Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

Journal of Computational and Applied Mathematics

Volumn 181, Issue 1, 2005, Pages 24-45

  Doha, E H ^a   Abd Elhameed, W M ^a  

^a CAIRO UNIVERSITY   (Egypt)

Author keywords

Expansion coefficients; Orthogonal polynomials; Parabolic and elliptic partial differential equations; Spectral methods

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; MATRIX ALGEBRA; ORDINARY DIFFERENTIAL EQUATIONS; POISSON EQUATION; POLYNOMIALS; TENSORS; THEOREM PROVING; VECTORS;

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS; EXPANSION COEFICIENT; ORTHOGONAL POLYNOMIALS; PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS; SPECTRAL METHODS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 19444374070     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2004.11.015     Document Type: Article

References (32)

1. M. Abramowitz, I.A. Stegun, (Eds.), Handbook of mathematical functions, Applied Mathematical Series, vol. 55, National Bureau of Standards, New York, 1970.
2. R. Bellman K.L. Cooke Differential-Difference Equations 1963 Academic Press New York
3. C. Canuto M.Y. Hussaini A. Quarteroni T.A. Zang Spectral Methods in Fluid Dynamics 1988 Springer Berlin
4. P.M. Dew R.E. Scraton Chebyshev methods for the numerical solution of parabolic partial differential equations in two and three space variables J. Inst. Math. Appl. 16 1975 121-131
5. E.H. Doha An accurate double Chebyshev approximation for Poisson's equation Ann. Univ. Sci. Budapest. Sect. Comp. 10 1990 243-276
6. E.H. Doha An accurate solution of parabolic equations by expansion in ultraspherical polynomials Comput. Math. Appl. 19 4 1990 75-88
7. E.H. Doha The coefficients of differentiated expansions and derivatives of ultraspherical polynomials Comput. Math. Appl. 21 2-3 1991 115-122
8. E.H. Doha The Chebyshev coefficients of general-order derivatives of an infinitely differentiable function in two or three variables Ann. Univ. Sci. Budapest. Sect. Comp. 13 1992 83-91
9. E.H. Doha On the coefficients of differentiated expansions of double and triple Legendre polynomials Ann. Univ. Sci. Budapest. Sect. Comp. 15 1995 25-35
10. E.H. Doha The coefficients of differentiated expansions of double and triple ultraspherical polynomials Ann. Univ. Sci. Budapest. Sect. Comp. 19 2000 57-79
11. E.H. Doha M.A. Helal An accurate double Chebyshev spectral approximation for parabolic partial differential equations J. Egypt. Math. Soc. 5 1 1997 83-101
12. E.H. Doha F.M.R. Al-Kholi An efficient double Legendre spectral method for parabolic and elliptic partial differential equations Internat. J. Comput. Math. 78 2001 413-432
13. G. Fairweather Finite Element Galerkin Methods for Differential Equations 1978 Marcel Dekker, Inc. New York
14. L. Fox I.B. Parker Chebyshev Polynomials in Numerical Analysis 1972 Oxford University Press Oxford
15. D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 26, SIAM, Philadelphia, 1977.
16. A. Graham Kronecker Product and Matrix Calculus With Applications 1981 Ellis Horwood Ltd. London
17. D.B. Haidvogel T. Zang The accurate solution of Poisson's equation by Chebyshev polynomials J. Comput. Phys. 30 1979 167-180
18. T.S. Horner A double Chebyshev series method for elliptic partial differential equations in: J. Noye (Ed.) Numerical Solution of Partial Differential Equations 1982 North-Holland Amsterdam
19. A. Karageorghis A note on the Chebyshev coefficients of the general order derivative of an infinitely differentiable function J. Comput. Appl. Math. 21 1988 129-132
20. D. Knibb R.E. Scraton On the solution of parabolic partial differential equation in Chebyshev series Comput. J. 14 1971 428-432
21. C. Lanczos Applied Analysis 1957 Pitman London
22. W.A. Light Are Chebyshev expansions really better Bull. Inst. Math. Appl. 22 1986 180-181
23. C.V. Loan Computational Frameworks for Fast Fourier Transform 1992 SIAM Philadelphia, USA
24. Y.L. Luke, The Special Functions and Their Approximations, vols. 1 and 2, Academic Press, New York, 1969.
25. A.R. Mitchell D.F. Griffiths The Finite Difference Methods in Partial Differential Equations 1980 Wiley New York
26. S.A. Orszag Accurate solution of the orr-Sommer field stability equation J. Fluid Mech. 50 1971 689-703
27. T.N. Phillips On the Legendre coefficients of a general order derivative of an infinitely differentiated function IMA J. Numer. Anal. 8 1988 455-459
28. H. Schlichting Boundary-Layer Theory 1979 McGraw-Hill New York
29. C.L. Streett et al. Spectral multigrid methods with applications to transonic potential flow J. Comput. Phys. 57 1985 43-76
30. Szegö, G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, 1985.
31. H. Tal-Ezer A pseudospectral Legendre method for hyperbolic equations with an improved stability condition J. Comput. Phys. 67 1986 145-172
32. R.S. Varga Matrix Iterative Analysis 1962 Prentice-Hall, Inc. Englewood Cliffs, NJ