Properties Of Boundary Value Problems Research Articles

OSCILLATION PROPERTIES FOR NON-CLASSICAL STURM-LIOUVILLE PROBLEMS WITH ADDITIONAL TRANSMISSION CONDITIONS

This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP’s) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP’s). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm’s theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm’s classical theorems on comparison and oscillation.

Inverse Sturm-Liouville problems with two supplementary discontinuous conditions on two symmetric disjoint intervals

In this paper, we consider Sturm-Liouville problems on two symmetric disjoint intervals with two supplementary discontinuous conditions at an interior point. First, we investigate some spectral properties of boundary value problems, and obtain the asymptotic form of the eigenvalues and the eigenfunctions. Then, the eigenfunction expansion of Green’s function is presented and we prove the uniqueness theorems for the solution of the inverse problem, and reconstruct the Sturm-Liouville operator and the coefficients of boundary conditions using the Weyl m-function and spectral data. Also, numerical examples are presented.

Spectral properties of boundary-value problem with a shift for wave equation

We consider a differential operator determined by wave equation with potential in characteristic triangle, and boundary-value conditions with shift on the characteristics, and with oblique derivative on non-characteristic part of a boundary. We obtain condition for validity of the Volterra property, and show completeness of the root functions in the rest cases. We study basis property for the system of root functions under assumption that the potential depends on a single variable.

Abstract Differential Equations with VMO Coefficients in Half Space and Applications

In this paper, L p -separability properties of boundary value problems for elliptic and parabolic differential-operator equations with VMO coefficients are studied in half plane. In application, the maximal regularity properties of the anisotropic elliptic equation and system of equations with VMO coefficients are derived.

Some Properties of Boundary Value Problem for Radiative Transfer Equation with Diffuse Reflection and Refraction Conditions

We study the boundary value problem for the radiative transfer equation with diffuse reflection and refraction conditions. We prove the continuous dependence of the solution on the data and show that the resolving operator of the conjugate problem is the adjoint of the resolving operator of the original problem and , where Uf(ω, x) = f(−ω, x). Bibliography: 10 titles.

Spectral Analysis of Integro–Differential Equations in Viscoelasticity Theory

We study spectral properties of boundary value problems for integro-differential equations in viscoelasticity theory. The sum of N exponential functions with negative exponents is taken for the convolution kernel. We prove that the spectrum of the boundary value problem is the union of N sequences of real numbers and two sequences all elements, except for finitely many ones, are not real. In particular, we obtain conditions under which these two sequences contains no real numbers. Bibliography: 8 titles.

Uniqueness studies in boundary value problems involving some second gradient models

Enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies. New attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations, especially the ones not obeying a normality rule, induce ill-posed boundary value problems. However little work has been done to study the well-posedness properties of boundary value problems involving these enhanced models. The aim of this paper is a partial study of this point. Only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered. We extend to second gradient models a numerical search method to find non-uniqueness, already established for classical media. In the case of loss of uniqueness for the underlying model, our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well. It seems that second gradient extensions of classical models do not restore the uniqueness properties, but only the objectivity of the numerical computation. Finally our numerical experiments show once more that post-peak behavior is a structural effect. They also suggest that some constitutive equations often used in geomaterials modeling may be questionable, and may not be complete because even adding second gradient enhancement to these models yields non-unique solutions.

Uniqueness studies in boundary value problems involving some second gradient models

Enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies. New attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations, especially the ones not obeying a normality rule, induce ill-posed boundary value problems. However little work has been done to study the well-posedness properties of boundary value problems involving these enhanced models. The aim of this paper is a partial study of this point. Only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered. We extend to second gradient models a numerical search method to find non-uniqueness, already established for classical media. In the case of loss of uniqueness for the underlying model, our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well. It seems that second gradient extensions of classical models do not restore the uniqueness properties, but only the objectivity of the numerical computation. Finally our numerical experiments show once more that post-peak behavior is a structural effect. They also suggest that some constitutive equations often used in geomaterials modeling may be questionable, and may not be complete because even adding second gradient enhancement to these models yields non-unique solutions.

A multiresolution homogenization of modal analysis with application to layered media

We apply multiresolution techniques to study the effective properties of boundary value problems. Conditions under which boundary values are preserved in the effective (`homogenized') formulation are developed and discussed. Relations between the eigenfunctions and eigenvalues of the generic formulation and those of the effective formulation are explored. Applications to the construction of effective Green function in a complex lamination are discussed. The analytic results are demonstrated via numerical computations.