novel fictitious time integration method for solving the discretized inverse Sturm-Liouville problems, for specified eigenvalues. CMES, Comput. Model. Eng. Sci. 36 (2008), 261-285. DOI 10.3970/cmes.2008.036.261 | MR 2489473 | Zbl 1232.74007
[17] C.-S. Liu, S. N. Atluri: A novel time integration method for solving a large system of non-linear algebraic equations. CMES, Comput. Model. Eng. Sci. 31 (2008), 71-83. MR 2450570 | Zbl 1152.65428
[18] C.-S. Liu, J.-R. Chang, J.-H. Shen, Y.-W. Chen: A boundary shape function method for computing eigenvalues and eigenfunctions of Sturm-Liouville problems. Mathematics 10 (2022), Article ID 3689, 22 pages. DOI 10.3390/math10193689
[19] C.-S. Liu, B. Li: An upper bound theory to approximate the natural frequencies and parameters identification of composite beams. Composite Struct. 171 (2017), 131-144. DOI 10.1016/j.compstruct.2017.03.014
[20] C.-S. Liu, B. Li: Reconstructing a second-order Sturm-Liouville operator by an energetic boundary function iterative method. Appl. Math. Lett. 73 (2017), 49-55. DOI 10.1016/j.aml.2017.04.023 | MR 3659907 | Zbl 1375.65100
[21] C.-S. Liu, B.-T. Li: An $R(x)$-orthonormal theory for the vibration performance of non-smooth symmetric composite beam with complex interface. Acta Mech. Sin. 35 (2019), 228-241. DOI 10.1007/s10409-018-0799-3 | MR 3908891
[22] B. van Brunt: The Calculus of Variations. Universitext. Springer, New York (2004). DOI 10.1007/b97436 | MR 2004181 | Zbl 1039.49001
[23] G. Vanden Berghe, M. Van Daele: Exponentially-fitted Numerov methods. J. Comput. Appl. Math. 200 (2007), 140-153. DOI 10.1016/j.cam.2005.12.022 | MR 2276821 | Zbl 1110.65071
Affiliations: Chein-Shan Liu, Center of Excellence for Ocean Engineering, National Taiwan Ocean University, No. 2, Beining Rd., Zhongzheng Dist., Keelung City 202301, Taiwan, e-mail: csliu@ntou.edu.tw; Botong Li (corresponding author), School of Mathematics and Physics, University of Science and Technology Beijing, Haidian District, Beijing 100083, P. R. China, e-mail: libotong@ustb.edu.cn
