JACOBI SPECTRAL GALERKIN METHODS FOR VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNEL

Yang, Yin

doi:10.4134/BKMS.2016.53.1.247

HOME > Journal Browse > About Journal > Journal Vol & Issue > Full Record

JACOBI SPECTRAL GALERKIN METHODS FOR VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNEL

Journal title : Bulletin of the Korean Mathematical Society
Volume 53, Issue 1, 2016, pp.247-262
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2016.53.1.247

Title & Authors

JACOBI SPECTRAL GALERKIN METHODS FOR VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNEL
Yang, Yin;

Abstract

We propose and analyze spectral and pseudo-spectral Jacobi-Galerkin approaches for weakly singular Volterra integral equations (VIEs). We provide a rigorous error analysis for spectral and pseudo-spectral Jacobi-Galerkin methods, which show that the errors of the approximate solution decay exponentially in ${$L^{\infty}$}$ norm and weighted ${$L^2$}$ -norm. The numerical examples are given to illustrate the theoretical results.

Keywords

spectral Galerkin methods;Jacobi polynomial;Volterra integral equations with weakly singular kernels;

Language

English

Cited by

3time(s) in CrossRef

Spectral Collocation Methods for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels, Bulletin of the Malaysian Mathematical Sciences Society, 2017

Numerical solutions for solving time fractional Fokker–Planck equations based on spectral collocation methods, Journal of Computational and Applied Mathematics, 2017

Spectral collocation method for the time-fractional diffusion-wave equation and convergence analysis, Computers & Mathematics with Applications, 2017, 73, 6, 1218

References

P. Baratella and A. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math. 163 (2004), no. 2, 401-418.

C. Canuto, M. Y. Hussaini, and A. Quarteroni, Spectral Methods, Fundamentals in single domains, Sci. Comput., Springer-Verlag, Berlin, 2006

Y. Chen and T. Tang, Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math. 233 (2009), no. 4, 938-950.

Y. Chen and T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel, Math. Comp. 79 (2010), no. 269, 147-167.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Appl. Math. Sci., Springer-Verlag, Heidelberg, 2nd Edition, 1998.

J. Douglas, T. Dupont, and L. Wahlbin, The stability in $L^q$ of the $L^2$-projection into finite element function spaces, Numer. Math. 23 (1974), no. 3, 193-197.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981.

A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.

G. Mastroianni and D. Occorsto, Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey, J. Comput. Appl. Math. 134 (2001), no. 1-2, 325-341.

10.

P. Nevai, Mean convergence of Lagrange interpolation: III, Trans. Amer. Math. Soc. 282 (1984), no. 2, 669-698.

11.

D. L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150 (1970), 41-53.

12.

D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc. 162 (1971), 157-170.

13.

S. Jie, T. Tao, and L. Wang, Spectral methods. Algorithms, analysis and applications, Springer Ser. Comput. Math., 41. Springer, Heidelberg, 2011.

14.

T. Tang, X. Xu, and J. Cheng. On Spectral methods for Volterra integral equation and the convergence analysis, J. Comput. Math. 26 (2008), no. 6, 825-837.

15.

X. Tao, Z. Xie, and X. Zhou, Spectral Petrov-Galerkin methods for the second kind Volterra type integro-differential equations, Numer. Math. Theory Methods Appl. 4 (2011), no. 2, 216-236.

16.

Z. Wan, Y. Chen, and Y. Huang, Legendre spectral Galerkin method for second-kind Volterra integral equations, Front. Math. China 4 (2009), no. 1, 181-193.

17.

Y. Wei and Y. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech. 4 (2012), no. 1, 1-20.

18.

Z. Xie and X. Li, and T. Tang, Convergence analysis of spectral Galerkin methods for Volterra type integral equations, J. Sci. Comput. 53 (2012), no. 2, 414-434.

19.

Y. Yang, Y. Chen, and Y. Huang, Convergence analysis of the Jacobi spectral-collocation method for fractional integro-differential equations, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 3, 673-690.

20.

Y. Yang, Y. Chen, and Y. Huang, Convergence analysis of Legendre-collocation methods for nonlinear Volterra type integro equations, Adv. Appl. Math. Mech. 7 (2015), no. 1, 74-88.