Discontinuous Operators Research Articles

Investigation of the Spectrum of Nonself-Adjoint Discontinuous Sturm-Liouville Operator

In this paper, we study nonself-adjoint Sturm-Liouville operator containing both the discontinuous coefficient and discontinuity conditions at some point on the positive half-line. The eigenvalues and the spectral singularities of this problem are examined and it is proved that this problem has a finite number of spectral singularities and eigenvalues with finite multiplicities under two different additional conditions. Furthermore, the principal functions corresponding to the eigenvalues and the spectral singularities of this operator are determined.

High‐order in‐cell discontinuous reconstruction path‐conservative methods for nonconservative hyperbolic systems–DR.MOOD method

AbstractIn this work, we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in‐cell discontinuous reconstruction operator are the key points to develop a new family of high‐order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two‐Layer Shallow Water system.

A Birkhoff–Kellogg Type Theorem for Discontinuous Operators with Applications

By means of fixed point index theory for multivalued maps, we provide an analogue of the classical Birkhoff–Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in detail this fact for a class of second-order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory.

Discontinuous optimal operator for the P1 Galerkin method approximation of elliptic PDEs with a weakly regular source term

Discontinuous optimal operator for the P1 Galerkin method approximation of elliptic PDEs with a weakly regular source term

Inverse problems for the eigenparameter Dirac operator with complex weight

Abstract Inverse spectral problems are considered for the discontinuous Dirac operator with complex-value weight and the spectral parameter boundary conditions. We investigate some properties of spectral characteristics and show that the potential can be uniquely determined by the Weyl-type function or by two spectra on the whole interval.

On determination of discontinuous Sturm-Liouville operator fromWeyl function

In this paper, the Weyl function for the Sturm-Liouville operator which contains the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval is defined and examined. The uniqueness theorem of solution of the inverse spectral problem for the discontinuous Sturm-Liouville operator according to Weyl function is proved.

Inverse problems for discontinuous Dirac operator with eigenparameter dependent boundary and transmission conditions

AbstractIn this study, we consider the discontinuous Dirac equations system with eigenparameter dependent boundary and finite number of transmission conditions. First, the space that corresponds to problem is introduced, the norm on this space is defined and the operator model that corresponds to the given problem is constructed on this space. Then the integral equations and asymptotics of eigenfunctions of the problem are obtained. The characteristic function is defined and the asymptotic formula of the characteristic function is given by using obtained asymptotics of eigenfunctions. After the Weyl solution and the Weyl function of the problem are formed. Finally, some uniqueness theorems are proved by using Weyl function and some spectral data.

Control and perturbation in Sturm — Liouville’s problem with discontinuous nonlinearity

We consider the Sturm — Liouville problem with discontinuous nonlinearity, control and perturbation. Previously obtained results for equations with a spectral parameter and a discontinuous operator are applied to this problem. By the variational method, we have established theorems on the existence of solutions to the Sturm — Liouville problem with discontinuous nonlinearity and to the optimal control problem, as well as on topological properties of the set of the acceptable “control — state” pairs. A one-dimensional analog of the Gol’dshtik model for separated flows of an incompressible fluid with control and perturbation is given as an application.

Spectral analysis of discontinuous Sturm-Liouville operators with Herglotzs transmission

<abstract><p>In this paper, we study the spectral properties of the Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions. In details, we introduce a Hilbert space formula, so that the problem we consider can be interpreted as an eigenvalue problem of an self-adjoint operator. Moreover, the Green's function and the resolvent of the related linear operator are obtained.</p></abstract>

RESTORATION OF DISCONTINUOUS FUNCTIONS BY DISCONTINUOUS INTERLINATION SPLINES

Context. The problem of development and research of methods for approximation of discontinuous functions by discontinuous interlination splines and its further application to problems of computed tomography. The object of the study was the modeling of objects with a discontinuous internal structure.
 Objective. The aim of this study is to develop a general method for constructing discontinuous interlining polynomial splines, which, as a special case, include discontinuous and continuously differentiated splines.
 Method. Modern methods of restoring functions are characterized by new approaches to obtaining, processing and analyzing information. There is a need to build mathematical models in which information can be represented not only by function values at points, but also in the form of a set of function traces on planes or straight lines.
 At the same time, practice shows that among the multidimensional objects that need to be investigated, more problems are described by a discontinuous functions.
 The paper develops a general method for constructing discontinuous interlining polynomial splines, which, as a special case, include discontinuous and continuously differentiable splines. It is considered that the domain of the definition of the required twodimensional function is divided into rectangular elements. Theorems on interlination and approximation properties of such discontinuous constructions are formulated and proved. The method is developed for approximating discontinuous functions of two variables based on the constructed discontinuous splines. The input data are the traces of an unknown function along a given system of mutually perpendicular straight lines. The proposed method has not only theoretical significance but also practical application in the IT domain, especially in computing tomography, allowing more accurately restore the internal structure of the body.
 Results. The discontinuous interlination operator from known traces of the function of two variables on a system of mutually perpendicular straight lines is researched.
 Conclusions. The functions of two variables that are discontinuous at some points or on some lines are better approximated by discontinuous spline interlinants. At the same time, equally high approximation estimates can be obtained. The results obtained have significant advantages over existing methods of interpolation and approximation of discontinuous functions. In further research, the authors plan to develop a theory of discontinuous splines on areas of complex shape bounded by arcs of known curves.

Approximation of discontinuous inverse operators with neural networks

In this work we deal with parametric inverse problems, which consist in recovering a finite number of parameters describing the structure of an unknown object, from indirect measurements. State-of-the-art methods for approximating a regularizing inverse operator by using a dataset of input–output pairs of the forward model rely on deep learning techniques. In these approaches, a neural network (NN) is trained to predict the value of the sought parameters directly from the data. In this paper, we show that these methods provide suboptimal results when a regularizing inverse operator is discontinuous with respect to the Euclidean topology. Hence, we propose a two-step strategy for approximating it by means of a NN, which works under general topological conditions. First, we embed the parameters into a subspace of a low-dimensional Euclidean space; second, we use a NN to approximate a homeomorphism between the subspace and the image of the parameter space through the forward operator. The parameters are then retrieved by applying the inverse of the embedding to the network predictions. The results are shown for the problem of x-ray imaging of solar flares with data from the Spectrometer/Telescope for Imaging X-rays. In this case, the parameter space is homeomorphic to a Moebius strip. Our simulation studies show that the use of a NN for predicting the parameters directly from the data yields systematic errors due to the non-Euclidean topology of the parameter space. The proposed strategy overcomes the discontinuity issues and furnishes stable and accurate reconstructions.

Multiple Solutions for a Class of BVPs of Second-Order Discontinuous Differential Equations with Impulse Effects

This paper deals with a class of boundary value problems of second-order differential equations with impulses and discontinuity. The existence of single or multiple positive solutions to discontinuous differential equations with impulse effects is established by using the nonlinear alternative of Krasnoselskii’s fixed point theorem for discontinuous operators on cones. Finally, an example is given to illustrate the main results.

Dynamical System of a Quadratic Stochastic Operator with Two Discontinuity Points

We consider a population consisting of two species whose dynamics is determined by a quadratic stochastic operator with variable coefficients, which makes it a discontinuous operator at two points. This operator depends on three parameters. The set of these parameters is divided into seven subsets. For each subset of parameters, we find fixed points, periodic points, and the set of limit points of trajectories generated by the respective quadratic stochastic operators.

The weak eigenfunctions of boundary-value problem with symmetric discontinuities

Abstract The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions. We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.

On the basis properties of a system of eigenfunctions of a spectral problem for a second-order discontinuous differential operator in the weighted grand-Lebesgue space with a general weight

The question of the basis property of a system of eigenfunctions of one spectral problem for a discontinuous second-order differential operator with a spectral parameter under discontinuity conditions is considered in the weighted grand-Lebesgue spaces Lp),?(0, 1), 1 < p < +?, with a general weight ?(?). These spaces are non-separable and therefore it is necessary to define its subspace associated with differential equation. In this paper, using the shift operator, a subspace Gp),?(0, 1) is considered, in which the basis property of exponentials and trigonometric systems of sines and cosines is established when the weight function ?(?) satisfies the Muckenhoupt condition. It is proved that the system of eigenfunctions and associated functions of the discontinuous differential operator corresponding to the given problem forms a basis in the weighted space Gp),?(0, 1) ? C,1 < p < +? with the weight ?(?) satisfying the Muckenhoupt condition. The question of the defect basis property of the system of eigenfunctions and associated functions of the given problem in the weighted spaces Gp),?(0, 1),1 < p < +?, is considered.

Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

<abstract><p>In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.</p></abstract>

Approximate controllability of neutral delay integro-differential inclusion of order $ \alpha\in (1, 2) $ with non-instantaneous impulses

<p style='text-indent:20px;'>This paper aims to establish the approximate controllability results for fractional neutral integro-differential inclusions with non-instantaneous impulse and infinite delay. Sufficient conditions for approximate controllability have been established for the proposed control problem. The tools for study include the fixed point theorem for discontinuous multi-valued operators with the <inline-formula><tex-math id="M3">\begin{document}$ \alpha- $\end{document}</tex-math></inline-formula>resolvent operator. Finally, the proposed results are illustrated with the help of an example.</p>

RECONSTRUCTION OF THE TWO VARIABLES DISCONTINUOUS FUNCTION BY DIFFERENT INFORMATION OPERATORS USING TRIANGULAR ELEMENTS

The paper examines methods for constructing mathematical models of two variables discontinuous functions using various information about them: one-sided values at points and one-sided traces along a given system of lines. The case is considered when the domain of the required function is triangulated by right-angled triangles. If interpolation or approximation methods are used, then for their construction the values of the function at given points must be given; if we use interlination methods, then traces of the desired function along a given system of lines. In this work, we construct a discontinuous interpolation and approximation splines for approximating a discontinuous function of two variables with given one-sided values in a given system of points (in our case, at the vertices of right-angled triangles), and prove theorems on the estimation of the approximation error by constructed discontinuous structures. In the paper a discontinuous interlination spline, which uses completely different information about the discontinuous function, namely one-sided traces along a given system of lines (in our case, along the sides of right-angled triangles) is also built. Interlination of functions can find wide application in the aircraft and automobile body design automation; when receiving and processing the results of sonar and radar, when solving problems of computed tomography, in digital signal processing and in many other areas. In the paper theorems on the integral form and an estimate of the approximation error by the constructed discontinuous interlination operator are also proved. Computational experiments that compare the results of the approximation of a discontinuous function of two variables by different information operators using triangular elements are presented. In the future, it is planned to apply the constructed operators of discontinuous approximation and interlination to solve a two-dimensional problem of computed tomography with a significant use of the inhomogeneity of the internal structure of the body, which must be reconstructed.

Development and analysis of entropy stable no-slip wall boundary conditions for the Eulerian model for viscous and heat conducting compressible flows

Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Phys A Stat Mech Appl 506:350–375, 2018). The spatial discretization is based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier–Stokes models. The numerical results obtained with the two models are compared, and similarities and differences are then highlighted. However, the differences are very small and probably smaller than what the current experimental technology allows to measure.

On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

<p>A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ) type non-expansive mappings is established.</p>