Lidstone polynomials and boundary value problems

Agarwal, Ravi P.; Wong, Patricia J. Y.

doi:10.1016/0898-1221(89)90023-0

Cited by 73 publications

(41 citation statements)

References 11 publications

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“…From Theorem 1, general results on the existence and uniqueness of solution of problem (2) by standard techniques [2,3] can be obtained. In the following, we will not linger over them, but we will outline the close relationship between interpolation and differential equations.…”

Section: The Fredholm Integral Equation For Problem (2)mentioning

confidence: 99%

“…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%

“…Later, Agarwal ([2], p. 2), Agarwal and Wong ( [3], pp. 21, 151, 186) considered some separate boundary value problems and the related Fredholm integral equation, using only polynomial interpolation, without taking into account the related Peano kernel.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Fredholm Integral Equation For Problem (2)mentioning

confidence: 99%

“…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%

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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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“…The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [4][5][6][7] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an even order (2mth order) differential equation and the Lidstone boundary conditions…”

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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“…In the past twenty years, the existence of solutions, especially the existence of positive solutions, of (1.1), (1.2) and its general cases, has been extensively studied by using the Leray-Schauder degree and the fixed point theorem in cones, see Agarwal [1], Agarwal and Wong [2], Aftabizadeh [3], Yang [4], Del Pino and Manásevich [5], Ma and Wang [6], Ma, Zhang and Fu [7], Bai and Wang [8], Bai and Ge [9], Yao [10], Y. Li [11], F. Li et al [12] and references therein. Also, the global structure of positive solution set (and nodal solutions set) are investigated by several authors, see for example, the interesting contributions [13]- [15] by Bari and Rynne. Very recently Ma [16]- [18] studied the global bifurcation phenomena of nodal solutions of (1.1), (1.2) when m = 2 and f 0 ∈ (0, ∞), where f 0 = lim…”

Section: Introductionmentioning

confidence: 99%

Global structure of positive solutions for superlinear 2mth-boundary value problems

2010

Czech Math J

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Lidstone polynomials and boundary value problems

Cited by 73 publications

References 11 publications

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

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

Eigenvalues of complementary Lidstone boundary value problems

Global structure of positive solutions for superlinear 2mth-boundary value problems

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