Stationary Boundary Value Problem Research Articles

SIMULATION OF DYNAMIC PROCESSES IN A p - i - n DIODE BY THE METHODS OF PERTURBATION THEORY

In the paper a mathematical model of the dynamic process of the electron-hole plasma formation in the active region ( i - region) of a semiconductor p - i - n - diode in the direct bias mode with a harmonic signal fed on the diode is developed. The basis of the model is a nonlinear nonstationary sin- gularly perturbed boundary value problem for a system of continuity equations of electron-hole currents and Poisson's equation. The algorithm for searching the charge carrier concentration distributions and the potential distribution is based on the asymptotic method of boundary functions and the Fourier method. In the course of the research, a technique for decomposing a nonlinear problem based on the development of perturbation theory me- thods was proposed. The initial nonlinear problem is reduced to a recurrent sequence of linear stationary boundary value problems, solved by classical and partially original analytical-numerical methods. The identification of the boundary functions in the solution of the problem provides, in particular, in contrast to the classical approximation of ambipolar diffusion, an adequate description of the electric field strength behavior in the active region of p - i - n - diodes. The obtained research results shed light on the peculiarities of the formation of the impedance characteristics of p - i - n - structures. The results of the work reveal more deeply the nature of the physical processes in the studied technical system and are aimed at improving the methodology for modeling and designing control semiconductor devices.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ В СТАТИЧНИХ ТРИВИМІРНИХ КРАЙОВИХ ЗАДАЧАХ — КОСОСИМЕТРИЧНА ЗАДАЧА ДЛЯ ШАРУ, ПОСЛАБЛЕНОГО НАСКРІЗНИМ ОТВОРОМ І КОВЗНИМ ЗАЩЕМЛЕННЯМ ТОРЦІВ

Spatial static boundary-value problems of mathematical physics for a layer with a non-circular cylindrical through hole have hardly been solved using the method of singular integral equations (SIE) despite the fact that numerous fundamental theoretical issues have been developed. Methods for the calculation of spectral characteristics are also absent. The paper provides an overview of the methods for solving these problems. A new mathematical model has been constructed, and a new method based on a system of three SIEs has been developed and tested numerically. As a result of a high-precision numerical study, it was found that with an increase in the thickness of the layer, an increase in the relative circumferential stress occurs. In the case of a circular hole, a shift of the maximum relative circumferential stress from the ends to the depth of the layer is observed. In the case of an elliptical hole, with a decrease in one of the radii, an increase in the relative circumferential stress is also observed. Keywords: three-dimensional boundary-value problems, singular integral equations, numerical experiment, static bending, a through hole.

Influence of Nonisolated Singularities in a Lower-Order Coefficient of the Bitsadze Equation on the Statement of Boundary Value Problems

We study the influence of nonisolated singularities (i.e., singularities along closed lines lying inside the domain) in the lower-order coefficients of the Bitsadze equation on the statement of boundary value problems. We discover that the conditions on the boundary of the domain in the Riemann–Hilbert problem are not sufficient for the solution; therefore, we consider a problem that combines elements of the Riemann–Hilbert problem on the boundary of the domain and the linear conjugation problem on the circles that support the singularities of the coefficients lying inside the domain. Using an appropriate refinement of Kellogg’s theorem on the conformal mapping of this domain onto a disk, we study the solvability of the problem

Influence of Nonisolated Singularities in a Lower-Order Coefficient of the Bitsadze Equation on the Statement of Boundary Value Problems

Influence of Nonisolated Singularities in a Lower-Order Coefficient of the Bitsadze Equation on the Statement of Boundary Value Problems

Non-self-adjoint model statement of boundary problem of acoustics. Part 1

Based on the mathematical definition of a non-self-adjoint operator and the physical meaning of a specific boundary value problem, a non-self-adjoint model statement of boundary value problems in acoustics is formulated. As an example, boundary value problems for the reflection of a plane wave and a spherical wave at the interface between two liquid media are considered. A new definition of the reflection coefficient of a spherical wave is introduced. In the region of subcritical angles of incidence, the new definition takes into account the appearance in the total sound field of converging recoil waves corresponding to the eigenfunctions of the adjoint operator. In the region of supercritical angles of incidence, the participation of the eigenfunctions of two conjugate operators in the total sound field forms a nonzero power flux through the interface and its transformation into the vortex component of the intensity vector at the total internal reflection horizon. The definition of the total internal reflection horizon is given. Experimental data are presented that confirm the new definition of the reflection coefficient and the physical correctness of the non-self-adjoint model statement.

Solution of the problems of thermoelasticity for a circle with double porosity

The main purpose of this work presents some explicit solutions of boundary value problems in the theory of thermoelasticity for solids with double porosity. Special representations are constructed for the general solution of basic equations. They are expressed through of elementary functions, whose properties are well known. Applying these concepts, in the proposed work, the boundary value problems of statics of the theory of thermoelasticity for an elastic circle with double porosity are solved explicitly, in the form of absolutely and uniformly converging series.

A scheme for the analysis of primal stationary boundary value problems based on FE/FD multi-method

We present a non-iterative scheme for solving general linear stationary primal boundary value problems with an arbitrary number of degrees of freedom. The main goal is to combine the finite element method performance for domains with arbitrary geometry with the finite difference method efficiency related to its low computational cost. We apply a domain decomposition scheme to define rectangular subdomains (2D case) or rectangular hexahedral subdomains (3D case). A finite element mesh is generated in such a way that the elements in the regular regions are squares (2D case) or cubes (3D case). A suitable overlap scheme allows the finite element method to be used in regions of non-regular elements, while the finite difference method can be used in regions with regular elements (squares for 2D case or cubes for 3D case). We present six computational simulations showing the best performance in the CPU time of the proposed scheme compared to the classic finite element method while maintaining solution accuracy. • Reduction in the computational cost of assembling the matrices. • Solution time of the system of equations is preserved. • Feasible for problems with arbitrary degrees of freedom and non-homogeneous media. • Able to deal with several regular finite difference subdomains.

РЕЗОНАНСНЫЕ ХАРАКТЕРИСТИКИ СОРБЦИОННОГО ОПТОВОЛОКОННОГО ПЬЕЗОЭЛЕКТРОЛЮМИНЕСЦЕНТНОГО ДАТЧИКА

A mathematical electro-mechanical model of a representative fragment of a gas-analytical optical fiber piezoelectroluminescent sensor with a special external absorption layer for monitoring extended areas has been developed. Informative light signal is generated as a result of mechanoluminescent effect caused by contact interaction of piezoelectric and electroluminescent cylindrical layers (coatings) of fiber in case of forced electromechanical vibrations of sensor under action of applied harmonic control electric voltage. The algorithm for finding the sorption spectrum (changes in the density of the absorption layer along the length of the sensor) is presented through the solution of the Fredholm integral equation based on the results of measuring the informative light spectrum of the intensity of the light at the output from the fiber of the sensor. Numerical solution of coupled stationary boundary value problem of electroelasticity for representative fragment of sensor is implemented. Natural frequencies and shapes of sensor oscillations were found taking into account geometric shape, relative location and anisotropy of electroelastic properties of its structural elements, including characteristics of absorption layer. Regularities of influence of diagnosed value of absorption layer density on informative values of resonance frequencies of different forms of sensor oscillations were revealed.

On one case of quasi-correctness of the static boundary value problem for shells of rotation

Within the framework of the membrane theory of convex shells, a static boundary value problem is studied for one class of spherical domes.

MODEL OF VOLUMETRIC HEAT SOURCES IN ANODE OF MICROFOCUS X-RAY TUBE

The work is developed a numerical-analytical approach to modeling thermal processes in a composite anode of microfocus X-ray tubes. At the first stage of modeling, with the help of well-known analytical approximations, the distribution function of volumetric heat sources is built. At the second stage, this function is used when solving a stationary boundary value problem with the Poisson equation by the Finite Elements Method. Using the proposed technique, an adequate assessment of the maximum power of the tube with the transmission type W-Be anode was developed.

Exact Solutions to the Navier–Stokes Equations with Couple Stresses

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

Unique solvability of a stationary radiative‐conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies

We consider a stationary boundary value problem describing a radiative‐conductive heat transfer in a system consisting of one absolutely black body and several semitransparent bodies. To describe the radiative transfer, the integro‐differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. Besides, we proved the comparison theorems and established the results on improving the properties of solutions with increasing exponents of data summability.

Unique solvability of a stationary radiative–conductive heat transfer problem in a semitransparent body with absolutely black inclusions

We consider a stationary boundary value problem describing a radiative–conductive heat transfer in a semitransparent body with absolutely black inclusions. To describe the radiative transfer, the integro-differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. Besides, we proved the comparison theorems and established the results on improving the properties of solutions with increasing exponents of data summability.

Variational interior and interface transmission conditions: multidomain mixed Darcy/Stokes control problems

Multidomain mixed pairs of coupled stationary constrained boundary value problems, are formulated variationally in frameworks of reflexive Banach spaces as macro-hybrid systems, and analyzed. Interior synchronizing and interface coupling transmission constraints are modeled in terms of Lagrange multipliers as solutions of dual variational subpotential maximal monotone inclusions. Existence and uniqueness results are established via resolvent fixed-point characterizations of the corresponding primal and dual macro-hybridized problems. Multimedia mixed mechanical subsurface-surface Darcy/Stokes incompressible flow coupled pairs with intrinsic distributed control constraints exemplify the general macro-hybrid mixed systems. Coupling dual-primal and primal-dual interface macro-hybrid mixed variational conditions are lastly developed and applied to the Darcy/Stokes multimedia pair model.

Некоторые математические вопросы ионизации плазмы

A number of mathematical problems of the theory of ionization in relation to processes in stationary plasma thrusters (SPT) are solved in this work. Two main one-dimensional mathematical models of ionization are considered – hydrodynamic and kinetic. The main question is the existence of ionization oscillations (breathing modes). On the basis of a hydrodynamic model, a boundary value problem for stationary ionization equations is solved. Its unique solvability and the absence of breathing modes are proved. In the case when the ion velocity in the flow region has a single zero with a positive derivative, it is proved that the stationary boundary value problem has a countable number of solutions, and a necessary and sufficient condition for the existence of breathing modes is formulated. Finally, an analytical solution of the ionization equations is given in the case of constant velocities of atoms and ions, and the formulas obtained are applied to the solution of the Cauchy problem, boundary value and mixed problems in the simplest regions. In the case of the kinetic model of ionization, the existence of breathing modes is numerically shown and the results obtained are compared with the hydrodynamic case.

The role of mechanics in the growth and homeostasis of the intestinal crypt

We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us to precisely characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous “treadmilling” state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis.

Comparing the best-reply strategy and mean-field games: The stationary case

Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity

In this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is introduced. The studied class of solids is therefore related to Korteweg or Cahn–Hilliard fluids. The postulated energy naturally induces the space in which the aforementioned well-posedness result can be formulated. In this energy space, the introduced norm does involve the linear combination of some specific higher-order derivatives only: it is, in fact, a particular example of anisotropic Sobolev space. It is also proven that aforementioned weak solutions belongs to the space H^1(div,V), i.e. the space of H^1 functions whose divergence belongs to H^1. The proposed mathematical frame is essential to conceptually base, on solid grounds, the numerical integration schemes required to investigate the properties of dilatational strain gradient elastic bodies. Their energy, as studied in the present paper, has manifold interests. Mathematically speaking, its singularity causes interesting mathematical difficulties whose overcoming leads to an increased understanding of the theory of second gradient continua. On the other hand, from the mechanical point of view, it gives an example of energy for a second gradient continuum which can sustain externally applied surface forces and double forces but cannot sustain externally applied surface couples. In this way, it is proven that couple stress continua, introduced by Toupin, represent only a particular case of the more general class of second gradient continua. Moreover, it is easily checked that for dilatational strain gradient continua, balance of force and balance of torques (or couples) are not enough to characterise equilibrium: to this aim, externally applied surface double forces must also be specified. As a consequence, the postulation scheme based on variational principles seems more suitable to study second gradient continua. It has to be remarked finally that dilatational strain gradient seems suitable to model the experimentally observed behaviour of some material used in 3D printing process.

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

Аналитическое и численное моделирование стационарной краевой задачи диффузии-конвекции-распада для однородного слоя на основе уравнений турбулентной диффузии

For comparison and testing purposes, an analytical and numerical method has been developed for solving the boundary value problem of convection-diffusion-decay for a homogeneous layer. A model stationary boundary value problem of the third kind is described by the equations of turbulent diffusion. In the analytical model, the Fourier symbol of the Green's function for the boundary value problem is constructed, the calculation of the Fourier integral is based on the theory of residues. In the numerical model, the Fourier integral is calculated using an algorithm based on the Gauss-Kronrod formulas.In the two-dimensional case, comparative calculations for the near and middle zones showed good agreement between the results. The accuracy of calculations within each model can be changed by several orders of magnitude. Calculations of the substance concentration function for two plane problems of convection-diffusion-decay are given as examples. The implementation of the analytical method in the flat case is relatively straightforward. Numerical calculation is much simpler and more convenient for engineering and serial scientific calculations. Numerical integration is realized just as easily in the spatial case as in the two-dimensional case, but the counting time increases significantly. Therefore, a balance is required between the required accuracy and the calculation time. For the spatial case, an analytical approach based on the theory of residues was fundamentally developed in the works of other authors. However, the method is rather cumbersome and better suited for theoretical research.