A Legendre Petrov-Galerkin method for fourth-order differential equations

Computers and Mathematics with Applications

Volumn 61, Issue 1, 2011, Pages 8-16

  Shen, Ting Ting ^a   Xing, Kang Zheng ^a   Ma, He Ping ^a  

^a SHANGHAI UNIVERSITY   (China)

Author keywords

Fourth order differential equations; KuramotoSivashinsky equation; Legendre PetrovGalerkin method; Optimal error estimates

Indexed keywords

CHEBYSHEV COLLOCATION METHOD; FOURTH-ORDER DIFFERENTIAL EQUATIONS; KURAMOTO-SIVASHINSKY EQUATIONS; KURAMOTOSIVASHINSKY EQUATION; L2-NORM; LEGENDRE; NUMERICAL RESULTS; OPTIMAL ERROR ESTIMATE; OPTIMAL RATE; PETROV-GALERKIN; PETROV-GALERKIN METHODS;

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; GALERKIN METHODS; OPTIMIZATION;

NONLINEAR EQUATIONS;

EID: 78649448704     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2010.10.025     Document Type: Article

References (23)

1. J. Shen Efficient spectralGalerkin method I. Direct solvers of second- and fourth-order equations using Legendre polynomials SIAM J. Sci. Comput. 15 1994 1489 1505
2. J. Shen Efficient spectralGalerkin method II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials SIAM J. Sci. Comput. 16 1995 74 87
3. B. Bialecki, and A. Karageorghis A Legendre spectral Galerkin method for the biharmonic Dirichlet problem SIAM J. Sci. Comput. 22 2000 1549 1569
4. B. Bialecki, and A. Karageorghis A Legendre spectral collocation method for the biharmonic Dirichlet problem Math. Model. Numer. Anal. 34 2000 637 662
5. E.H. Doha, and A.H. Bhrawy Efficient spectralGalerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials Appl. Numer. Math. 58 2008 1224 1244
6. C. Bernardi, Y. Maday, Some spectral approximations of one-dimensional fourth-order problems, ICASE Report No.8936, ICASE, NASA-Langley Research Center, Hampton, VA, 1989.
7. C. Bernardi, and Y. Maday Spectral methods P.G. Ciarlet, J.L. Lions, Techniques of Scientific Computing (Part 2) Handbook of Numerical Analysis vol. V 1997 Elsevier Amsterdam 209 486
8. D. Funaro, and W. Heinrichs Some results about the pseudospectral approximation of one dimensional fourth order problems Numer. Math. 58 1990 399 418
9. C. Canuto, M.Y. Hussaini, A. Quartesoni, and T.A. Zang Spectral Methods: Fundamentals in Single Domains 2006 Springer Berlin
10. D. Gottlieb, and S.A. Orszag Numerical Analysis of Spectral Methods, Theory and Applications SIAM-CBMS, Philadelphia 1977 143 144
11. T.T. Shen, Z.Q. Zhang, and H.P. Ma Optimal Error Estimates of the Legendre Tau Method for Second-order Differential Equations J. Sci. Comput. 42 2010 198 215
12. E. Elbarbary Efficient ChebyshevPetrovGalerkin method for solving second-order equations J. Sci. Comput. 34 2008 113 126
13. M.S. Ismail Numerical solution of complex modified Kortewegde Vries equation by PetrovGalerkin method Appl. Math. Comput. 202 2008 520 531
14. H.P. Ma, and W.W. Sun A LegendrePetrovGalerkin and Chebyshev collocation method for third-order differential equations SIAM J. Numer. Anal. 38 2000 1425 1438
15. H.P. Ma, and W.W. Sun Optimal error estimates of the LegendrePetrovGalerkin method for the Kortewegde Vries equation SIAM J. Numer. Anal. 39 2001 1380 1394
16. W.Z. Huang, H.P. Ma, and W.W. Sun Convergence analysis of spectral collocation methods for a singular differential equation SIAM J. Numer. Anal. 41 2003 2333 2349
17. J. Shen, and L.L. Wang Sparse spectral approximations of high-dimensional problems based on hyperbolic cross SIAM J. Numer. Anal. 48 2010 1087 1109
18. Siraj-ul-Islam, Ikram A. Tirmizi, and Saadat Ashraf A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications Appl. Math. Comput. 174 2006 1169 1180
19. M. Uddin, S. Haq, and S. Islam A mesh-free numerical method for solution of the family of KuramotoSivashinsky equations Appl. Math. Comput. 212 2009 458 469
20. L.L. Wang, B.Y. Guo, and Z.Q. Wang A generalized Gauss type quadrature formula and its applications to pseudospectral method Numer. Math.: A Journal of Chinese Universitéies 11 2002 179 196
21. C. Canuto, M.Y. Hussaini, A. Quartesoni, and T.A. Zang Spectral Methods in Fluid Dynamics 1987 Springer Berlin
22. H.Y. Li Super spectral viscosity methods for nonlinear conservation Laws Chebyshev Collocation Methods and their Applications 2002 Shanghai Universitéy Press Shanghai
23. H.P. Ma ChebyshevLegendre spectral viscosity method for nonlinear conservation laws SIAM J. Numer. Anal. 35 1998 869 892