A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

Cai, Haotao

doi:10.1007/s11425-014-4806-2

Cited by 9 publications

(4 citation statements)

References 26 publications

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“…Fractional-order differential equations were solved using these polynomials via the Tau method in [19]. Cai [20] and Behzadi [21] employed the Jacobi method for analyzing Fredholm integral and Fredholm-Volterra integro-differential equations, respectively. In the last few years, Bhrawy and co-workers [22] used Jacobi polynomials to solve numerically various problems by spectral method, tau method and collocation methods.…”

Section: Orthogonal Jacobi Polynomialsmentioning

confidence: 99%

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

Bahşı¹,

Çevik²,

Sezer³

2018

ntmsci

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An advanced matrix method is formulated for the solution of Volterra integro-differential equations with weakly singular kernel by using orthogonal Jacobi polynomials. Employing this practical method, it is possible to solve various types of these equations routinely in a systematic fashion. An error estimation procedure is prescribed to estimate the error of the basic Jacobi method and then the error correction term is added to the basic method to obtain more accurate results. Four test experiments are provided to confirm the validity and systematic approach of the advanced method. These experiments also certified that this advanced method surpasses the basic Jacobi method, as well as several alternative approaches.

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Section: Orthogonal Jacobi Polynomialsmentioning

confidence: 99%

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

Bahşı¹,

Çevik²,

Sezer³

2018

ntmsci

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“…To this end, we introduce the result proposed in [9][10][11]: for any ∈ ( ), there exists a positive constant independent of ,…”

Section: Journal Of Function Spacesmentioning

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%

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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“…In [14], Wei and Chen considered a spectral Jacobi-collocation method for solving Volterra type integrodifferential equation. In [15], Cai considered a Jacobicollocation method for solving Fredholm integral equations of second kind with weakly singular kernels.…”

Section: Introductionmentioning

confidence: 99%

A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

Guo

Cai

Xin

2014

Abstract and Applied Analysis

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The purpose of this paper is to provide a Jacobi-collocation method for solving second kind Volterra integral equations with a smooth kernel. This method leads to a fully discrete integral operator. First, it is shown that the fully discrete integral operator is stable in bothL∞and weightedL2norms. Then, the proposed approach is proved to arrive at an optimal (the most possible) convergent order in both norms. One numerical example demonstrates the efficiency and accuracy of the proposed method.

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A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

Cited by 9 publications

References 26 publications

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

Jacobi polynomial solutions of Volterra integro-differential equations with weakly singular kernel

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

A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

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