Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations

Xie, Zhongliang; Li, Xianjuan; Tang, Tao

doi:10.1007/s10915-012-9577-8

Cited by 80 publications

(32 citation statements)

References 9 publications

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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%

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Cai

2017

Journal of Function Spaces

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We develop a generalized Jacobi-Galerkin method for second kind Volterra integral equations with weakly singular kernels. In this method, we first introduce some known singular nonpolynomial functions in the approximation space of the conventional JacobiGalerkin method. Secondly, we use the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation so as to obtain high-order accuracy for the approximation. Then, we establish that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order. One numerical example is presented to demonstrate the effectiveness of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Cai

2017

Journal of Function Spaces

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“…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%

Jacobi Spectral Galerkin Methods for Volterra Integral Equations With Weakly Singular Kernel

Yang¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. We propose and analyze spectral and pseudo-spectral JacobiGalerkin approaches for weakly singular Volterra integral equations (VIEs). We provide a rigorous error analysis for spectral and pseudospectral Jacobi-Galerkin methods, which show that the errors of the approximate solution decay exponentially in L ∞ norm and weighted L 2 -norm. The numerical examples are given to illustrate the theoretical results.

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“…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%

Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods

Tang

2015

J Sci Comput

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In this paper we present and analyze Chebyshev and Legendre pseudo-spectral methods for the second kind Volterra integral equations with weakly singular kernel (x − s) −μ , 0 < μ < 1. The proposed methods are based on the Gauss-type quadrature formula for approximating the integral operators involved in the equations. The present work is an extension of the earlier proposed spectral Jacobi-Galerkin method for the second kind Volterra integral equations with regular kernels (Xie et al. in J Sci Comput 53(2):414-434, [21]). A detailed convergence analysis is carried out, and several error estimates in L ∞ and L 2 ω norms are obtained. Numerical examples are considered to verify the theoretical predictions.

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Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations

Cited by 80 publications

References 9 publications

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

Jacobi Spectral Galerkin Methods for Volterra Integral Equations With Weakly Singular Kernel

Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods

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