An extension of trapezoidal type product integration rules

Rza̧dkowski, Grzegorz

doi:10.1016/j.cam.2009.07.001

Cited by 5 publications

(8 citation statements)

References 7 publications

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“…In [19] we have proved, that if F 0 (x) is a continuous and positive function and also (x) is twice continuously differentiable in D then Q n (F 0 , ) is convergent to the integral with the error O(1/n 2 ). We will prove now that the quadrature is convergent in the case when F 0 (x) is absolutely integrable and (x) is continuous in D.…”

Section: Weakly Singular Fredholm Integral Equationsmentioning

confidence: 98%

“…The first order Bernoulli function F 1(k) (x) (see [18], [19]), associated with F 0 (x) in the interval […”

Section: Preliminariesmentioning

confidence: 99%

“…Let us observe that quadrature (19) is in fact the trapezoidal product integration rule. In order to see this, we will express it in an integral form.…”

Section: Preliminariesmentioning

confidence: 99%

“…Corollary 3.4. The sum of the weights of quadrature (19) is uniformly bounded with the following estimation…”

Section: Weakly Singular Fredholm Integral Equationsmentioning

confidence: 99%

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Convergence analysis of the generalized Euler-Maclaurin quadrature rule for solving weakly singular integral equations

Rza̧dkowski

Tohidi

2019

Filomat

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In the present paper we use the generalized Euler-Maclaurin summation formula to study the convergence and to solve weakly singular Fredholm and Volterra integral equations. Since these equations have different nature, the proposed convergence analysis for each equation has a different structure. Moreover, as an application of this summation formula, we consider the numerical solution of the fractional ordinary differential equations (FODEs) by transforming FODEs into the associated weakly singular Volterra integral equations of the first kind. Some numerical illustrations are designed to depict the accuracy and versatility of the idea.

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Section: Weakly Singular Fredholm Integral Equationsmentioning

confidence: 98%

“…The first order Bernoulli function F 1(k) (x) (see [18], [19]), associated with F 0 (x) in the interval […”

Section: Preliminariesmentioning

confidence: 99%

“…Let us observe that quadrature (19) is in fact the trapezoidal product integration rule. In order to see this, we will express it in an integral form.…”

Section: Preliminariesmentioning

confidence: 99%

“…Corollary 3.4. The sum of the weights of quadrature (19) is uniformly bounded with the following estimation…”

Section: Weakly Singular Fredholm Integral Equationsmentioning

confidence: 99%

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Convergence analysis of the generalized Euler-Maclaurin quadrature rule for solving weakly singular integral equations

Rza̧dkowski

Tohidi

2019

Filomat

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“…In [3], the discrete Galerkin method has been proposed and analysed for obtaining the numerical solution of these equations. The trapezoidal method was applied for approximating singular or nonsingular integral equations [4][5][6][7]. In [8], a generalization of the Euler-Maclaurin summation formula for solving WSFIEs of the second kind was introduced.…”

Section: Introductionmentioning

confidence: 99%

Reproducing kernel method for a class of weakly singular Fredholm integral equations

Alvandi

Paripour

2018

Journal of Taibah University for Science

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Numerical methods for solving integral equations have been the focus of much research, including reproducing kernel methods. We present a new algorithm to solve weakly singular Fredholm integral equations (WSFIEs). The advantage of this method is that it is possible to pick any point in the interval of integration and also the approximate solution. The advantage of this method is used to remove singularity and reproducing kernel functions are used as a basis. The convergence of approximation solution to the exact solution is also proved. Some examples are displayed to demonstrate that the method is accurate and efficient for WSFIEs.

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