×

Multidomain spectral method for the Gauss hypergeometric function. (English) Zbl 1462.65092

Summary: We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line \(\mathbb{R}\cup \infty\), except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier-ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
33C05 Classical hypergeometric functions, \({}_2F_1\)
65D20 Computation of special functions and constants, construction of tables
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards (1970)
[2] Atkinson, K.; Han, W., On the numerical solution of some semilinear elliptic problems, Electron Trans. Numer. Anal., 17, 206-217 (2004) · Zbl 1065.65133
[3] Auzinger, W.; Karner, E.; Koch, O.; Weinmüller, EB, Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations, Opuscula Math., 26, 29-41 (2006)
[4] Berrut, J-P; Trefethen, LN, Barycentric Lagrange interpolation, SIAM Rev., 46, 3, 501-517 (2004) · Zbl 1061.65006 · doi:10.1137/S0036144502417715
[5] Boyd, JP; Yu, F., Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions, J. Comput. Phys., 230, 4, 1408-1438 (2011) · Zbl 1210.65192 · doi:10.1016/j.jcp.2010.11.011
[6] Clarkson, PA, Painlevé Equations - Nonlinear Special Functions, Lecture Notes in Mathematics, 1883-1883 (2006), Berlin: Springer, Berlin
[7] Driscoll, TA; Hale, N.; Trefethen, LN, Chebfun Guide (2014), Oxford: Pafnuty Publications, Oxford
[8] Dubrovin, B.; Grava, T.; Klein, C., On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation, J. Nonl. Sci., 19, 1, 57-94 (2009) · Zbl 1220.37048 · doi:10.1007/s00332-008-9025-y
[9] Fiziev, PP; Staicova, DR, Solving systems of transcendental equations involving the Heun functions, Am. J. Comput. Math., 2, 2 (2012) · doi:10.4236/ajcm.2012.22013
[10] Fornberg, B.; Weideman, JAC, A numerical methodology for the Painlevé equations, J. Comp. Phys., 230, 5957-5973 (2011) · Zbl 1220.65092 · doi:10.1016/j.jcp.2011.04.007
[11] Frauendiener, J.; Klein, C., Computational approach to hyperelliptic Riemann surfaces, Lett. Math. Phys., 105, 3, 379-400 (2015) · Zbl 1317.14127 · doi:10.1007/s11005-015-0743-4
[12] Frauendiener, J., Klein, C. In: A. Bobenko, C. Klein (eds.) Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics, vol. 2013. Springer (2011) · Zbl 1317.14127
[13] Frauendiener, J.; Klein, C., Computational approach to compact Riemann surfaces, Nonlinearity, 30, 1, 138-172 (2017) · Zbl 1360.14142 · doi:10.1088/1361-6544/30/1/138
[14] Klein, C.; Stoilov, N., Numerical approach to Painlevé transcendents on unbounded domains., SIGMA, 14, 68-78 (2018) · Zbl 1397.34157
[15] Mason, J.C., Hanscomb, D.C.: Chebyshev Polynomials. Chapman and Hall/CRC (2002)
[16] Olver, F W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V.: NIST digital library of mathematical functions, https://dlmf.nist.gov, Release 1.0.22 of 2019-03-15
[17] Olver, S.; Townsend, A., A fast and well-conditioned spectral method, SIAM Rev., 55, 3, 462-489 (2013) · Zbl 1273.65182 · doi:10.1137/120865458
[18] Pearson, JW; Olver, S.; Porter, MA, Numerical methods for the computation of the confluent and Gauss hypergeometric functions, Numer Algor., 74, 821-866 (2017) · Zbl 1360.33009 · doi:10.1007/s11075-016-0173-0
[19] Seaborn, J.B.: Hypergeometric Functions and their Applications. Springer (1991) · Zbl 0744.33001
[20] Trefethen, LN; Weideman, JAC, Two results on polynomial interpolation in equally spaced points, J. Approx. Theory, 65, 3, 247-260 (1991) · Zbl 0736.41005 · doi:10.1016/0021-9045(91)90090-W
[21] Trefethen, LN, Spectral Methods in Matlab (2000), Philadelphia: SIAM, Philadelphia · Zbl 0953.68643
[22] Trefethen, L.N.: Approximation Theory and Approximation Practice, vol. 128. SIAM (2013) · Zbl 1264.41001
[23] Wilber, HD, Numerical Computing with Functions on the Sphere and Disk (2016), Master’s thesis: Boise State University, Master’s thesis
[24] Wilber, HD; Townsend, A.; Wright, GB, Computing with functions in spherical and polar geometries II. The disk, SIAM J. Sci. Comput., 39, 3, C238-C262 (2017) · Zbl 1368.65026 · doi:10.1137/16M1070207
[25] Weideman, JAC; Reddy, SC, A Matlab differentiation matrix suite, ACM TOMS, 26, 465-519 (2000) · doi:10.1145/365723.365727
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.