Minimality Properties of Sturm-Liouville Problems with Increasing Affine Boundary Conditions

Aliyev, Yagub N.

doi:10.1007/978-3-030-51945-2_3

Cited by 4 publications

(3 citation statements)

References 6 publications

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“…The cases that were not mentioned here doesn't contain any necessary and sufficient conditions in them. Similar problem was studied in [14].…”

Section: Affine Casementioning

confidence: 94%

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Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

Aliyev

2023

Preprint

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In the paper we study Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter. The necessary and sufficient conditions for minimality and completeness of the chosen system of root functions of the corresponding operator were given in two forms, one of which can be done by direct computations. This computationally simpler way was known for the affine case and even found some applications in the literature. But, until now, it was not developed for the quadratic case. The aim of the present paper is to fill this gap for the quadratic case and to consider the affine and quadratic cases together in a unified way. 2020 MSC: 34B24, 34L10.

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“…The cases that were not mentioned here doesn't contain any necessary and sufficient conditions in them. Similar problem was studied in [14].…”

Section: Affine Casementioning

confidence: 94%

“…In [11], [14], [19] many examples of the affine case were given. In those examples both ways of describing the necessary and sufficient conditions were presented.…”

Section: Examplesmentioning

confidence: 99%

Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

Aliyev

2023

Preprint

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“…In general, the target function is different from the characteristic function, which is usually not attainable in the closed form. The above work was extended in [10] to a boundary condition linearly dependent on the eigenparameter. In this paper, we also set the target equation f (λ) = a 2 (λ)w(1; λ) + b 2 (λ)w ′ (1; λ) = 0 as a mathematical tool to determine the eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

Liu,

Chang,

Kuo

2024

Mathematics

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For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions.

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Necessary and Sufficient Conditions for Basis Properties of the System of Root Functions of Sturm-Liouville Boundary Value Problems with Eigenparameter Dependent Boundary Conditions

Aliyev

2024

Trends in Mathematics

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Minimality Properties of Sturm-Liouville Problems with Increasing Affine Boundary Conditions

Cited by 4 publications

References 6 publications

Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

Sturm-Liouville problems with a boundary condition depending affinely or quadratically on an eigenparameter

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

Necessary and Sufficient Conditions for Basis Properties of the System of Root Functions of Sturm-Liouville Boundary Value Problems with Eigenparameter Dependent Boundary Conditions

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