scholarly journals The problem with fuzzy eigenvalue parameter in one of the boundary conditions

2020 ◽  
Vol 10 (2) ◽  
pp. 159-165
Author(s):  
Hülya Gültekin Çitil
Keyword(s):  
Boundary Conditions ◽  
Eigenvalue Parameter ◽  
Fuzzy Eigenvalues

In this work, we study the problem with fuzzy eigenvalue parameter in one of the boundary conditions. We find fuzzy eigenvalues of the problem using the Wronskian functions \underline{W}_{\alpha }\left( \lambda \right) and \overline{W}_{\alpha }\left( \lambda \right). Also, we find eigenfunctions associated with eigenvalues. We draw graphics of eigenfunctions.

Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions

1980 ◽  
Vol 87 (1-2) ◽  
pp. 1-34 ◽  
Author(s):  
Charles T. Fulton
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Spectral Function ◽  
Total Variation ◽  
Eigenvalue Problems ◽  
Left Endpoint ◽  
Order Zero ◽  
Limit Circle ◽  
Weyl Theory ◽  
Eigenvalue Parameter

SynopsisIn this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.

Perturbation of two-point boundary value problems with eigenvalue parameter in the boundary conditions

1983 ◽  
Vol 94 (3-4) ◽  
pp. 213-219 ◽  
Author(s):  
Lawrence Turyn
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Banach Spaces ◽  
Boundary Value ◽  
Simple Eigenvalue ◽  
Point Boundary ◽  
Eigenvalue Parameter

SynopsisWe discuss smooth changes of eigenvalues under perturbation of the boundary value problems given in the title. The simple eigenvalue criterion is developed in the setting of Banach spaces, so very general perturbations of both the differential equation and the boundary conditions are allowed. Further, we need no assumptions about self-adjointness of the original or perturbed problems. The discussion is concluded with the application of the simple eigenvalue criterion to two examples.

scholarly journals Indefinite eigenvalue problem with eigenparameter in the two boundary conditions

1998 ◽  
Vol 21 (4) ◽  
pp. 775-784
Author(s):  
S. F. M. Ibrahim
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Selfadjoint Operator ◽  
Order Differential Equation ◽  
Expansion Theorem ◽  
Compact Resolvent ◽  
Second Order Differential Equation ◽  
Eigenvalue Parameter

The object of this paper is to establish an expansion theorem for a regular indefinite eigenvalue problem of second order differential equation with an eigenvalue parameter,λin the two boundary conditions. We associated with this problem aJ-selfadjoint operator with compact resolvent defined in a suitable Krein space and then we develop an associated eigenfunction expansion theorem.

scholarly journals Essentially isospectral transformations and their applications

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Conditions ◽  
Existence Theorems ◽  
Dirichlet Boundary Conditions ◽  
Trace Formulas ◽  
Asymptotics Of Eigenvalues ◽  
Regularized Trace ◽  
Eigenvalue Parameter ◽  
Nevanlinna Functions ◽  
Sturm Liouville ◽  
Uniqueness And Existence

We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

scholarly journals Approximation of Eigenvalues of Sturm-Liouville Problems by Using Hermite Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
M. M. Tharwat ◽  
S. M. Al-Harbi
Keyword(s):  
Boundary Conditions ◽  
Error Bounds ◽  
Eigenvalue Problems ◽  
Sampling Theorem ◽  
Eigenvalue Parameter ◽  
Derivative Sampling ◽  
Sturm Liouville ◽  
Truncation And Amplitude Errors ◽  
Computable Error Bounds ◽  
Hermite Interpolations

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper, we use the derivative sampling theorem “Hermite interpolations” to compute approximate values of the eigenvalues of Sturm-Liouville problems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Also, using computable error bounds, we obtain eigenvalue enclosures. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc method and explain that the Hermite interpolations method gives remarkably better results.

scholarly journals Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Manfred Möller ◽  
Bertin Zinsou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Imaginary Axis ◽  
Order Ordinary Differential Equation ◽  
Eigenvalue Parameter ◽  
Upper Half Plane ◽  
Parameter Dependent ◽  
Asymptotics Of The Eigenvalues

Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameterλand which has separable boundary conditions depending linearly onλ. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.

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