Finite-Part Integrals and Modified Splines

Demichelis, Vittoria; Rabinowitz, Paul H.

doi:10.1023/b:bitn.0000039419.61024.0d

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…The obtained uniform convergence results (42) and (43), together with interpolation conditions 1. of Theorem 4 at t 0 and t Rn , allow the use of Martensen splines M R f for the numerical evaluation of certain finite-part integrals [7,8,12].…”

Section: Discussionmentioning

confidence: 99%

Smoothness and error bounds of Martensen splines

Demichelis

Sciarra

2015

Journal of Computational and Applied Mathematics

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Martensen splines M f of degree n interpolate f and its derivatives up to the order n − 1 at a subset of the knots of the spline space, have local support and exactly reproduce both polynomials and splines of degree ≤ n. An approximation error estimate has been provided for f ∈ C n+1 . This paper aims to clarify how well the Martensen splines M f approximate smooth functions on compact intervals. Assuming that f ∈ C n−1 , approximation error estimates are provided for D j f, j = 0, 1, . . . , n − 1, where D j is the jth derivative operator. Moreover, a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of D j M f to D j f , for j = 0, 1, . . . , n − 1.

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Section: Discussionmentioning

confidence: 99%

Smoothness and error bounds of Martensen splines

Demichelis

Sciarra

2015

Journal of Computational and Applied Mathematics

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“…In [4] two examples of sequences { f N } based on locally uniform partitions and satisfying (4)-(6) are provided for any positive integer p. These are the modified approximating splines and the modified optimal nodal splines, which are obtained by modifying the approximating splines [8] as well as the optimal nodal splines [1,2,3] in such a way that condition (5) is true for any positive integer p. In this paper, we consider sequences of approximating splines for which we can prove (4)-(6) without modifying their definition on [a, b]. In particular, we shall consider the Martensen spline operator, introduced in [9] and recently studied in [15,16].…”

Section: G(x + H) − G(x)| G ∈ C(j)mentioning

confidence: 99%

Martensen splines and finite-part integrals

Demichelis

Sciarra

2014

Numer Algor

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We state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We prove that a sequence of Martensen splines, based on locally uniform meshes, satisfies the sufficient conditions required by the theorem. We construct the quadrature rules based on such splines and illustrate their behaviour by presenting some numerical results and comparisons with composite midpoint, Simpson and Newton-Cotes rules.

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“…In recent decades, there have been a lot of works in developing efficient quadrature methods for hypersingular integrals on an interval such as Gaussian method [10,11,20,22,25,26], Newton-Cotes method [24, 27-29, 31, 36-38], transformation method [2,7,8] and some others [4,9,12]. Relatively speaking, the quadrature method for (1.1) has less been studied, references [7,8,14,34,37,38] may in fact be the entire literatures on the subject.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of a certain hypersingular integral equation of the first kind

2011

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In this paper, we first discuss the midpoint rule for evaluating hypersingular integrals with the kernel sin −2 [(x − s)/2] defined on a circle, and the key point is placed on its pointwise superconvergence phenomenon. We show that this phenomenon occurs when the singular point s is located at the midpoint of each subinterval and obtain the corresponding supercovergence analysis. Then we apply the rule to construct a collocation scheme for solving the relevant hypersingular integral equation, by choosing the midpoints as the collocation points. It's interesting that the inverse of coefficient matrix for the resulting linear system has an explicit expression, by which an optimal error estimate is established. At last, some numerical experiments are presented to confirm the theoretical analysis.

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Finite-Part Integrals and Modified Splines

Cited by 4 publications

References 14 publications

Smoothness and error bounds of Martensen splines

Smoothness and error bounds of Martensen splines

Martensen splines and finite-part integrals

Numerical solution of a certain hypersingular integral equation of the first kind

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