Discrete Legendre Collocation Methods for Fredholm–Hammerstein Integral Equations with Weakly Singular Kernel

2018 ◽  
pp. 315-328
Author(s):  
Bijaya Laxmi Panigrahi
Keyword(s):  
Integral Equations ◽  
Collocation Methods ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Hammerstein Integral Equations ◽  
Weakly Singular

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.

scholarly journals A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel

1981 ◽  
Vol 22 (4) ◽  
pp. 431-438 ◽  
Author(s):  
G. Vainikko ◽  
P. Uba
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Collocation Methods ◽  
Piecewise Polynomial ◽  
Arbitrary Degree ◽  
Weakly Singular Kernel ◽  
Piecewise Polynomial Approximation ◽  
Collocation Points ◽  
Weakly Singular ◽  
Singular Kernels

AbstractWe construct collocation methods with an arbitrary degree of accuracy for integral equations with logarithmically or algebraically singular kernels. Superconvergence at collocation points is obtained. A grid is used, the degree of non-uniformity of which is in good conformity with the smoothness of the solution and the desired accuracy of the method.

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