Spectral and pseudospectral approximations using Hermite functions: Application to the Dirac equation

Advances in Computational Mathematics

Volumn 19, Issue 1-3, 2003, Pages 35-55

  Guo, Ben Yu ^a   Shen, Jie ^b,c   Xu, Cheng Long ^d  

^a SHANGHAI NORMAL UNIVERSITY   (China)

^b XIAMEN UNIVERSITY   (China)

^c PURDUE UNIVERSITY   (United States)

^d TONGJI UNIVERSITY   (China)

Author keywords

Dirac equation; Hermite approximation; Spectral and pseudospectral

Indexed keywords

EID: 0037244340     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022892132249     Document Type: Conference Paper

References (24)

1. A. Alvarez and B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A 86 (1981) 327-332.
2. A. Alvarez, K. Pen-yu and L. Vazquez, The numerical study of a nonlinear one-dimensional Dirac equation, Appl. Math. Comput. 13 (1983) 1-15.
3. C. Bernardi and Y. Maday, Spectral methods, in: Handbook of Numerical Analysis, Vol. 5, Techniques of Scientific Computing, eds. P.G. Ciarlet and J.L. Lions (Elsevier, Amsterdam, 1997) pp. 209-486.
4. J.P. Boyd, The rate of convergence of Hermite function series, Math. Comp. 35 (1980) 1309-1316.
5. J.P. Boyd, The asymptotic coefficients of Hermite series, J. Comput. Phys. 54 (1984) 382-410.
6. J.P. Boyd, Chebyshev and Fourier Spectral Methods (Springer, Berlin, 1989).
7. J.P. Boyd and D.W. Moore, Summability methods for Hermite functions, Dynam. Atomos. Sci. 10 (1986) 51-62.
8. W.S. Don and D. Gottlieb, The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points, SIAM J. Numer. Anal. 31 (1994) 1519-1534.
9. D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp. 57 (1990) 597-619.
10. B.-y. Guo, Spectral Methods and Their Applications (World Scientific, Singapore, 1998).
11. B.-y. Guo, Error estimation for Hermite spectral method for nonlinear partial differential equations, Math. Comp. 68 (1999) 1067-1078.
12. B.-y. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math. 86 (2000) 635-654.
13. B.-y. Guo and C.-I. Xu, Hermite pseudospectral method for nonlinear partial differential equations, Math. Model. Numer. Anal. 34 (2000) 859-872.
14. E. Hill, Contributions to the theory of Hermitian series, Duke Math. J. 5 (1939) 875-936.
15. A.L. Levin and D.S. Lubinsky, Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights, Constr. Approx. 8 (1992) 461-533.
16. Y. Maday, B. Pernaud-Thomas and H. Vandeven, Une réhabilitation des méthodes spectrales de type Laguerre, Rech. Aérospat. 6 (1985) 353-379.
17. V.G. Makhankov, Dynamics of classical solutions, in non-integrable systems, Phys. Rep. 35 (1978) 1-128.
18. R.J. Nessel and G. Wilmes, On Nikolskii-type inequalities for orthogonal expansion, in: Approximation Theory, Vol. II, eds. G. Vorentz, C.K. Chui and L.L. Schumaker (Academic Press, New York, 1976) pp. 479-484.
19. J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal. 38 (2000) 1113-1133.
20. G. Szegö, Orthogonal Polynomials (Amer. Math. Soc., New York, 1967).
21. T. Tang, The Hermite spectral method for Gaussian-type functions, SIAM J. Sci. Comput. 14 (1993) 594-606.
22. J.A.C. Weideman, The eigenvalues of Hermite and rational differentiation matrices, Numer. Math. 61 (1992) 409-431.
23. C.-I. Xu and B.-y. Guo, Laguerre pseudospectral method for nonlinear partial differential equations in unbounded domains, J. Comput. Math. 16 (2002) 77-96.
24. D. Zwillinger, Handbook of Differential Equations, 3rd ed. (Academic Press, New York, 1997).