New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials

• Autores: E. Doha, W.M. Abd Elhameed, A. H. Bhrawy
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 64, Fasc. 3, 2013, págs. 373-394
• Idioma: inglés
• DOI: 10.1007/s13348-012-0067-y
• Texto completo no disponible (Saber más ...)
• Resumen

  • A family of symmetric generalized Jacobi polynomials (GJPs) is introduced and used for solving high even-order linear elliptic differential equations in one variable by the Galerkin method. Some efficient and accurate algorithms are developed and implemented for solving such equations. The use of GJPs leads to a simplified analysis, more precise error estimates and very efficient numerical algorithms. The methods lead to linear systems with specially structured matrices that can be efficiently inverted. Two numerical examples are presented aiming to demonstrate the accuracy and the efficiency of the proposed algorithms.

• Referencias bibliográficas
  • Abd-Elhameed, W.M.: Efficient spectral Legendre dual-Petrov–Galerkin algorithms for the direct solution of (2n+1)th-order linear differential...
  • Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Applied Mathematical Series, vol. 55. National Bureau of Standards,...
  • Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
  • Aslanov, A.: A general formula for the series solution of high-order linear and nonlinear boundary value problems. Math. Comput. Model. 55,...
  • Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for elliptic boundary value problems: a survey. Numer. Algorithms...
  • Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover Publications, Mineola (2001)
  • Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)
  • Chandrasekaran, S., Ipsen, I.C.F.: On the sensitivity of solution components in linear systems of equations. SIAM J. Matrix Anal. Appl. 16(1),...
  • Coutsias, E.A., Hagstrom, T., Torres, D.: An efficient spectral method for ordinary differential equations with rational function. Math. Comput....
  • Doha, E.H.: An accurate solution of parabolic equations by expansion in ultraspherical polynomials. Comput. Math. Appl. 19(4), 75–88 (1990)
  • Doha, E.H.: The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. Comput. Math. Appl. 21(2–3), 115–122...
  • Doha, E.H.: On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential...
  • Doha, E.H.: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J....
  • Doha, E.H., Abd-Elhameed, W.M.: Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical...
  • Doha, E.H., Abd-Elhameed, W.M.: Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical...
  • Doha, E.H., Abd-Elhameed, W.M.: Efficient spectral ultraspherical-dual-Petrov–Galerkin algorithms for the direct solution of (2n+1)th-order...
  • Doha, E.H., Bhrawy, A.H.: Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials....
  • Doha, E.H., Abd-Elhameed, W.M., Bhrawy, A.H.: Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order...
  • Doha, E.H., Abd-Elhameed, W.M., Youssri, Y.H.: Efficient spectral-Petrov–Galerkin methods for the integrated forms of third- and fifth-order...
  • Funaro, D.: Polynomial Approximation of Differential Equations. Lecturer Notes in Physics. Springer, Heidelberg (1992)
  • Gheorghiu, C.I.: Spectral Methods for Differential Problems. “T. Popoviciu” Institute of Numerical Analysis, Cluj-Napoca (2007)
  • Gheorghiu, C.I., Dragomirescu, F.-I.: Spectral methods in linear stability. Applications to thermal convection with variable gravity field....
  • Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)
  • Grisvard, P.: Elliptic Problems in Nonsmooth Domains. SIAM, Philadelphia (2011)
  • Guo, B.-Y., Shen, J., Wang, L.-L.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006)
  • Guo, B.-Y., Shen, J., Wang, L.-L.: Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 59(5), 1011–1028 (2009)
  • Heinrichs, W.: Spectral methods with sparse matrices. Numer. Math. 56, 25–41 (1989)
  • Heinrichs, W.: Algebraic spectral multigrid methods. Comput. Methods Appl. Mech. Eng. 80, 281–286 (1990)
  • Noor, M.A., Mohyud-Din, S.T.: Homotopy perturbation method for solving sixth-order boundary value problems. Comput. Math. Appl. 55, 2953–2972...
  • Noor, M.A., Noor, K.I., Mohyud-Din, S.T.: Variational iteration method for solving sixth-order boundary value problems. Commun. Nonlinear...
  • Schatzman, M., Taylor, J.: Numerical Analysis: A Mathematical Introduction (translated). Clarendon Press, Oxford (2002)
  • Siddiqi, S.S., Akram, G.: Solution of 10th-order boundary value problems using non-polynomial spline technique. Appl. Math. Comput. 190, 641–651...
  • Siddiqi, S.S., Twizell, E.H.: Spline solutions of linear tenth-order boundary value problems. Int. J. Comput. Math. 68, 345–362 (1998)
  • Szegö, G.: Orthogonal Polynomials. Am. Math. Soc. Colloq. Pub., vol. 23 (1985)
  • Wang, W., Zhao, Y.B., Wei, G.W.: A note on the numerical solution of high-order differential equations. J. Comput. Appl. Math. 159, 387–398...