Discontinuity Curves Research Articles

Singularity extraction for elliptic equations with coefficients with jumps and Dirac sources

We explore the mathematical structure of the solution to an elliptic diffusion problem with point-wise Dirac sources. The conductivity parameter is space-varying, may have jumps and the Dirac sources may be located along the discontinuity curves of that parameter. The variational problem, issued by duality, is proven to be well posed using a sharp elliptic regularity result by Di-Giorgi [Mem. Accad. Sci. Torino, 3, 1957]. The paper is aimed at a key expansion into a split singular/regular contributions. The singular part is calculated by an explicit formula, while the regular correction can be computed as the solution to a standard variational Poisson problem. The latter can be successfully approximated by most of the numerical methods practiced nowadays. Some analytical examples are discussed at last to assess the minimality of the assumptions we use to establish our theoretical results.

An Algorithm for Construction of the Asymptotic Approximation of a Stable Stationary Solution to a Diffusion Equation System with a Discontinuous Source Function

An algorithm is presented for the construction of an asymptotic approximation of a stable stationary solution to a diffusion equation system in a two-dimensional domain with a smooth boundary and a source function that is discontinuous along some smooth curve lying entirely inside the domain. Each of the equations contains a small parameter as a factor in front of the Laplace operator, and as a result, the system is singularly perturbed. In the vicinity of the curve, the solution of the system has a large gradient. Such a problem statement is used in the model of urban development in metropolitan areas. The discontinuity curves in this model are the boundaries of urban biocenoses or large water pools, which prevent the spread of urban development. The small parameter is the ratio of the city’s outskirts linear size to the whole metropolis linear size. The algorithm includes the construction of an asymptotic approximation to a solution with a large gradient at the media interface as well as the steps for obtaining the existence conditions. To prove the existence and stability theorems, we use the upper and lower solutions, which are constructed as modifications of the asymptotic approximation to the solution. The latter is constructed using the Vasil’yeva algorithm as an expansion of a small parameter exponent.

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments.

The Numerical Solution of the External Dirichlet Generalized Harmonic Problem for a Sphere by the Method of Probabilistic Solution

In the present paper, an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given, where “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. The algorithm consists of the following main stages: (1) the transition from an infinite domain to a finite domain by an inversion; (2) the consideration of a new Dirichlet generalized harmonic problem on the basis of Kelvin’s theorem for the obtained finite domain; (3) the numerical solution of the new problem for the finite domain by the MPS, which in turn is based on a computer simulation of the Weiner process; (4) finding the probabilistic solution of the posed generalized problem at any fixed points of the infinite domain by the solution of the new problem. For illustration, numerical examples are considered and results are presented.

A fault detection method based on partition of unity and kernel approximation

In this paper, we present a scattered data approximation method for detecting and approximating the discontinuities of a bivariate function and its gradient. The new algorithm is based on partition of unity, polyharmonic kernel interpolation, and principal component analysis. Localized polyharmonic interpolation in partition of unity setting is applied for detecting a set of fault points on or close to discontinuity curves. Then a combination of partition of unity and principal component regression is used to thinning the detected points by moving them approximately on the fault curves. Finally, an ordered subset of these narrowed points is extracted and a parametric spline interpolation is applied to reconstruct the fault curves. A selection of numerical examples with different behaviors and an application for solving scalar conservation law equations illustrate the performance of the algorithm.

Inverse Problem for the Integral Dynamic Models with Discontinuous Kernels

The objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves which correspond to integration bounds. A direct method of discretization with a posteriori verification of calculations is proposed. The family of quadrature rules is employed for approximation purposes. It is shown that the arithmetic complexity of the proposed numerical method is O(N3). The method has first-order convergence. A generalization of the method is also proposed for the case of an arbitrary number of discontinuity curves. The illustrative examples are included to demonstrate the efficiency and accuracy of proposed solver.

Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution

In this article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve inx−yx-yplane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.

Crossing limit cycles for discontinuous piecewise differential systems formed by linear Hamiltonian saddles or linear centers separated by a conic

The extension of the 16th Hilbert problem to discontinuous piecewise linear differential systems asks for an upper bound for the maximum number of crossing limit cycles that such systems can exhibit. The study of this problem is being very active, especially for discontinuous piecewise linear differential systems defined in two zones and separated by one straight line. In the case that the differential systems in these zones are formed either by linear centers or linear Hamiltonian saddles it is known that there are no crossing limit cycles. However it is also known that the number of crossing limit cycles can change if we change the shape of the discontinuity curve. In this paper we study the maximum number of crossing limit cycles of discontinuous piecewise differential systems formed by either linear Hamiltonian saddles or linear centers and separated by a conic which intersect the conic in two points. For this class of discontinuous piecewise differential systems we solve the extended 16th Hilbert problem.

ASYMPTOTICS OF A MULTIZONAL INTERNAL LAYER SOLUTION TO A PIECEWISE-SMOOTH SINGULARLY PERTURBED EQUATION WITH A TRIPLE ROOT OF THE DEGENERATE EQUATION

A singularly perturbed boundary value problem for a stationary equation of reaction-diffusion type in the case when reactive term undergoes discontinuity along some curve that is independent of the small parameter is studied. This is a new class of problems with triple roots of the degenerate equation, which leads to the formation of complex multizonal internal layers in the neighborhood of the discontinuity curve. By the method of asymptotic differential inequalities and matching asymptotic expansion, the existence of a contrast structure solution is proved. Using a different modified boundary layer function method, the asymptotic representation of point itself and this solution are constructed.

Study on Gravity Sedimentation of Multicomponent Hygroscopic Aerosols in Reactor Severe Accident

In the event of a severe accident, the multicomponent hygroscopic aerosols in the containment will absorb water under the high humidity condition, thereby influencing the gravity sedimentation behavior. In this study, a physical model of the equilibrium particle diameter of the multicomponent hygroscopic aerosol particles was developed through theoretical analysis, and it was also validated by experimental results. The model focuses on the effect of solubility on the hygroscopic process and explains the reason why the multicomponent hygroscopic particles grow along a discontinuity curve. Based on a typical gigawatt-class pressurized water reactor, the effects of relative humidity, dry particle diameter and mass fraction of hygroscopic components on the removal coefficient of the gravity sedimentation were investigated. The results show that the velocity of the gravity sedimentation will significantly increase, only if the aerosol particles grow to a certain degree. Only when the humidity is more than a certain value, the sedimentation process of the pure hygroscopic aerosol particles with a dry particle diameter exceeding 0.1 μm will accelerate due to hygroscopicity, and this humidity limit will decrease as the dry particle diameter increases. With the progress of the accident, the mass fraction of non-hygroscopic components in the aerosol particles gradually decreases, leading the above-mentioned humidity limit increasing and the acceleration of gravity sedimentation due to the hygroscopicity decreasing in the same humidity.

Existence and stability of periodic contrast structure in reaction-advection-diffusion equation with discontinuous reactive and convective terms

In this project, we study the periodic Dirichlet boundary value problem for a singularly perturbed reaction-advection-diffusion equation on the segment in case of discontinuous reactive and convective terms. Applying the boundary function method, we construct the asymptotic approximation of the periodic solution with internal transition layer located in the vicinity of a curve of discontinuity of the mentioned terms. For the problem here we prove the existence of the periodic solution, estimate the accuracy of the asymptotical approximation and investigate the stability of the periodic solution as solutions of the corresponding initial boundary value problems for the reaction-advection-diffusion equation.

Crossing limit cycles for a class of piecewise linear differential centers separated by a conic

In previous years the study of the version of Hilbert's 16th problem for piecewise linear differential systems in the plane has increased. There are many papers studying the maximum number of crossing limit cycles when the differential system is defined in two zones separated by a straight line. In particular in [11,13] it was proved that piecewise linear differential centers separated by a straight line have no crossing limit cycles. However in [14,15] it was shown that the maximum number of crossing limit cycles of piecewise linear differential centers can change depending of the shape of the discontinuity curve. In this work we study the maximum number of crossing limit cycles of piecewise linear differential centers separated by a conic.differential centers separated by a conic
 For more information see https://ejde.math.txstate.edu/Volumes/2020/41/abstr.html

The Riemann solutions to the compressible ideal fluid flow

In this paper, we study the exact solutions of Riemann problem for the compressible ideal fluid flow, where the external force is a continuous function of time. The exact expressions for contact discontinuity curve, shock wave curve and rarefaction wave curve are provided. Six types of exact solutions of the Riemann problem are also obtained. In particular, a vacuum arises at t>0 although the initial data never involves the vacuum, and these solutions do not possess self-similarity in the (t,x)-plane.

A Research for Implementing Image Interpolation using Inpainting and Shearlet Transform

This paper proposes a way to improve the compression ratio of images by expunging some parts of the image prior transmission. The remaining data besides essential details for recovering the removed regions are encoded to produce the final data. At the decoding side an inpainting method is applied to retrieve the removed region. The Shearlet Transform is used for the smoothing purpose of the recovered image. This transform can identify the location of singularities of a function and also the orientation of discontinuity curves. The Shearlet Transform has the ability to provide a very accurate geometrical characterization of general discontinuity occurring in images.

Асимптотичний аналіз сингулярно збуреного рівняння Кортевега-де Фріза

The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

The method of probabilistic solution for 3D Dirichlet ordinary and generalized harmonic problems in finite domains bounded with one surface

The Dirichlet ordinary and generalized harmonic problems for some 3D finite domains are considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. An algorithm of numerical solution by the method of probabilistic solution (MPS) is given, which in its turn is based on a computer simulation of the Wiener process. Since, in the case of 3D generalized problems there are none exact test problems, therefore, for such problems, the way of testing of our method is suggested. For examining and to illustrate the effectiveness and simplicity of the proposed method five numerical examples are considered on finding the electric field. In the role of domains are taken ellipsoidal, spherical and cylindrical domains and both the potential and strength of the field are calculated. Numerical results are presented.

Adaptive anisotropic Petrov–Galerkin methods for first order transport equations

This paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of Petrov–Galerkin type. The key issues can be summarized as follows. The variational formulations allow us to employ meshes with cells of arbitrary aspect ratios. We develop a refinement scheme generating highly anisotropic partitions that is inspired by shearlet systems. We establish L2 approximation rates for N-term approximations from corresponding piecewise polynomials for certain compact cartoon classes of functions. In contrast to earlier results in a curvelet or shearlet context the cartoon classes are concisely defined through certain characteristic parameters and the dependence of the approximation rates on these parameters is made explicit here. The approximation rate results serve then as a benchmark for subsequent applications to adaptive Galerkin solvers for transport equations. We outline a new class of directionally adaptive, Petrov–Galerkin discretizations for such equations. In numerical experiments, the new algorithms track C2 discontinuity curves stably and accurately, and realize essentially optimal rates. Finally, we treat parameter dependent transport problems, which arise in kinetic models as well as in radiative transfer. In heterogeneous media these problems feature propagation of singularities along curved characteristics precluding, in particular, fast marching methods based on ray-tracing. Since now the solutions are functions of spatial variables and parameters one has to address the curse of dimensionality. As for non-adaptive schemes considered in Grella and Schwab (2011) and Grella (2014), we show computationally, for a model parametric transport problem in heterogeneous media in 2+1 dimension, that sparse tensorization of the presently proposed directionally adaptive variational discretization scheme in physical space combined with hierarchic collocation in ordinate space can overcome the curse of dimensionality when approximating averaged bulk quantities.

$$C^*$$ C ∗ -algebras of Bergman Type Operators with Piecewise Continuous Coefficients Over Domains with Dini-Smooth Corners

For any simply connected domain \(U\subset \mathbb {C}\) with a piecewise Dini-smooth boundary \(\partial U\) which admits Dini-smooth corners of openings lying in \((0,2\pi ]\), the \(C^*\)-algebra \({\mathfrak B}_U=\mathrm{alg}\{aI,B_U,\widetilde{B}_U:a\in PC({\mathfrak L})\}\) generated by the operators of multiplication by piecewise continuous functions with discontinuities on a finite union \({\mathfrak L}\subset U\) of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to \(\partial U\) at the points of \({\mathfrak L}\cap \partial U\), and by the Bergman projection \(B_U\) and anti-Bergman projection \(\widetilde{B}_U\) acting on the Lebesgue space \(L^2(U)\) is studied. A Fredholm symbol calculus for the \(C^*\)-algebra \({\mathfrak B}_U\) is constructed and a Fredholm criterion for the operators \(A\in {\mathfrak B}_U\) in terms of their symbols is established. Results essentially depend on openings of corners and angles between curves of discontinuity of coefficients at the points \(z\in \partial U\cap {\mathfrak L}\).

Investigation and numerical solution of some 3D internal Dirichlet generalized harmonic problems in finite domains

A Dirichlet generalized harmonic problem for finite right circular cylindrical domains is considered. The term “generalized” indicates that a boundary function has a finite number of first kind discontinuity curves. It is shown that if a finite domain is bounded by several surfaces and the curves are placed in arbitrary form, then the generalized problem has a unique solution depending continuously on the data. The problem is considered for the simple case when the curves of discontinuity are circles with centers situated on the axis of the cylinder. An algorithm of numerical solution by a probabilistic method is given, which in its turn is based on a computer simulation of the Wiener process. A numerical example is considered to illustrate the effectiveness and simplicity of the proposed method.

Shearlet approximation of functions with discontinuous derivatives

We demonstrate that shearlet systems yield superior N-term approximation rates compared with wavelet systems of functions whose first or higher order derivatives are smooth away from smooth discontinuity curves. We will also provide an improved estimate for the decay of shearlet coefficients that intersect a discontinuity curve non-tangentially.