Abstract
The theory of Gegenbauer (ultraspherical) polynomial approximation has received considerable attention in recent decades. In particular, the Gegenbauer polynomials have been applied extensively in the resolution of the Gibbs phenomenon, construction of numerical quadratures, solution of ordinary and partial differential equations, integral and integro-differential equations, optimal control problems, etc. To achieve better solution approximations, some methods presented in the literature apply the Gegenbauer operational matrix of integration for approximating the integral operations, and recast many of the aforementioned problems into unconstrained/constrained optimization problems. The Gegenbauer parameter α associated with the Gegenbauer polynomials is then added as an extra unknown variable to be optimized in the resulting optimization problem as an attempt to optimize its value rather than choosing a random value. This issue is addressed in this article as we prove theoretically that it is invalid. In particular, we provide a solid mathematical proof demonstrating that optimizing the Gegenbauer operational matrix of integration for the solution of various mathematical problems by recasting them into equivalent optimization problems with α added as an extra optimization variable violates the discrete Gegenbauer orthonormality relation, and may in turn produce false solution approximations.
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References
Abramowitz, M., Stegun, V.: Handbook of Mathematical Functions. Dover (1965)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Area, I., Dimitrov, D.K., Godoy, E., Ronveaux, A.: Zeros of Gegenbauer and Hermite polynomials and connection coefficients. Math. Comput. 73, 1937–1951 (2004)
Bayin, S.S.: Mathematical Methods in Science and Engineering. Wiley-Interscience (2006)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, NY (2000)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin (1988)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains, Scientific Computation. Springer, Berlin, NY (2006)
Doha, E.H.: An accurate solution of parabolic equations by expansion in ultraspherical polynomials. Comput. Math. Appl. 19, 75–88 (1990)
El-Gendi, S.E.: Chebyshev solution of differential, integral, and integro-differential equations. Comput. J. 12, 282–287 (1969)
El-Hawary, H.M., Salim, M.S., Hussien, H.S.: An optimal ultraspherical approximation of integrals. Int. J. Comput Math. 76, 219–237 (2000)
El-Hawary, H.M., Salim, M.S., Hussien, H.S.: Ultraspherical integral method for optimal control problems governed by ordinary differential equations. J. Glob. Optim. 25, 283–303 (2003)
El-Kady, M.M., Hussien, H.S., Ibrahim, M.A.: Ultraspherical spectral integration method for solving linear integro-differential equations. World Acad. Sci. Eng. Technol. 33, 880–887 (2009)
Elbarbary, E.M.E.: Integration preconditioning matrix for ultraspherical pseudospectral operators. SIAM J. Sci. Comput. 28, 1186–1201 (2006)
Elbarbary, E.M.E.: Pseudospectral integration matrix and the boundary value problems. Int. J. Comput. Math. 84, 1851–1861 (2007)
Elgindy, K.T.: Generation of higher order pseudospectral integration matrices. Appl. Math. Comput. 209, 153–161 (2009)
Elgindy, K.T., Smith-Miles, K.A.: Optimal Gegenbauer quadrature over arbitrary integration nodes. J. Comput. Appl. Math. 242, 82–106 (2013). ISSN 0377-0427. doi:10.1016/j.cam.2012.10.020
Elgindy, K.T., Smith-Miles, K.A.: Solving boundary value problems, integral, and integrodifferential equations using Gegenbauer integration matrices. J. Comput. Appl. Math. 237(1), 307–325 (2013). ISSN 0377-0427. doi:10.1016/j.cam.2012.05.024
Elgindy, K.T., Smith-Miles, K.A.: Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method. Submitted (2012)
Elgindy, K.T., Smith-Miles, K.A., Miller, B.: Solving optimal control problems using a Gegenbauer transcription method. In: Proceedings of 2012 Australian Control Conference, AUCC 2012. University of New South Wales, Sydney, Australia, 15–16 November 2012
Fornberg, B.: A practical guide to pseudospectral methods. In: Cambridge Monographs on Applied and Computational Mathematic, vol. 1. Cambridge University Press, Cambridge (1996)
Funaro, D.: A preconditioning matrix for the Chebyshev differencing operator. SIAM J. Numer. Anal. 24, 1024–1031 (1987)
Ghoreishi, F., Hosseini, S.M.: Integration matrix based on arbitrary grids with a preconditioner for pseudospectral method. J. Comput. Appl. Math. 214, 274–287 (2008)
Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS, Philadelphia (1977)
Greengard, L.: Spectral integration and two-point boundary value problems. SIAM J. Numer. Anal. 28, 1071–1080 (1991)
Guo, B.Y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)
Hendriksen, E., van Rossum, H.: Electrostatic interpretation of zeros. In: Alfaro, M., Dehesa, J., Marcellan, F., de Francia, J.R., Vinuesa, J. (eds.) Orthogonal Polynomials and their Applications. Lecture Notes in Mathematics, vol. 1329, pp. 241–250. Springer Berlin / Heidelberg (1988). doi:10.1007/BFb0083363
Hesthaven, J.S.: Integration preconditioning of pseudospectral operators. I. Basic linear operators. SIAM J. Numer. Anal. 35, 1571–1593 (1998)
Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral methods for time-dependent problems. In: Cambridge Monographs on Applied and Computational Mathematics, vol. 21. Cambridge University Press (2007)
Lundbladh, A., Henningson, D.S., Johansson, A.V.. An efficient spectral integration method for the solution of the Navier Stokes equations, Technical Report FFA TN 1992-28. Aeronautical Research Institute of Sweden, Bromma (1992)
Mai-Duy, N., Tanner, R.I.: A spectral collocation method based on integrated Chebyshev polynomials for two-dimensional biharmonic boundary-value problems. J. Comput. Appl. Math. 201, 30–47 (2007)
Mercier, B.: An Introduction to the Numerical Analysis of Spectral Methods, vol. 318. Lecture Notes in Physics, Springer-Verlag, Berlin, NY (1989)
Mihaila, B., Mihaila, I.: Numerical approximations using Chebyshev polynomial expansions: El-gendi’s method revisited, J. Phys. A: Math. Gen. 35, 731 (2002)
Szeg\(\ddot{\rm o}\), G.: Orthogonal Polynomials, 4th edn. AMS Coll. Publ. (1975)
Tang, T., Trummer, M.R.: Boundary layer resolving pseudospectral methods for singular perturbation problems, SIAM J. Sci. Comput. 17, 430–438 (1996)
Tian, H.: Spectral Methods for volterra integral equations, M.Sc. thesis. Harbin Institute of Technology, Harbin, P.R. China (1989)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)
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Communicated by: Leslie Greengard.
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Elgindy, K.T., Smith-Miles, K.A. On the optimization of Gegenbauer operational matrix of integration. Adv Comput Math 39, 511–524 (2013). https://doi.org/10.1007/s10444-012-9289-5
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DOI: https://doi.org/10.1007/s10444-012-9289-5