2012
DOI: 10.1007/s10915-012-9577-8
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Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations

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Cited by 79 publications
(31 citation statements)
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“…There are many numerical attempts based on the spline approximation to overcome the difficulty caused by the singularity of the solution of (2) (see [1][2][3][4][5][6][7][8]). Recently, spectral methods using Jacobi polynomial basis have received considerable attention to approximating the solution of integral equations due to their high accuracy and easy implementation (see [9][10][11][12][13][14][15][16][17]). In particular, Chen and Tang in [11] proposed a Jacobi-collocation spectral method for second kind Volterra integral equations with weakly singular kernels.…”
Section: Introductionmentioning
confidence: 99%
“…There are many numerical attempts based on the spline approximation to overcome the difficulty caused by the singularity of the solution of (2) (see [1][2][3][4][5][6][7][8]). Recently, spectral methods using Jacobi polynomial basis have received considerable attention to approximating the solution of integral equations due to their high accuracy and easy implementation (see [9][10][11][12][13][14][15][16][17]). In particular, Chen and Tang in [11] proposed a Jacobi-collocation spectral method for second kind Volterra integral equations with weakly singular kernels.…”
Section: Introductionmentioning
confidence: 99%
“…In [20], the authors extended the Legendre-collocation methods to nonlinear Volterra integral equations. Recently, in [16], the authors provided a Legendre spectral Galerkin method for second-kind Volterra integral equations, [15,18] provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential equations. Inspired by those work, we extend spectral Galerkin approach to Volterra integral equations with weakly singular kernels and provide a rigorous convergence analysis for the spectral and pseudo-spectral Jacobi-Galerkin methods, which indicates that the proposed methods converges exponentially provided that the data in the given VIEs are smooth.…”
Section: Introductionmentioning
confidence: 99%
“…The work more relevant to the present one is given by Xie et al in [21] which investigated a spectral Jacobi-Galerkin approach for second kind VIEs. The Gauss-Legendre quadrature formula was used to approximate the integral operator.…”
Section: Introductionmentioning
confidence: 99%