Piecewise Constant Coefficients Research Articles

О деформировании решетчатой пластинки глаза

Several problems pertaining to deformation of the circular elastic isotropic plate of variable rigidity in the presence of elastic support on the edge were study. In the first problem Timoshenko model was considered. In the second problem the deformation of a multilayer plate with piecewise constant Lame coefficient along the thickness coordinate which are assumed to be constant within one layer was researched, the Zig-zag method is used for modeling. The basis for constructing a solution for both problems is the use of Lagrange’s variation principle of plates, numerical solution constructed using the Ritz method. During the research work the deflection and inflection points of the lamina cribrosa were compared for a healthy eye and for an eye with different stages of primary open-angle glaucoma for constant Lame coefficient as well as for their exponentially decreasing above. The problem of deflection of the lamina cribrosa when tacking into account the prelaminar layer was solved based on the Zig-zag method, the solution to this problem was also compared for a healthy eye and for an eye with various stages of primary open-angle glaucoma.

Generalizations of the Jacobi identity to the case of the Lauricella function F D ( N )

ABSTRACT The paper considers the issue of constructing differential relations for the Lauricella hypergeometric function F D ( N ) . The formulas found give explicit expressions for the derivative of the product of the function F D ( N ) and some binomials through the product of other binomials and a combination of adjacent Lauricella functions whose parameters differ by 1 or − 1 . The derived differential relations generalize the Jacobi identity and other differential formulas known in the theory of the Gauss hypergeometric function on the multidimensional case. With special parameter values F D ( N ) , the right-hand side of such Jacobi-type formulas are simplified and have the form of a product of binomials and a polynomial. Such formulas can be used to transform a Cauchy-type integral to the form of the Schwartz–Christoffel integral and have an application to the Riemann–Hilbert problem with piecewise constant coefficients.

Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients

We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.

ON AN INVERSE SCATTERING PROBLEM FOR THE QUADRATIC PENCIL OF THE STURM-LIOUVILLE OPERATOR WITH A PIECE-WISE CONSTANT COEFFICIENT

In this work, direct and inverse scattering problems on the real axis for the quadratic pencil of the Sturm-Liouville operator with piece-wise constant coefficient are studied. The new integral representations for solutions are given, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proven, the necessary and sufficient conditions for recovering of the potentials are examined.

Bounded Solutions of a Difference Equation with Piecewise Constant Operator Coefficients

Bounded Solutions of a Difference Equation with Piecewise Constant Operator Coefficients

A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.

Mixed Contact Problem for the Lamé System with Piecewise Constant Main Coefficients

Mixed Contact Problem for the Lamé System with Piecewise Constant Main Coefficients

Interaction of Boundary Singular Points in an Elliptic Boundary Value Problem

The paper continues the construction of the Lp-theory of elliptic Dirichlet and Neumann boundary value problems with discontinuous piecewise constant coefficients in divergent form for an unbounded domain R2 with a piecewisesmooth noncompact Lipschitz boundary and C1 smooth discontinuity lines of the coefficients. An earlier constructed Lp-theory is generalized to the case of different smallest eigenvalues corresponding to a finite and an infinite singular point, and the effect of their interaction is further studied in the class of functions with first derivatives from Lp( ) in the entire range of the exponent p (1, )..

High-Order Grid-Characteristic Method for Systems of Hyperbolic Equations with Piecewise Constant Coefficients

High-Order Grid-Characteristic Method for Systems of Hyperbolic Equations with Piecewise Constant Coefficients

Algebraic Multigrid Using a Stencil–CSR Hybrid Format on GPUs

We propose a new sparse matrix format which captures the matrix structure typical for discretized partial differential equations with piecewise-constant coefficients. The format uses a stencil representation for some blocks of matrix rows, typically corresponding to regions with constant coefficients, whereas other rows are encoded in the general compressed sparse row format. The stencil representation saves memory and is suitable for SIMD-like parallelism as available on GPUs. Further, this format is well suited for the implementation of algebraic multigrid methods, and we present a proof-of-concept GPU-accelerated aggregation-based algebraic multigrid solver based on this format. This solver is compared on a few model problems (2-dimension and 3-dimension Poisson-like) with the compressed sparse row–based solver AmgX from NVIDIA and with the CUDA version of the BoomerAMG solver. For the considered tested problems with one million unknowns or more, the presented solver outperforms AmgX and BoomerAMG in terms of both run time and memory usage, and the performance gap increases with the system size.

On the Jost Solutions of A Class of the Quadratic Pencil of the Sturm-Liouville Equation

In this study we construct new integral representations of Jost-type solutions of the quadratic pencil of the Sturm-Liouville equation with the piece-wise constant coefficient on the entire real line. Our aim is to express the special solutions of the Sturm-Liouville quadratic pencil in the form of some integral operators which kernels is related with the potential function of the Sturm-Liouville equation. This problem is technically diffucult due to the discontinous coefficient which causes the kernel function to also have a jump discontinuity.

Constructing basises in solution space of the system of equations for the Lauricella Function FD (N)

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function , depending on N complex variables and having complex parameters , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points and in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons.

Reduced order modeling for elliptic problems with high contrast diffusion coefficients

We consider a parametric elliptic PDE with a scalar piecewise constant diffusion coefficient taking arbitrary positive values on fixed subdomains. This problem is not uniformly elliptic, as the contrast can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is com- monly made on parametric elliptic PDEs. We construct reduced model spaces that approximate uniformly well all solutions with estimates in relative error that are independent of the contrast level. These estimates are sub-exponential in the reduced model dimension, yet exhibiting the curse of dimensionality as the number of subdomains grows. Similar es- timates are obtained for the Galerkin projection, as well as for the state estimation and parameter estimation inverse problems. A key ingredient in our construction and analysis is the study of the convergence towards limit solutions of stiff problems when diffusion tends to infinity in certain domains.

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

Abstract We study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacle Ω − ⊂ R d \Omega^{-}\subset\mathbb{R}^{d} , d = 2 , 3 d=2,3 . By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on ∂ Ω − \partial\Omega^{-} and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

Convergence for an Immersed Finite Volume Method for Elliptic and Parabolic Interface Problems

In this article we analyze an immersed interface finite volume method for second order elliptic and   parabolic interface problems. We show the optimal convergence of the elliptic interface problem in L^2 and energy norms.
 For the parabolic interface problem, we prove the optimal order in L^2 and energy norms for piecewise constant and variable diffusion coefficients respectively. Furthermore, for the elliptic interface problem, we demonstrate super convergence at element nodes when the diffusion coefficient  is a piecewise constant.  Numerical examples  are also provided to confirm the optimal error estimates.

Bounded Solutions of a Differential Equation with Piecewise Constant Operator Coefficients

Bounded Solutions of a Differential Equation with Piecewise Constant Operator Coefficients

Boundary and interface methods for energy stable finite difference discretizations of the dynamic beam equation

We consider energy stable summation by parts finite difference methods (SBP-FD) for the homogeneous and piecewise homogeneous dynamic beam equation (DBE). Previously the constant coefficient problem has been solved with SBP-FD together with penalty terms (SBP-SAT) to impose boundary conditions. In this work, we revisit this problem and compare SBP-SAT to the projection method (SBP-P). We also consider the DBE with discontinuous coefficients and present novel SBP-SAT, SBP-P, and hybrid SBP-SAT-P discretizations for imposing interface conditions. To demonstrate the methodology for two-dimensional problems, we also present a discretization of the piecewise homogeneous dynamic Kirchoff-Love plate equation based on the hybrid SBP-SAT-P method. Numerical experiments show that all methods considered are similar in terms of accuracy, but that SBP-P can be more computationally efficient (less restrictive time step requirement for explicit time integration methods) for both the constant and piecewise constant coefficient problems.

Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|<\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Modeling of Mechanisms of Wave Formation for COVID-19 Epidemic

Two modifications with variable coefficients of the well-known SEIR model for epidemic development in the application to the modeling of the infection curves of COVID-19 are considered. The data for these models are information on the number of infections each day obtained from the Johns Hopkins Coronavirus Resource Center database. In our paper, we propose special methods based on Tikhonov regularization for models’ identification on the class of piecewise constant coefficients. In contrast to the model with constant coefficients, which cannot always accurately describe some of infection curves, the first model is able to approximate them for different countries with an accuracy of 2–8%. The second model considered in the article takes into account external sources of infection in the form of an inhomogeneous term in one of the model equations and is able to approximate the data with a slightly better accuracy of 2–4%. For the second model, we also consider the possibility of using other input data, namely the number of infected people per day. Such data are used to model infection curves for several waves of the COVID-19 epidemic, including part of the Omicron wave. Numerical experiments carried out for a number of countries show that the waves of external sources of infection found are ahead of the wave of infection by 10 or more days. At the same time, other piecewise constant coefficients of the model change relatively slowly. These models can be applied fairly reliably to approximate many waves of infection curves with high precision and can be used to identify external and hidden sources of infection. This is the advantage of our models.

Крайова задача з м’якими межами для рівнянь параболічного типу з операторами Бесселя — Ейлера — Лежандра на сегменті кусково-однорідної полярної осі

Composite materials are widely used in a variety of technological process-es, building, energy saving, in connection with which there is a need to pose and solve problems of thermal conductivity in environments that are heteroge-neous in their structure (multilayer bodies). At the same time, the heterogeneity of the medium leads to the consideration of boundary value problems with piecewise continuous or piecewise constant coefficients [3] and differential op-erators of the Bessel, Euler, Legendre, and Fouriertype, which model the het-erogeneity of the medium in terms of a geometric variable.In the classical setting, the processes of heat propagation were studied under the assumption that the boundary of the medium is rigid in relation to the reflection of waves. However, if we assume that wave absorption can occur at the boundary of the medium (soft boundary), we obtain a boundary value problem containing a time derivative in the operators of boundary conditions and conjugation conditions of the form (1).The analytical solution of the corresponding boundary value problem can be obtained using integral transformations with a spectral parameter, which work for problems with soft boundaries according to the same logi-cal scheme as integral transformations without a spectral parameter in problems with hard boundaries.This paper is devoted to the construction of one class of such hybrid in-tegral transformations generated by a hybrid differential operator of the Bessel-Euler-Legendre type on the polar axis.In this article an integral image of the exact analytical solution of a mixed problem for parabolic equations on a three-complex segment of the polar axis with soft boundaries is obtained by the method of a hybrid inte-gral transformation of the Bessel-Euler-Legendre type under the assump-tion that the boundary conditions and conjugation conditions contain a de-rivative with respect to the time variable