Convergence analysis of the spectral methods for weakly singular volterra integro-differential equations with smooth solutions

Advances in Applied Mathematics and Mechanics

Volumn 4, Issue 1, 2012, Pages 1-20

  Wei, Yunxia ^a   Chen, Yanping ^b  

^a SUN YAT SEN UNIVERSITY   (China)

^b SOUTH CHINA NORMAL UNIVERSITY   (China)

Author keywords

Convergence analysis; Spectral methods; Volterra integro differential equations; Weakly singular kernels

Indexed keywords

EID: 84862506497     PISSN: 20700733     EISSN: 20751354     Source Type: Journal    
DOI: 10.4208/aamm.10-m1055     Document Type: Article

References (32)

1. I. ALI, H. BRUNNER AND T. TANG, A spectral method for pantograph-type delay differential equations and its convergence analysis, J. Comput. Math., 27 (2009), pp. 254-265.
2. I. ALI, H. BRUNNER AND T. TANG, Spectral methods for pantograph-type differential and integral equations with multiple delays, Front. Math. China., 4 (2009), pp. 49-61.
3. C. BERNARDI AND Y. MADAY, Spectral Methods, in: P. G. Ciarlet and J. L. Lions (Eds.), Handbook of Numerical Analysis, Vol. 5, Techniques of Scientific Computing, Elsevier, Amsterdam, 1997, 209-486.
4. M. BOLOGNA, Asymptotic solution for first and second order linear Volterra integro-differential equations with convolution kernels, J. Phys. A. Math. Theor., 43 (2010), pp. 1-13.
5. H. BRUNNER, Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6 (1986), pp. 221-239.
6. H. BRUNNER, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004.
7. H. BRUNNER, A. PEDAS AND G. VAINIKKO, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), pp. 957-982.
8. H. BRUNNER AND D. SCHÖ TZAU, hp-discontinuous Galerkin time-stepping for Volterra Integrodifferential equations, SIAM J. Numer. Anal., 44 (2006), pp. 224-245.
9. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI AND T. A. ZANG, Spectral Methods Fundamentals in Single Domains, Springer-Verlag, 2006.
10. Y. CHEN AND T. TANG, Spectral methods for weakly singular Volterra Integral equations with smooth solutions, J. Comput. Appl. Math., 233 (2009), pp. 938-950.
11. Y. CHEN AND T. TANG, Convergence analysis of the Jacobi spectral-collocation methods for Volterra Integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147-167.
12. D. COLTON AND R. KRESS, Inverse Coustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Springer-Verlag, Heidelberg, 2nd Edition, 1998.
13. B. Y. GUO, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl., 243 (2000), pp. 373-408.
14. B. Y. GUO AND J. SHEN, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numer. Math., 86 (2000), pp. 635-654.
15. B. Y. GUO AND L. L. WANG, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory., 128 (2004), pp. 1-41.
16. Y. J. JIANG, On spectral methods for Volterra-type Integro-differential equations, J. Comput. Appl. Math., 230 (2009), pp. 333-340.
17. A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.
18. W. B. LIU AND T. TANG, Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems, Appl. Numer. Math., 38 (2001), pp. 315-345.
19. G. MASTROIANNI AND D. OCCORSIO, Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey, J. Comput. Appl. Math., 134 (2001), pp. 325-341.
20. G. MASTROIANNI AND G. MONEGATO, Nystrom interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations, IMA J. Numer. Anal., 17 (1997), pp. 621-642.
21. P. NEVAI, Mean convergence of Lagrange interpolation: III, Trans. Am. Math. Soc., 282 (1984), pp. 669-698.
22. D. L. RAGOZIN, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Am. Math. Soc., 150 (1970), pp. 41-53.
23. D. L. RAGOZIN, Constructive polynomial approximation on spheres and projective spaces, Trans. Am. Math. Soc., 162 (1971), pp. 157-170.
24. E. RAWASHDEH, D. MCDOWELL AND L. RAKESH, Polynomial spline collocation methods for second-order Volterra integro-differential equations, IJMMS, 56 (2004), pp. 3011-3022.
25. J. SHEN, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), pp. 1113-1133.
26. J. SHEN AND T. TANG, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.
27. T. TANG, Superconvergence of numerical solutions to weakly singular Volterra integrodifferential equations, Numer. Math., 61 (1992), pp. 373-382.
28. T. TANG, X. XU AND J. CHEN, On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math., 26 (2008), pp. 825-837.
29. M. TARANG, Stability of the spline collocation method for second order Volterra integrodifferential equations, Math. Model. Anal., 9 (2004), pp. 79-90.
30. Y. X. Wei and Y. Chen, Convergence analysis of the legendre spectral collocation methods for second order Volterra integro-differential equations, Numer. Math. Theor. Meth. Appl., 4 (2011), pp. 419-438.
31. C. L. XU AND B. Y. GUO, Laguerre pseudospectral method for nonlinear partial differential equations, J. Comput. Math., 20 (2002), pp. 413-428.
32. M. ZAREBNIA AND Z. NIKPOUR, Solution of linear Volterra integro-differential equations via Sinc functions, Int. J. Appl. Math. Comput., 2 (2010), pp. 1-10.