Separated Boundary Conditions Research Articles

Oscillatory properties of selfadjoint boundary problems of the fourth order

A series of results and methods is presented, which make it possible to trace the relationship between the number of inner zeros of nontrivial solutions of fourth order selfadjoint boundary problems with separated boundary conditions and the negative inertia index.

Variational approach to a symmetric boundary value problem generated by a system of equations and separated boundary conditions

This work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second‐order Sturm–Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions.

Lyapunov-type inequalities for Hilfer fractional boundary value problems

This paper deals with fractional boundary value problems involving the Hilfer–Hadamard fractional differential operator of order [Formula: see text] and type [Formula: see text]. We derive the corresponding Lyapunov-type inequalities for two prominent classes of Hilfer–Hadamard fractional boundary value problems (HFBVPs) involving separated and anti-periodic boundary conditions. For this purpose, we construct the associated Green’s functions and deduce their important properties.

Асимптотики спектральных данных краевых задач четвертого порядка

In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods rely on the recent results concerning regularization and asymptotic analysis for higher-order differential operators with distribution coefficients. The results of this paper have applications to the theory of inverse spectral problems as well as a separate significance.

Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators

We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties.

The lower bounds of non-real eigenvalues for singular indefinite Sturm–Liouville problems

The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.

Uniqueness of the potential in conformable Sturm‐Liouville problem

This study is about the inverse problem of the conformable Sturm‐Liouville problem SLP. Namely, we first defined the problem with separable boundary conditions and gave some of its spectral properties. Then, by spectral data, we showed that the potential and the constants in the boundary conditions are coincident for two different problems. Here, the potential is in the space . We have also shown that if the potential function has the property of being symmetrical, only one spectrum is sufficient to find this function as uniquely. It is worth remembering that the methods and calculations used here are similar to those in the SLP with classical derivative. However, we think results will be valuable in Sturm‐Liouville theory.

A Comparison Result for the Nabla Fractional Difference Operator

This article establishes a comparison principle for the nabla fractional difference operator ∇ρ(a)ν, 1<ν<2. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green’s function. I prove that this Green’s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem.

Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions

Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions

Finite Spectrum of 2nth Order Boundary Value Problems with Transmission Conditions

For any positive integer 2n and any positive integer m, l, a class of regular self-adjoint and non-self-adjoint boundary value problems whose spectrum consists of at most eigenvalues is constructed. The key to this analysis is the division of intervals and an iterative construction of the characteristic function. In the self-adjoint case with separated boundary conditions this upper bound can be improved to

Inverse Sturm–Liouville problems with polynomials in nonseparated boundary conditions

An nonself-adjoint Sturm–Liouville problem with two polynomials in nonseparated boundary conditions are considered. It is shown that this problem have an infinite countable spectrum. The corresponding inverse problems is solved. Criterions for unique reconstruction of the nonself-adjoint Sturm-Liouville problem by eigenvalues of this problem and the spectral data of an additional problem with separated boundary conditions are proved. Schemes for unique reconstruction of the Sturm-Liouville problems with polynomials in nonseparated boundary conditions and corresponding examples are given

SPECTRAL DATA ASYMPTOTICS FOR THE HIGHER-ORDER DIFFERENTIAL OPERATORS WITH DISTRIBUTION COEFFICIENTS

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author’s knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance.

The numerical solution of inverse nodal problem for integro-differential operator by Legendre wavelet method

In this study, integro-differential equation under separated boundary conditions is considered. In fact, we study the inverse nodal problem, which includes differential part of Sturm–Liouville equation and Volterra type integral to obtain approximative solutions by Legendre polynomials. And, we use Legendre wavelets method to solve these types of problems. In addition, we present the error bound of for continuous derivatives with its approximative solution. Once and for all, the strength of the method can be seen in a few examples.

Формула первого регуляризованного следа для дифференциального оператора с разрывной весовой функцией

The author study the spectral properties of an eighth-order differential operator with a piecewise-smooth potential and a discontinuous weight function. For large values of the spectral parameter, the asymptotics of solutions of differential equations defining the operator under study is studied. With the help of the obtained asymptotics, the conditions of “conjugation” at the point of discontinuity of the coefficients, the necessity of which follows from physical considerations, are studied. The separated boundary conditions that define the operator are studied. An indicator diagram of an equation whose roots are the eigenvalues of the operator is investigated. The asymptotics of the eigenvalues of the differential operator under study is found. Using the Lidskyi - Sadovnichyi method, the first regularized trace of the differential operator is calculated.

A Two-Temperature Model Based on Fin-Approximation for Transient Longitudinal Heat Transfer in Unidirectional Composite

The reliability of a two-temperature model is assessed in the case of longitudinal heat transfer in unidirectional composites. One interest is that it makes it possible to apply separate boundary conditions or source terms on the fibre and the matrix (emissivity for example), without necessitating an explicit description of the fibre and matrix domains. For the sake of simplicity, the model under study is based on a fin-approximation in both fibre and matrix, which implies a high interfacial thermal resistance. The range of validity of this assumption is investigated by comparing the model to an axisymmetric one-temperature model, using non-dimension variables and Dirichlet boundary conditions. It turns out that this range of validity is strongly dependent on the parameters.

On a second order eigenvalue problem and its application

A linear eigenvalue problem governed by a second order differential equation with separate and general boundary conditions is considered and a new monotonicity result on the principal eigenvalue with respect to the coefficient of the advection term is established. The main approach is based on the functional proposed by Liu and Lou [16] and a key finding lies in the nice properties of the associated Fréchet operator when confined at suitable points and function spaces. As an application, this monotonicity result is used to study a class of competitive parabolic systems and the so-called “competitive exclusion principle” is observed in a larger parameter region than several existing works, which is a nontrivial improvement.

Enhancement of constant normal stiffness direct shear testing protocols for determining geomechanical properties of fractures

Discontinuity behaviour can have a large impact on geotechnical engineering design; therefore, it is essential to determine their geomechanical properties to predict rockmass behaviour and mitigate any potential failure that may affect personnel safety or damage property. Geotechnical numerical software programmes that discretely simulate discontinuities rely on direct shear laboratory tests to provide the properties needed as inputs to these tools. This study presents valuable direct shear laboratory test results for the mechanical properties and behaviours of fresh, unweathered, rough fractures in the Pointe du Bois granite. The testing programme considers three separate boundary conditions: constant normal stress (CNL*) and two variations of constant normal stiffness (CNS). A new machine stiffness model is proposed to estimate the machine stiffness for tunnelling applications. Measurements of shear strength and a proposed secant dilation angle are assessed and critically evaluated. It is found in this study that there is no significant difference between shear strength parameters determined from direct shear data from CNL* and CNS boundary conditions when comparing maximum strength and residual strength. In addition, it is found that the proposed secant dilation angle, critical shear displacement, and total dilation potential all have a negative relationship with increasing normal stress.

Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented.

Спектральные свойства дифференциального оператора четного порядка с разрывной весовой функцией

This article proposes a new method for studying differential operators with a discontinuous weight function. It is assumed that the potential of the operator is a piecewise smooth function on the segment of the operator definition. The conditions of «conjugation» at the point of discontinuity of the weight function are required. The spectral properties of a differential operator defined on a finite segment with separated boundary conditions are studied. The asymptotics of the fundamental system of solutions of the corresponding differential equation for large values of the spectral parameter is obtained. With the help of this asymptotics, the «conjugation» conditions of the differential operator in question are studied. The boundary conditions of the operator under study are investigated. As a result, we obtain an equation for the eigenvalues of the operator, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular polygon, is studied. In various sectors of the indicator diagram, the asymptotics of the eigenvalues of the investigated differential operator is found. The formula for the first regularized trace of this operator by using the found asymptotics of the eigenvalues by the Lidsky–Sadovnichy method is obtained. In the case of the passage to the limit, the resulting formula leads to the trace formula for the classical operator with a smooth potential and constant weight function.

Единственность восстановления оператора Штурма-Лиувилля со спектральным параметром, квадратично входящим в граничное условие

The work is devoted to the study of the inverse problem for the Sturm-Liouville operator with a real square-integrable potential. The boundary conditions are non-separated. One of these boundary conditions includes a quadratic function of the spectral parameter. A uniqueness theorem is proved and an algorithm for solving the inverse problem is constructed. As spectral data, we use the spectrum of the considered boundary value problem, the constant term of the quadratic function of the spectral parameter included in the boundary condition, and some special sequence of signs. From these spectral data, the characteristic function of the boundary value problem is first reconstructed in the form of an infinite product and the parameters of the boundary conditions, and then the problem is reduced to the inverse problem of reconstructing the potential of the Sturm-Liouville operator from the spectra of two boundary value problems with separated boundary conditions. The results of the article can be used for solving various versions of inverse problems of spectral analysis for differential operators, as well as for integrating some nonlinear equations of mathematical physics.