Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values

Costabile, F.; Dell’Accio, Francesco; Luceri, R.

doi:10.1016/j.cam.2004.07.004

Cited by 32 publications

(20 citation statements)

References 6 publications

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“…with α 0 , α h , β h , h ¼ 1;…, n real constants, then P r−1 is the complementary Lidstone interpolating polynomial [27] of degree 2n [3,24,27,28].…”

Section: Dynamical Systems -Analytical and Computational Techniquesmentioning

confidence: 99%

Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Costabile¹,

Gualtieri²,

Napoli³

2017

Dynamical Systems - Analytical and Computational Techniques

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In this chapter, the connection between general linear interpolation and initial, boundary and multipoint value problems is explained. First, a result of a theoretical nature is given, which highlights the relationship between the interpolation problem and the Fredholm integral equation for high-order differential problems. After observing that the given problem is equivalent to a Fredholm integral equation, this relation is used in order to determine a general procedure for the numerical solution of high-order differential problems by means of appropriate collocation methods based on the integration of the Fredholm integral equation. The classical analysis of the class of the obtained methods is carried out. Some particular cases are illustrated. Numerical examples are given in order to illustrate the efficiency of the method.

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“…with α 0 , α h , β h , h ¼ 1;…, n real constants, then P r−1 is the complementary Lidstone interpolating polynomial [27] of degree 2n [3,24,27,28].…”

Section: Dynamical Systems -Analytical and Computational Techniquesmentioning

confidence: 99%

Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Costabile¹,

Gualtieri²,

Napoli³

2017

Dynamical Systems - Analytical and Computational Techniques

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“…Several interpolation operators have been introduced and studied to increase the degree of exactness of the Shepard operator: Taylor [7,8,14,15,24], Lagrange [8,9], Hermite [5,8], Birkhoff [6,8], least square approximations [20,21,22,25], and splines [8]. The present paper is devoted to a further combination of the Shepard operator with a new interpolation operator, the generalized Taylor polynomial, recently proposed by F. Costabile [10]. In section 2 we recall some known properties of the generalized Taylor polynomial and give new results on the error of approximation that will be used later in the paper.…”

Section: The Problemmentioning

confidence: 99%

“…Clearly a polynomial can be defined by means of equation (2.3) also in the case when a > b; a property of P m [f, a, b] (x) not stated in [10] is that this polynomial does not depend on the order of the points a, b.…”

Section: For Functions In the Classmentioning

confidence: 99%

Shepard--Bernoulli operators

Caira

Dell’Accio

2007

Math. Comp.

Self Cite

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Abstract. We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given.

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“…The complementary Lidstone interpolation and boundary value problems are very recently introduced in [1], and studied by Agarwal et. al.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of complementary Lidstone boundary value problems

Agarwal

Wong

2012

Bound Value Probl

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We consider the following complementary Lidstone boundary value problemwhere l > 0. The values of l are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of l such that for any l in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y' and this derivative dependence is seldom investigated in the literature.

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Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values

Cited by 32 publications

References 6 publications

Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Relationship between Interpolation and Differential Equations: A Class of Collocation Methods

Shepard--Bernoulli operators

Eigenvalues of complementary Lidstone boundary value problems

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