initial boundary value problem
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Для моделирования процессов в ВЧ-плазме пониженного давления с продувом газа разработана гибридная математическая модель при числах Кнудсена - для несущего газа. Модель включает начально-краевую задачу для кинетического уравнения Больцмана, описывающего функцию распределения несущего нейтрального газа, краевые задачи для уравнения неразрывности электронной, ионной и метастабильной компонент, уравнения сохранения энергии электронов, для ВЧ-уравнений Максвелла в форме телеграфных уравнений и уравнения Пуассона для потенциальной составляющей поля. Приводятся результаты расчета электрической напряженности, концентрации электронов, ионов и метастабилей, потенциальной составляющей электромагнитного поля в цилиндрической вакуумной камере. A hybrid mathematical model for the Knudsen numbers - for the carrier gas has been developed to simulate processes in a low pressure RF plasma with gas flow. The model includes an initial boundary value problem for the kinetic Boltzmann equation describing the distribution function of the carrier neutral gas, boundary value problems for the continuity equation of the electronic, ionic and metastable components, the electron energy conservation equations, for Maxwell’s RF equations in the form of telegraphic equations and the Poisson equation for the potential part of field. The results of the calculation of the electric intensity, the concentration of electrons, iones and metastables, the potential component of the electromagnetic field in a cylindrical vacuum chamber are presented.

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

<p style='text-indent:20px;'>This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.</p>

The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.

A numerical calculation of the evolution of the temperature distribution in the longitudinal section of a niobium nitride membrane when it is heated by an electric current pulse is performed. Mathematical modeling was carried out on the basis of a two-dimensional initial-boundary value problem for an inhomogeneous heat equation. In the initial boundary value problem, it was taken into account that current and potential contacts to the membrane serve simultaneously as contacts for heat removal. The case was considered for the third from the left and the first from the right initial-boundary value problem. Analysis of the numerical solution showed that effective heat removal from the membrane can be provided by current-carrying and potential clamping contacts made, for example, of beryllium bronze. This makes it possible to study the current-voltage characteristics of superconducting membranes near the critical temperature of the transition to the superconducting state by currents close to the critical density without significant heating.

We consider some initial boundary value problems for the Burgers equation in a rectangular domain, which in a sense can be taken as a model one. The fact is that such a problem often arises when studying the Burgers equation in domains with moving boundaries. Using the methods of functional analysis, priori estimates, and Faedo-Galerkin in Sobolev spaces and in a rectangular domain, we show the correctness of the initial boundary value problem for the Burgers equation with nonlinear boundary conditions of the Neumann type.

This paper deals with numerical prediction of temperature (weakening) effects on the tensile strength of granitic rock. A 3D numerical approach based on the embedded discontinuity finite elements is developed for this purpose. The governing thermo-mechanical initial/boundary value problem is solved with an explicit (in time) staggered method while using extreme mass scaling to increase the critical time step. Rock fracture is represented by the embedded discontinuity concept implemented here with the linear (4-node) tetrahedral elements. The rock is modelled as a linear elastic (up to fracture by the Rankine criterion) heterogeneous material consisting of Quartz, Feldspar and Biotite minerals. Due to its strong and anomalous temperature dependence upon approaching the α-β transition at the Curie point (~573 °C), only Quartz in the numerical rock depends on temperature in the present approach. In the numerical testing, the sample is first volumetrically heated to a target temperature. Then, the uniaxial tension test is performed on the cooled down sample. The simulations demonstrate the validity of the proposed approach as the experimental deterioration, by thermally induced cracking, of the rock tensile strength is predicted with a good accuracy.

Abstract A time-explicit Runge-Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection-reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ≥ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results demonstrate the robustness of the scheme.

In this paper, the physics-informed neural networks (PINNs) are applied to high-dimensional system to solve the [Formula: see text]-dimensional initial-boundary value problem with [Formula: see text] hyperplane boundaries. This method is used to solve the most classic (2+1)-dimensional integrable Kadomtsev–Petviashvili (KP) equation and (3+1)-dimensional reduced KP equation. The dynamics of (2+1)-dimensional local waves such as solitons, breathers, lump and resonance rogue are reproduced. Numerical results display that the magnitude of the error is much smaller than the wave height itself, so it is considered that the classical solutions in these integrable systems are well obtained based on the data-driven mechanism.

The numerical method for simulation dynamics of nonlinear epidemic model of age-structured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (HPV) is developed (SIPCV model). Cell population dynamics is described by the initial-boundary value problem for the delayed semi-linear hyperbolic equations with age- and time-dependent coefficients and HPV dynamics is described by the initial problem for nonlinear delayed ODE. The model considers two time-delay parameters: the time between viral entry into a target susceptible cell and the production of new virus particles, and duration of the first stage of delayed immune response to HPV population growing. Using the method of characteristics and method of steps we obtain the exact solution of the SIPCV epidemic model in the form of explicit recurrent formulae. The numerical method designed for this solution and used the trapezoidal rule for integrals in recurrent formulae has a second order of accuracy. Numerical experiments with vanished mesh spacing illustrate the second order of accuracy of numerical solution with respect to the benchmark solution and show the dynamical regimes of cell-HPV population with the different phase portraits.