Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel. Issue 6 (28th November 2017)

• Record Type: Journal Article
• Authors: Liu, Xiong; Chen, Yanping
• Is Part Of: Advances in applied mathematics and mechanics. Volume 9:Issue 6(2017)
• Page Start: 1506
• Page End: 1524
• Publication Date: 2017-11-28
• Subjects: 65M10, -- 78A48; Chebyshev collocation method, -- Volterra integral equations, -- spectral rate of convergence, -- Hölder continuity; Engineering mathematics -- Periodicals; Mechanics -- Periodicals; 620.00151825
• DOI: 10.4208/aamm.OA-2016-0049
• Languages: English
• ISSNs: 2070-0733

Abstract

Abstract: In this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in the L ∞ -norm and weighted L 2 -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.