Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations

Discrete and Continuous Dynamical Systems - Series B

Volumn 6, Issue 6, 2006, Pages 1381-1402

  Shen, Jie ^a   Wang, Li Lian ^b  

^a PURDUE UNIVERSITY   (United States)

^b NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

Author keywords

Composite Legendre Laguerre approximation; Domain decomposition; Dual Petrov Galerkin method; KDV equation; Laguerre functions

Indexed keywords

EID: 33750509139     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2006.6.1381     Document Type: Article

References (21)

1. J. L.Bona, and W. G. Pritchard, and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), no. 1471, 457-510.
2. J. L. Bona, and S. M. Sun, and B. Y Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc., 354 (2002), no. 2, 427-490 (electronic).
3. J. Bona and R. Winther, The Korteweg-de Vries equation, posed in a quarter-plane, SIAM J. Math. Anal., 14 (1983), no. 6, 1056-1106.
4. Goubet Olivier and Jie Shen, On the dual Petrov-Galerkin formulation of the KdV equation in a finite interval, submitted to DCDS (series A)
5. Qiang Du, Benyu Guo and Jie Shen, Fourier spectral approximation to a dissipative system modeling the flow of liquid crystals, SIAM J. Numer. Anal., 39 (2001), no. 3, 735-762 (electronic).
6. Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Second edition, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL, 1984.
7. D. Funaro, Polynomial Approxiamtions of Differential Equations, Springer-verlag, 1992.
8. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, 1977,
9. Ben-yu Guo and He-ping Ma, Composite Legendre-Laguerre approximation in unbounded domains, J. Comput. Math., 19 (2001), no. 1, 101-112.
10. Benyu Guo and Jie Shen, Laguerre-Galerkin Method for Nonlinear Partial Differential Equations on a Semi-Infinite Interval, Numer. Math., 86 (2000), 635-654.
11. Ben-yu Guo and Jie Shen, On spectral approximations using modified Legendre rational functions: application to the Korteweg-de Vries equation on the half line, Indiana Univ. Math. J., 50 (2001), Special Issue, 181-204, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000)
12. Ben-yu Guo, Numerical solution of an initial-boundary value problem of the Korteweg-de Vries equation, Acta Math. Sci. (English Ed.), 5 (1985), no. 3, 337-348.
13. Joseph L. Hammack and Harvey Segur, The Korteweg-de Vries equation and water waves. II. Comparison with experiments, J. Fluid Mech., 65 (1974), 289-313.
14. J. E. Littlewood G. H. Hardy and G. Pólya, Inequalities, Cambridge University Press, UK, 1952.
15. Y. Maday, B. Pernaud-Thomas, and H. Vandeven, Reappraisal of Laguerre type spectral methods, La Recherche Aerospatiale, 6 (1985), 13-35.
16. Jie Shen, Stable and Efficient Spectral Methods in Unbounded Domains Using Laguerre functions, SIAM J. Numer. Anal, 38 (2000), 1113-1133.
17. Jie Shen, A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: application to the KDV equation, SIAM J. Numer. Anal, 41 (2003), 1595-1619.
18. Jie Shen, Efficient spectral-Galerkin method I. Direct solvers for second- and fourth-order equations by using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.
19. Jie Shen, Efficient spectral-Galerkin method II. Direct solvers for second- and fourth-order equations by using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), 74-87.
20. N. J. Zabusky and C. J. Galvin, Shallow water waves, the Korteveg-de-Vraie equation and solitons, J. Fluid Mech., 47 (1971), 811-824.
21. G. Szegö, Orthogonal Polynomials (fourth edition), 23, AMS Coll. Publ., 1975.