Equations Of Elliptic Type Research Articles

Differential Equations of Elliptic Type with Variable Operators and Homogeneous Robin Boundary Value Condition in UMD Spaces

Differential Equations of Elliptic Type with Variable Operators and Homogeneous Robin Boundary Value Condition in UMD Spaces

One class of quasilinear elliptic type equations with discontinuous nonlinearities

In a bounded domain $\Omega\subset \mathbb{R}^n$, a class of quasilinear elliptic type boundary problems with parameter and discontinuous nonlinearity is studied. This class of problems includes the H. J. Kuiper conductor heating problem in a homogeneous electric field. The topological method is applied to verify the existence of a continuum of generalized positive solutions from the Sobolev space $W_p^2(\Omega)$ ($p>n$) connecting $(0,0)$ with $\infty$ in the space $\mathbb R\times C^{1,\alpha}(\overline\Omega)$, $\alpha\in (0,(p-n)/p)$. A sufficient condition for semiregularity of generalized solutions of this problem is given. The constraints on the discontinuous nonlinearity are relaxed in comparison with those used by H. J. Kuiper and K. C. Chang.

One class of quasilinear elliptic type equations with discontinuous nonlinearities

В ограниченной области $\Omega\subset \mathbb{R}^n$ исследуется класс квазилинейных граничных задач эллиптического типа с параметром и разрывной нелинейностью. Рассматриваемый класс задач включает задачу Х. Дж. Купера о нагреве проводника в однородном электрическом поле. Топологическим методом устанавливается существование континуума обобщенных положительных решений из соболевского пространства $W_p^2(\Omega)$\enskip ($p>n$), соединяющего $(0,0)$ с $\infty$, в пространстве $\mathbb R\times C^{1,\alpha}(\overline\Omega)$, $\alpha\in (0,(p-n)/p)$. Приводится достаточное условие полуправильности обобщенных решений изучаемой задачи. По сравнению с работами Х. Дж. Купера и К.-Ч. Чанга ослаблены ограничения на разрывную нелинейность. Библиография: 26 наименований.

Численное решение прямой задачи электроимпедансной томографии в полной электродной постановке

This work is related to one of the methods of medical imaging - electrical impedance tomography (EIT). A feature of the considered two-dimensional mathematical formulation for the direct EIT problem is the use of an elliptic type equation with piecewise constant coefficients and a special integro-differential boundary condition at the contact boundary of the electrodes. For an approximate solution of the problem under consideration, a numerical method was developed using unstructured grids, the finite volume method for barycentric cells, and the Gaussian elimination method with the choice of the main element. The validation of the method was performed on a known analytical solution.

Numerical analysis of the stability of nonequilibrium 3D evolutionary transfer processes in the network

Analytical methods for solving various problems of an applied nature (for example, non-stationary transfer problems over network hydro, gas and heat carriers), whose mathematical models use the formalisms of evolutionary differential systems, are possible with rare exceptions. That is why the construction of numerical and simulation models for the use of quantitative analysis methods becomes a universal research tool, if at the same time the implementation of these models on a computer is carried out – in other words, a complex of software engineering of the process under study is formed. The study uses the method of semidiscretization by a time variable of the mathematical model of the evolutionary non-equilibrium process of continuous medium transfer, which remains one of the most effective methods for analyzing applied problems. In this case, the elliptic operator of the mathematical model has a special basis (a system of eigenfunctions), which is why the analysis is reduced to the study of a boundary value problem for elliptic-type equations with a spatial variable changing on a network-like domain. The paper presents the conditions for unambiguous weak solvability of a differential-difference system, which is a difference analogue in the time variable of the original system, and the way of constructing an algorithm for finding an approximate solution is indicated. The study contains an analysis of the stability and convergence of difference schemes of evolutionary network-like nonequilibrium processes of continuous media transfer over network carriers and includes an analysis of the correctness of the mathematical model of this process. The results of the work are applicable in the framework of oil and gas engineering to the study of issues of stabilization and parametric optimization of the processes of transportation of liquid media through spatial networks.

High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative Swift–Hohenberg model

In this study, we develop high-order time-accurate, efficient, and energy stable schemes for solving the conservative Swift–Hohenberg equation that can be used to describe the L2-gradient flow based phase-field crystal dynamics. By adopting a modified exponential scalar auxiliary variable approach, we first transform the original equations into an expanded system. Based on the expanded system, the first-, second-, and third-order time-accurate schemes are constructed using the backward Euler formula, second-order backward difference formula (BDF2), and third-order backward difference formula (BDF3), respectively. The energy dissipation law can be easily proved with respect to a modified energy. In each time step, the local variable is updated by solving one elliptic type equation and the non-local variables are explicitly computed. The whole algorithm is totally decoupled and easy to implement. Extensive numerical experiments in two- and three-dimensional spaces are performed to show the accuracy, energy stability, and practicability of the proposed schemes.

The Continuous Classical Boundary Optimal Control of Triple Nonlinear Elliptic Partial Differential Equations with State Constraints

Our aim in this work is to study the classical continuous boundary control vector problem for triple nonlinear partial differential equations of elliptic type involving a Neumann boundary control. At first, we prove that the triple nonlinear partial differential equations of elliptic type with a given classical continuous boundary control vector have a unique "state" solution vector, by using the Minty-Browder Theorem. In addition, we prove the existence of a classical continuous boundary optimal control vector ruled by the triple nonlinear partial differential equations of elliptic type with equality and inequality constraints. We study the existence of the unique solution for the triple adjoint equations related with the triple state equations.
 The Fréchet derivative is obtained. Finally we prove the theorems of both the necessary and sufficient conditions for optimality of the triple nonlinear partial differential equations of elliptic type through the Kuhn-Tucker-Lagrange's Multipliers theorem with equality and inequality constraints.

Existence of Solutions of Anisotropic Elliptic Equations with Variable Indices of Nonlinearity in ℝn

In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space ℝn, n ≥ 2. We prove the existence of solutions without restrictions to the growth rate as |x| → ∞.

Asymptotic derivation and simulations of a non-local Exner model in large viscosity regime

The present paper deals with the modeling and numerical approximation of bed load transport under the action of water. A new shallow water type model is derived from the stratified two-fluid Navier–Stokes equations. Its novelty lies in the magnitude of a viscosity term that leads to a momentum equation of elliptic type. The full model, sediment and water, verifies a dissipative energy balance for smooth solutions. The numerical resolution of the sediment layer is not trivial since the viscosity introduces a non-local term in the model. Adding a transport threshold makes the resolution even more challenging. A scheme based on a staggered discretization is proposed for the full model, sediment and water.

Gradient estimates for multi-phase problems

We prove optimal gradient estimates for distributional solutions to non-uniformly elliptic equations of multi-phase type in divergence form by investigating sharp conditions on such nonlinear operators for the Calderon-Zygmund theory.

Poly-Sinc Solution of Stochastic Elliptic Differential Equations

In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

A family of effective structure-preserving schemes with second-order accuracy for the undamped sine–Gordon equation

In this paper, a family of linear structure-preserving (energy conservation) schemes with second-order accuracy in the time direction is developed to numerically solve the undamped sine–Gordon equation. To be specific, first, transformation of the undamped sine–Gordon system into an equivalent new system is made by introducing an improved scalar auxiliary variable (SAV), and generalization of the conservative Crank–Nicolson scheme is made by applying a non-negative family parameter ϑ to discretize time-dependent variables at the time step (n+ϑ) instead of just (n+1∕2), thereby to establish a family of second-order conservative semi-discrete schemes. Further, based on the advantages of the improved SAV method, not only does the newly introduced scalar auxiliary variable be uncoupled with the original variables at the discrete level, it also requires the family of approximations, at each time step, no more efforts than the solution of a second-order linear differential equation of elliptic type with constant coefficients, making the computational cost of this method only half of the original one, which thus is particularly effective. Finally, several numerical experiments are presented to demonstrate the efficiency, stability, accuracy and energy conservation of the family of schemes developed herein.

Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N

We consider a class of fourth-order elliptic equations of Kirchhoff type with critical growth in \(R^N\). By using constrained minimization in the Nehari manifold, weestablish sufficient conditions for the existence of nodal (that is, sign-changing) solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/19/abstr.html

The Necessary Condition for Optimal Boundary Control Problems for Triple Elliptic Partial Differential Equations

In this work, we prove that the triple linear partial differential equations (PDEs) of elliptic type (TLEPDEs) with a given classical continuous boundary control vector (CCBCVr) has a unique "state" solution vector (SSV) by utilizing the Galerkin's method (GME). Also, we prove the existence of a classical continuous boundary optimal control vector (CCBOCVr) ruled by the TLEPDEs. We study the existence solution for the triple adjoint equations (TAJEs) related with the triple state equations (TSEs). The Fréchet derivative (FDe) for the objective function is derived. At the end we prove the necessary "conditions" theorem (NCTh) for optimality for the problem.

О приближении функций класса Зигмунда бигармоническими интегралами Пуассона

Работа посвящена изучению поведения верхнего предела отклонения функций класса Зигмунда от их бигармонических интегралов Пуассона. Исследования в данном направлении проводились и проводятся систематически как отечественными, так и зарубежными учеными. Большинство полученных результатов относится к оценке отклонений функций того или иного класса от операторов, построенных с помощью треугольных методов суммирования рядов Фурье (Фейера, Валле Пуссена, Рисса, Рогозинского, Стеклова, Фавара и др.). Что касается результатов относительно линейных методов суммирования рядов Фурье, заданных с помощью множества функций натурального аргумента (Абеля-Пуассона, Гаусса-Вейерштрасса, бигармонического и трехгармонического интегралов Пуассона), то здесь успехи менее заметны. Возможно, это связано с тем, что упомянутые выше линейные методы суммирования рядов Фурье являются решениями соответствующих интегрально-дифференциальных уравнений эллиптического типа и поэтому требуют более трудоемких вычислений с целью получения для них определенных оценок, пригодных для их непосредственное использование в прикладных целях. Исследования, проведенные в данной работе, относятся к изучению апроксимативных характеристик линейных положительных операторов типа Пуассона на классах функций Зигмунда. Согласно хорошо известным результатам П.П. Коровкина именно эти положительные линейные операторы осуществляют наилучшее асимптотическое приближение функций класса Зигмунда. Таким образом, полученная в данной работе оценка отклонения функций класса Зигмунда от их бигармонических интегралов Пуассона (наименее исследованных и наиболее востребованных среди всех линейных положительных операторов) актуальна с точки зрения прикладной математики.

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Exact method to solve of linear heat transfer problems

When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.

Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks

We propose a novel numerical method for high dimensional Hamilton--Jacobi--Bellman (HJB) type elliptic partial differential equations (PDEs). The HJB PDEs, reformulated as optimal control problems, are tackled by the actor-critic framework inspired by reinforcement learning, based on neural network parametrization of the value and control functions. Within the actor-critic framework, we employ a policy gradient approach to improve the control, while for the value function, we derive a variance reduced least-squares temporal difference method using stochastic calculus. To numerically discretize the stochastic control problem, we employ an adaptive step size scheme to improve the accuracy near the domain boundary. Numerical examples up to $20$ spatial dimensions including the linear quadratic regulators, the stochastic Van der Pol oscillators, the diffusive Eikonal equations, and fully nonlinear elliptic PDEs derived from a regulator problem are presented to validate the effectiveness of our proposed method.

THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS

For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

О ПРИБЛИЖЕНИИ ФУНКЦИЙ ИНТЕГРАЛАМИ ГАУССА–ВЕЙЕРШТРАССА

Рассматривая разные схемы и алгоритмы игровых задач динамики, исследователи часто сталкиваются с решением дифференциальных уравнений в частных производных. Особое место среди последних занимают так называемые уравнения эллиптического типа (согласно соответствующей классификации), с помощью которых наиболее полно и качественно можно описать естественные и социальные процессы. Кроме того, математический аппарат дифференциальных уравнений в част-ных производных эллиптического типа позволяет проникать в среду детерминированных явлений и предсказывать их будущее. В то же время одним из самых важных понятий прикладной математики является понятие модуля непрерывности. Термин «модуль непрерывности» и его определение был введен Анри Лебегом в начале прошлого века с целью изучения разнообразных свойств непрерывных функций. Используя понятие модуля непрерывности и его свойства, можно исследовать принадлежность изучаемого объекта к определенному классу функций: Гельдера, Липшица, Зигмунда и т.д. Это, несомненно, позволяет наиболее эффективно осуществлять приближение функций различного рода операторами. В данной работе на примере интеграла Гаусса-Вейерштрасса, как решение соответствующего дифференциального уравнения эллиптического типа, исследуется его скорость сходимости в терминах модуля непрерывности второго порядка к функции, по которой он фактически был построен. А именно, были изучены предельные свойства интеграла Гаусса-Вейерштрасса, как линейного положительного оператора, осуществляющего свое наилучшее приближение на функциях класса Зигмунда. Полученные в данной статье результаты в дальнейшем могут использоваться при решении многих задач прикладной математики.