Convergence analysis of spectral collocation methods for a class of weakly singular Volterra integral equations

2015 ◽  
Vol 250 ◽  
pp. 131-144
Author(s):  
Xiaohua Ma ◽  
Chengming Huang ◽  
Xin Niu
Keyword(s):  
Convergence Analysis ◽  
Integral Equations ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Spectral Collocation ◽  
Weakly Singular ◽  
Spectral Collocation Methods

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

2017 ◽  
Vol 9 (6) ◽  
pp. 1506-1524
Author(s):  
Xiong Liu ◽  
Yanping Chen
Keyword(s):  
Convergence Analysis ◽  
Integral Equations ◽  
Lebesgue Constant ◽  
Volterra Integral Equations ◽  
Collocation Methods ◽  
Singular Kernel ◽  
Quadrature Formulas ◽  
Weakly Singular Kernel ◽  
Chebyshev Collocation ◽  
Weakly Singular

AbstractIn 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 theL∞-norm and weightedL2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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