A mixed scheme of product integration rules in (−1,1)

Russo, Maria Grazia

doi:10.1016/j.apnum.2019.09.019

Cited by 4 publications

(6 citation statements)

References 25 publications

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“…Now, we introduce a Nyström method based on the extended product quadrature rule (8), calling it the Extended Nyström Method (ENM). Proceeding in analogy to the ONM, we begin by constructing the following sequence ( K 2m+1 f )(y) m :…”

Section: The Extended Nyström Methodsmentioning

confidence: 99%

“…In addition to the previous well-known product rule, we recall the following Extended Product Rule (see [8]) based on the zeros of p m (w)p m+1 (w). Denoted by {y k := x m+1,k (w) : k = 1, .…”

Section: Product Integration Rulesmentioning

confidence: 99%

“…As usual, the ordinary Modified Moments (MMs), which depend on the specific kernel we considered, are often derived by suitable recurrence relations (see, e.g., [13]). In [8] a general scheme for deriving GMMs starting from MMs was proposed. Alternatively, for very smooth kernels, Gaussian rules can be also used.…”

Section: Computational Detailsmentioning

confidence: 99%

“…Following a similar idea, we propose here a Mixed Nyström scheme based on product quadrature rules of the "extended" type, i.e., based on the zeros of the polynomial p m+1 (w)p m (w), where {p n (w)} n denotes the orthonormal sequence with respect to a suitable fixed Jacobi weight w [8]. The advantages of using a Nyström scheme with respect to a collocation one include several benefits.…”

Section: Introductionmentioning

confidence: 99%

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Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

Mezzanotte

Russo

2021

Mathematics

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In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given.

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Section: The Extended Nyström Methodsmentioning

confidence: 99%

Section: Product Integration Rulesmentioning

confidence: 99%

Section: Computational Detailsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

Mezzanotte

Russo

2021

Mathematics

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“…Remark 2 Equation ( 27) and (28) show that the weights of the quadrature rule Σ r ,n ( f , y) depends on y and therefore, changing y, they must be recomputed. Note that the dependence on y is typical of the product integration rules, that, in the face of fast convergent rules, requires more computational effort.…”

Section: Theoremmentioning

confidence: 99%

Product integration rules by the constrained mock-Chebyshev least squares operator

2023

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In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of “pathological” behavior, e.g. “nearly” singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates.

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The High Order Augmented Finite Volume Methods Based on Series Expansion for Nonlinear Degenerate Parabolic Equations

Zhao

Zhang

et al. 2021

J Sci Comput

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A mixed scheme of product integration rules in (−1,1)

Cited by 4 publications

References 25 publications

Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

Product integration rules by the constrained mock-Chebyshev least squares operator

The High Order Augmented Finite Volume Methods Based on Series Expansion for Nonlinear Degenerate Parabolic Equations

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