Computing the eigenvalues of a class of nonlocal Sturm-Liouville problems

Bilal Chanane

doi:10.1016/j.mcm.2008.10.021

Computing the eigenvalues of a class of nonlocal Sturm-Liouville problems

Bilal Chanane^*

^*Corresponding author for this work

King Fahd University of Petroleum & Minerals

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

In this paper, we shall use the regularized sampling method introduced recently to compute the eigenvalues of Sturm-Liouville problems with nonlocal conditions {(- y^″ + q (x) y = λ y, x ∈ [0, 1]; χ₀ (y) = 0, χ₁ (y) = 0,) where q ∈ L¹ and, χ₀ and χ₁ are continuous linear functionals defined by χ₀ (y) = ∫₀¹ [y (t) d ψ₁ (t) + y^′ (t) d ψ₂ (t)], χ₁ (y) = ∫₀¹ [y (t) d φ{symbol}₁ (t) + y^′ (t) d φ{symbol}₂ (t)], where χ₀ and χ₁ are independent, and ψ₁,ψ₂, φ{symbol}₁ and φ{symbol}₂ are functions of bounded variations. Integration is in the sense of Riemann-Stieltjes. A few numerical examples will be presented to illustrate the merits of the method, and comparisons will be made with the exact eigenvalues when they are available.

Original language	English
Pages (from-to)	225-232
Number of pages	8
Journal	Mathematical and Computer Modelling
Volume	50
Issue number	1-2
DOIs	https://doi.org/10.1016/j.mcm.2008.10.021
State	Published - Jul 2009

Bibliographical note

Funding Information:
Part of this work was done while the author was visiting Cardiff University, UK. The author wishes to thank Professor W.D. Evans and the Mathematics Department for their hospitality and stimulating atmosphere during his stay and King Fahd University of Petroleum and Minerals for its constant support and by making this possible through the grant MS/SPECTRAL/269.

Keywords

Eigenvalue problems
Integral boundary condition
Nonlocal condition
Second order Sturm-Liouville problems
Whittaker-Shannon-Kotel'nikov theorem

ASJC Scopus subject areas

Modeling and Simulation
Computer Science Applications

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10.1016/j.mcm.2008.10.021

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title = "Computing the eigenvalues of a class of nonlocal Sturm-Liouville problems",

abstract = "In this paper, we shall use the regularized sampling method introduced recently to compute the eigenvalues of Sturm-Liouville problems with nonlocal conditions {(- y″ + q (x) y = λ y, x ∈ [0, 1]; χ0 (y) = 0, χ1 (y) = 0,) where q ∈ L1 and, χ0 and χ1 are continuous linear functionals defined by χ0 (y) = ∫01 [y (t) d ψ1 (t) + y′ (t) d ψ2 (t)], χ1 (y) = ∫01 [y (t) d φ{symbol}1 (t) + y′ (t) d φ{symbol}2 (t)], where χ0 and χ1 are independent, and ψ1,ψ2, φ{symbol}1 and φ{symbol}2 are functions of bounded variations. Integration is in the sense of Riemann-Stieltjes. A few numerical examples will be presented to illustrate the merits of the method, and comparisons will be made with the exact eigenvalues when they are available.",

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author = "Bilal Chanane",

note = "Funding Information: Part of this work was done while the author was visiting Cardiff University, UK. The author wishes to thank Professor W.D. Evans and the Mathematics Department for their hospitality and stimulating atmosphere during his stay and King Fahd University of Petroleum and Minerals for its constant support and by making this possible through the grant MS/SPECTRAL/269. ",

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N2 - In this paper, we shall use the regularized sampling method introduced recently to compute the eigenvalues of Sturm-Liouville problems with nonlocal conditions {(- y″ + q (x) y = λ y, x ∈ [0, 1]; χ0 (y) = 0, χ1 (y) = 0,) where q ∈ L1 and, χ0 and χ1 are continuous linear functionals defined by χ0 (y) = ∫01 [y (t) d ψ1 (t) + y′ (t) d ψ2 (t)], χ1 (y) = ∫01 [y (t) d φ{symbol}1 (t) + y′ (t) d φ{symbol}2 (t)], where χ0 and χ1 are independent, and ψ1,ψ2, φ{symbol}1 and φ{symbol}2 are functions of bounded variations. Integration is in the sense of Riemann-Stieltjes. A few numerical examples will be presented to illustrate the merits of the method, and comparisons will be made with the exact eigenvalues when they are available.

AB - In this paper, we shall use the regularized sampling method introduced recently to compute the eigenvalues of Sturm-Liouville problems with nonlocal conditions {(- y″ + q (x) y = λ y, x ∈ [0, 1]; χ0 (y) = 0, χ1 (y) = 0,) where q ∈ L1 and, χ0 and χ1 are continuous linear functionals defined by χ0 (y) = ∫01 [y (t) d ψ1 (t) + y′ (t) d ψ2 (t)], χ1 (y) = ∫01 [y (t) d φ{symbol}1 (t) + y′ (t) d φ{symbol}2 (t)], where χ0 and χ1 are independent, and ψ1,ψ2, φ{symbol}1 and φ{symbol}2 are functions of bounded variations. Integration is in the sense of Riemann-Stieltjes. A few numerical examples will be presented to illustrate the merits of the method, and comparisons will be made with the exact eigenvalues when they are available.

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KW - Integral boundary condition

KW - Nonlocal condition

KW - Second order Sturm-Liouville problems

KW - Whittaker-Shannon-Kotel'nikov theorem

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Computing the eigenvalues of a class of nonlocal Sturm-Liouville problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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