TY - JOUR

T1 - Convergence analysis of the Jacobi spectral-collocation methods for volterra integral equations with a weakly singular kernel

AU - Chen, Yanping

AU - Tang, Tao

N1 - The first author is supported by Guangdong Provincial “Zhujiang Scholar Award Project”, National Science Foundation of China 10671163, the National Basic Research Program under the Grant 2005CB321703. The second author is supported by Hong Kong Research Grant Council, Natural Science Foundation of China (G10729101), and Ministry of Education of China through a Changjiang Scholar Program.

PY - 2010/1

Y1 - 2010/1

N2 - In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [-1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

AB - In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [-1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

UR - http://www.scopus.com/inward/record.url?scp=77952816520&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-09-02269-8

DO - 10.1090/S0025-5718-09-02269-8

M3 - Journal article

AN - SCOPUS:77952816520

SN - 0025-5718

VL - 79

SP - 147

EP - 167

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 269

ER -