Restoration of the linear potential in the Sturm-Liouville problem

2017 ◽  
Vol 12 (2) ◽  
pp. 152-156 ◽  
Author(s):  
A.M. Akhtyamov ◽  
I.M. Utyashev
Keyword(s):  
Boundary Value Problems ◽  
Variable Coefficient ◽  
Natural Frequencies ◽  
Boundary Value ◽  
Liouville Problem ◽  
Frequency Equation ◽  
Linear Potential ◽  
Linearly Independent ◽  
Sturm Liouville ◽  
Coefficient Of Elasticity

The problem of identifying the variable coefficient of elasticity of a medium with respect to natural frequencies of a string oscillating in this medium is considered. A method for solving the problem is found, based on the representation of linearly independent solutions of the differential equation in the form of Taylor series with respect to two variables, substituting them into the frequency equation, and determining the unknown coefficients of the linear function from this frequency equation. An analytical method has also been developed that allows one to prove the uniqueness or nonuniqueness of the restored polynomial coefficient of elasticity of a medium by a finite number of natural frequencies of oscillations of a string, and also to find a class of isospectral problems, that is, boundary value problems for which the eigenvalue spectra coincide. The latter is based on the method of variation of an arbitrary constant. We consider examples of finding isospectral classes, and also unique boundary value problems having a given spectrum.

scholarly journals Completeness of the system of eigenfunctions of the Sturm-Liouville problem with the singularity

2020 ◽  
Vol 30 (1) ◽  
pp. 59-63
Author(s):  
V.P. Tanana
Keyword(s):  
Thermal Conductivity ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Liouville Problem ◽  
Discontinuous Coefficients ◽  
Order Differential Operator ◽  
Temperature Conductivity ◽  
Inverse Boundary ◽  
Sturm Liouville ◽  
Inverse Boundary Value Problems

Mathematical modeling of composite materials plays an important role in modern technology, and the solution and study of inverse boundary value problems of heat transfer is impossible without the use of systems of eigenfunctions of the Sturm-Liouville problem for the differential equation with discontinuous coefficients. One of the most important properties of such systems is their completeness in the corresponding spaces. This property of systems allows to prove theorems of existence and uniqueness of both direct problems and inverse boundary value problems of thermal conductivity, and also to prove numerical methods of solving such problems. In this paper, we prove the completeness of the Sturm-Liouville problem in the space $L_2[r_0,r_2]$ for a second-order differential operator with a discontinuous coefficient. This problem arises when investigating and solving the inverse boundary problem of thermal conductivity for a hollow ball consisting of two balls with different temperature conductivity coefficients. Self-conjugacy, injectivity, and positive definiteness of this operator are proved.

scholarly journals Note on the conformable boundary value problems: Sturm’s theorems and Green’s function

2021 ◽  
Vol 67 (3 May-Jun) ◽  
pp. 471
Author(s):  
F. Martínez ◽  
I. Martínez ◽  
M. K. A. Kaabar ◽  
S. Paredes
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Boundary Value ◽  
Liouville Problem ◽  
Comparison Theorems ◽  
Conformable Derivative ◽  
Sturm Liouville ◽  
Linear Differential

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

scholarly journals Note on the Conformable Boundary Value Problems: Sturm’s Theorems and Green’s Function

2020 ◽  
Author(s):  
F. Martínez ◽  
Inmaculada Martínez ◽  
Mohammed K. A. Kaabar ◽  
Silvestre Paredes
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Boundary Value ◽  
Liouville Problem ◽  
Comparison Theorems ◽  
Conformable Derivative ◽  
Sturm Liouville ◽  
Linear Differential

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.

scholarly journals Degenerate boundary conditions on a geometric graph

2019 ◽  
Vol 485 (3) ◽  
pp. 272-275
Author(s):  
V. A. Sadovnichy ◽  
Ya. T. Sultanaev ◽  
A. M. Akhtyamov
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Liouville Problem ◽  
Geometric Graph ◽  
Entire Plane ◽  
Characteristic Determinant ◽  
Sturm Liouville ◽  
Shaped Graph

The boundary conditions of the Sturm-Liouville problem defined on a star-shaped geometric graph of three edges are studied. It is shown that if the lengths of the edges are different, then the Sturm-Liouville problem does not have degenerate boundary conditions. If the lengths of the edges and the potentials are the same, then the characteristic determinant of the Sturm-Liouville problem can not be equal to a constant different from zero. But the set of Sturm-Liouville problems for which the characteristic determinant is identically equal to zero is an infinite (continuum). In this way, in contrast to the Sturm-Liouville problem defined on an interval, the set of boundary-value problems on a star-shaped graph whose spectrum completely fills the entire plane is much richer. In the particular case when the minor A124 for matrix of coefficients is nonzero, it does not consist of two problems, as in the case of the Sturm-Liouville problem given on an interval, but of 18 classes, each containing two to four arbitrary constants.

scholarly journals A Study of the Eigenfunctions of the Singular Sturm–Liouville Problem Using the Analytical Method and the Decomposition Technique

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 415 ◽  
Author(s):  
Oktay Sh. Mukhtarov ◽  
Merve Yücel
Keyword(s):  
Boundary Value Problems ◽  
Differential Operators ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Problems ◽  
Adomian Decomposition ◽  
Interval Boundary ◽  
Sturm Liouville

The history of boundary value problems for differential equations starts with the well-known studies of D. Bernoulli, J. D’Alambert, C. Sturm, J. Liouville, L. Euler, G. Birkhoff and V. Steklov. The greatest success in spectral theory of ordinary differential operators has been achieved for Sturm–Liouville problems. The Sturm–Liouville-type boundary value problem appears in solving the many important problems of natural science. For the classical Sturm–Liouville problem, it is guaranteed that all the eigenvalues are real and simple, and the corresponding eigenfunctions forms a basis in a suitable Hilbert space. This work is aimed at computing the eigenvalues and eigenfunctions of singular two-interval Sturm–Liouville problems. The problem studied here differs from the standard Sturm–Liouville problems in that it contains additional transmission conditions at the interior point of interaction, and the eigenparameter λ appears not only in the differential equation, but also in the boundary conditions. Such boundary value transmission problems (BVTPs) are much more complicated to solve than one-interval boundary value problems ones. The major difficulty lies in the existence of eigenvalues and the corresponding eigenfunctions. It is not clear how to apply the known analytical and approximate techniques to such BVTPs. Based on the Adomian decomposition method (ADM), we present a new analytical and numerical algorithm for computing the eigenvalues and corresponding eigenfunctions. Some graphical illustrations of the eigenvalues and eigenfunctions are also presented. The obtained results demonstrate that the ADM can be adapted to find the eigenvalues and eigenfunctions not only of the classical one-interval boundary value problems (BVPs) but also of a singular two-interval BVTPs.

Boundary Value Problems of Electroelasticity with Concentrated Singularities

1994 ◽  
Vol 1 (5) ◽  
pp. 459-467
Author(s):  
T. Buchukuri ◽  
D. Yanakidi
Keyword(s):  
Singular Point ◽  
Boundary Value Problems ◽  
Power Function ◽  
Boundary Value ◽  
Linearly Independent

Abstract We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

scholarly journals On the analytic form of the discrete Kramer sampling theorem

2001 ◽  
Vol 25 (11) ◽  
pp. 709-715 ◽  
Author(s):  
Antonio G. García ◽  
Miguel A. Hernández-Medina ◽  
María J. Muñoz-Bouzo
Keyword(s):  
Orthogonal Polynomials ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Analytic Form ◽  
Sampling Theorem ◽  
Discrete Version ◽  
Moment Problems ◽  
Sturm Liouville ◽  
The Subject ◽  
Sampling Expansions

The classical Kramer sampling theorem is, in the subject of self-adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm-Liouville problems. In this paper a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.

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