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An exponentially convergent functional-discrete method for solving Sturm-Liouville problems with a potential including the Dirac \(\delta\)-function. (English) Zbl 1302.65172
Summary: We present a functional-discrete method for solving Sturm-Liouville problems with a potential that includes a function from \(L_1(0, 1)\) and the Dirac \(\delta\)-function. For both the linear and the nonlinear case sufficient conditions for an exponential rate of convergence of the method are obtained. The question of a possible software implementation of the method is discussed in detail. The theoretical results are successfully confirmed by a numerical example.
MSC:
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65Y15 Packaged methods for numerical algorithms
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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[1] Makarov, V., About functional-discrete method of arbitrary accuracy order for solving Sturm-Liouville problem with piecewise smooth coefficients, Dokl. Akad. Nauk. SSSR, 320, 34-39, (1991)
[2] Kryloff, N.; Bogolioubov, N., Sopra il metodo del coefficient constati (metodo del tronconi) per l’integrazione approssimate delle equazioni differenziali delle fisica Mathematica, Boll. Unione Mat. Ital., 7, 72-76, (1926) · JFM 54.0479.02
[3] Gordon, R., New method for constructing wavefunctions for bound states and scattering, J. Chem. Phys., 51, 14-25, (1969)
[4] S. Pruess, Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equations, Ph.D. Thesis, Purdue University, 1970. · Zbl 0224.65025
[5] Pryce, J., Numerical solution of Sturm-Liouville problems, (1993), Clarendon Press Oxford, New York, Tokyo, p. 322 · Zbl 0795.65053
[6] Ixaru, L. G., Metode numerice pentru ecuatii diferneţiale cu aplicaţii, (Editura Academiei Republicii Socialiste România, (1979), Bucharest), With an English summary · Zbl 0408.65044
[7] Makarov, V.; Rossokhata, N., A review of functional-discrete technique for eigenvalue problems, J. Numer. Appl. Math., 97, 97-102, (2009)
[8] Bandyrskii˘, B.; Gavrilyuk, I.; Lazurchak, I.; Makarov, V., Functional-discrete method (FD-method) for matrix Sturm-Liouville problems, Comput. Methods Appl. Math., 5, 1-25, (2005) · Zbl 1088.65074
[9] Makarov, V.; Ukhanev, O., FD-method for Sturm-Liouville problems. exponential rate of convergence, Appl. Math. Inform., 2, 1-19, (1997) · Zbl 0970.65089
[10] Makarov, V.; Rossokhata, N.; Bandyrskii˘, B., Functional-discrete method with a high order of accuracy for the eigenvalue transmission problem, Comput. Methods Appl. Math., 4, 369-381, (2004)
[11] Makarov, V.; Rossokhata, N.; Bandyrskii˘, B., Functional-discrete method for an eigenvalue transmission problem with periodic boundary conditions, Comput. Methods Appl. Math., 45, 201-220, (2005) · Zbl 1082.34027
[12] Makarov, V.; Rossokhata, N., Error estimates of convergence rate for FD-method for Sturm-Liouville problem with potential in \(L_1\), Collected Works of Institute of Mathematics NAS of Ukraine, 1, 1-16, (2005), (ukr)
[13] Makarov, V. L.; Rossokhata, N. O.; Dragunov, D. V., Exponentially convergent functional-discrete method for eigenvalue transmission problems with discontinuous flux and potential as a function in the space \(l_1\), Comput. Methods Appl. Math., 12, 46-72, (2012) · Zbl 1284.65091
[14] Gavrilyuk, I.; Klimenko, A.; Makarov, V.; Rossokhata, N., Exponentially convergent algorithm for nonlinear eigenvalue problems, IMA J. Numer. Anal., 27, 818-838, (2007) · Zbl 1130.65078
[15] Makarov, V.; Rossokhata, N., FD-method for nonlinear eigenvalue problems with discontinuous eigenfunctions, Nonlinear Oscil., 10, 126-143, (2007) · Zbl 1268.34041
[16] Makarov, V., FD-method for nonlinear eigenvalue problems for nonlinear differential equations, Dopov. Nats. Akad. Nauk Ukr., 8, 16-22, (2008), (ukr.) · Zbl 1164.65448
[17] Makarov, V. L.; Dragunov, D. V.; Klimenko, Y. V., The FD-method for solving Sturm-Liouville problems with special singular differential operator, Math. Comp., 82, 953-973, (2013) · Zbl 1262.65089
[18] Ixaru, L. G., (Numerical Methods for Differential Equations and Applications, Mathematics and its Applications (East European Series), (1984), D. Reidel Publishing Co. Dordrecht), Translated from the Romanian
[19] Ixaru, L. G., CP methods for the Schrödinger equation, Numerical Analysis 2000, Vol. VI, Ordinary Differential Equations and Integral Equations, J. Comput. Appl. Math., 125, 347-357, (2000) · Zbl 0971.65067
[20] Ixaru, L. G.; De Meyer, H.; Vanden Berghe, G., CP methods for the Schrödinger equation revisited, J. Comput. Appl. Math., 88, 289-314, (1998) · Zbl 0909.65045
[21] Ledoux, V.; Ixaru, L. G.; Rizea, M.; Van Daele, M.; Vanden Berghe, G., Solution of the Schrödinger equation over an infinite integration interval by perturbation methods, revisited, Comput. Phys. Comm., 175, 612-619, (2006) · Zbl 1196.81126
[22] Ledoux, V.; Rizea, M.; Van Daele, M.; Vanden Berghe, G.; Silişteanu, I., Eigenvalue problem for a coupled channel Schrödinger equation with application to the description of deformed nuclear systems, J. Comput. Appl. Math., 228, 197-211, (2009) · Zbl 1165.65046
[23] Ledoux, V.; Van Daele, M.; Vanden Berghe, G., MATSLISE: a MATLAB package for the numerical solution of Sturm-Liouville and Schrödinger equations, ACM Trans. Math. Software, 31, 532-554, (2005) · Zbl 1136.65327
[24] Savchuk, A. M.; Shkalikov, A. A., Sturm-Liouville operators with distribution potentials, Tr. Mosk. Mat. Obs., 64, 159-212, (2003) · Zbl 1066.34085
[25] Savchuk, A. M.; Shkalikov, A. A., Sturm-Liouville operators with singular potentials, Mat. Zametki, 66, 897-912, (1999) · Zbl 0968.34072
[26] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H., Solvable models in quantum mechanics, (2005), AMS Chelsea Publishing Providence, RI, With an appendix by Pavel Exner · Zbl 1078.81003
[27] Albeverio, S.; Kurasov, P., (Singular Perturbations of Differential Operators, London Mathematical Society Lecture Note Series, 271, (2000), Cambridge University Press Cambridge), Solvable Schrödinger type operators · Zbl 0945.47015
[28] Bracewell, R. N., (The Fourier Transform and its Applications, McGraw-Hill Series in Electrical Engineering. Circuits and Systems, (1986), McGraw-Hill Book Co. New York)
[29] F. Atkinson, Discrete and Continuous Boundary Problems, New York, 1964. · Zbl 0117.05806
[30] Vinokurov, V.; Sadovnichii, V., The asymptotics of eigenvalues and eigenfunctions and a trace formula for a potential with delta functions, Differential Equations, 38, 772-789, (2002) · Zbl 1043.34092
[31] Vinokurov, V. A.; Sadovnichiĭ, V. A., Analytic dependence of the eigenvalue and eigenfunction of the Sturm-Liouville problem on the integrable potential, Dokl. Akad. Nauk, 400, 439-443, (2005)
[32] Vinokurov, V. A., The eigenvalue and eigenfunction of the Sturm-Liouville problem as analytic functions of the integrable potential, Differ. Uravn., 41, 730-738, (2005), 861
[33] Seng, V.; Abbaoui, K.; Cherruault, Y., Adomian’s polynomials for nonlinear operators, Math. Comput. Modelling, 24, 59-65, (1996) · Zbl 0855.47041
[34] Hille, E., Analytic function theory, (1959), Ginn and Co. Boston, p. 308
[35] Nash, S.; Kahaner, D.; Moler, C., Numerical methods and software, (1989), Prentice-Hall, Inc. New Jersey, p. 495 · Zbl 0744.65002
[36] Stenger, F., (Numerical Methods Based on Sinc and Analytic Functions, Springer Series in Computational Mathematics, 20, (1993), Springer-Verlag New York) · Zbl 0803.65141
[37] T. Okayama, T. Matsuo, M. Sugihara, Error estimates with explicit constants for sinc approximation, sinc quadrature and sinc indefinite integration, Numerische Mathematik (2013) (in press). · Zbl 1281.65020
[38] Fousse, L.; Hanrot, G.; Lefèvre, V.; Pélissier, P.; Zimmermann, P., MPFR: a multiple-precision binary floating-point library with correct rounding, ACM Trans. Math. Software, 33, 15, (2007), Art. 13 · Zbl 1365.65302
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