Value Problem For Parabolic Equation Research Articles

To the Properties of One Fox Function

В работе рассматривается частный случай специальной функции Фокса с четырьмя параметрами, которая возникает в теории краевых задач для параболических уравнений c оператором Бесселя и дробной производной по времени. Целью исследования является получение некоторых рекуррентных соотношений, формул дифференцирования и интегрального преобразования рассматриваемой функции. При получении результатов работы в основном используется представление рассматриваемой функции через интеграл Меллина–Барнса. Также используются её асимптотические разложения при большом и малом значениях аргумента. С помощью указанного интегрального представления и некоторых известных формул для гамма-функции Эйлера, получены рекуррентные соотношения, связывающие функции с разными параметрами, а также функцию с её производной первого порядка. Получена формула дифференцирования n-го порядка. Исследуется несобственный интеграл первого рода, который содержит рассматриваемую функцию с двумя зависимыми параметрами из четырёх. Показывается, что этот несобственный интеграл может быть записан в терминах известной специальной функции Макдональда. При частных значениях параметров рассматриваемой в работе функции получаются некоторые известные элементарные и специальные функции. Результаты работы носят теоретический характер и будут полезны при исследовании краевых задач для вырождающихся параболических уравнений с производными дробного порядка по времени. The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.

A Note on Stability of Parabolic Difference Equations on Torus

The present article investigates nonlocal boundary value problems for parabolic equations of reverse type on torus. The first order of accuracy difference scheme for the numerical solution of nonlocal boundary value problems for parabolic equations on circle and torus are presented. For the solutions of the difference scheme, the stability estimates and coercivity estimates in various Hölder norms are established. Furthermore, theoretical results are supported by numerical experiments.

Some qualitative properties for the Kirchhoff total variation flow

The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem.

On Solving Initial Boundary Value Problems for a Parabolic Equation in Dini Classes

The method of potentials is one of the classical methods for solving initial boundary value problems for parabolic equations and systems. The method is based on studying the smoothness of parabolic potentials in various functional spaces and studying the solvability of the corresponding integral equations and systems to which the problems are reduced. This method allows one to find a constructive form of solutions to initial-boundary value problems and directly investigate the smoothness of solutions. This article continues a series of works on constructing the solvability theory of parabolic initial boundary value problems in domains with nonsmooth lateral boundaries in the time variable. The article considers the first and second initial boundary value problems with zero initial condition for a uniformly parabolic equation that is one-dimensional in the spatial variable. It is assumed that the equation coefficients are bounded and uniformly continuous with a modulus of continuity satisfying twice the Dini condition. The specific feature of the study is that the right side of the equation can grow to infinity in a certain way when approaching the initial data assignment line. The solvability (in the classical sense) of these problems is established in a semi-bounded curvilinear domain in a plane with a nonsmooth lateral boundary from the Dini-Hölder class, admitting, in particular, so-called "beaks". The solution is constructed as a sum of the plane parabolic potential and the single layer potential, the kernel of which is the fundamental solution of the equation. The smoothness of the solution of these problems in the Dini-Hölder classes is investigated, and the corresponding correctness estimates are obtained. These estimates characterize the behavior of solutions and their first-order spatial derivative in the closure of the domain. The results obtained can be used to study heat and mass transfer processes. For the proof, the smoothness in the Dini-Hölder space of a plane parabolic potential with an unlimited density of the Dini class in a curvilinear domain with a nonsmooth lateral boundary is investigated. The obtained result can be used in solving other initial boundary value problems for an inhomogeneous parabolic equation.

Trace theorem and non-zero boundary value problem for parabolic equations in weighted Sobolev spaces

We present weighted Sobolev spaces \(\widetilde{\mathfrak {H}}_{p, \theta }^{\gamma }(S, T)\) and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in particular, for the regularity theory of stochastic partial differential equations on bounded domains.

Mixed boundary value problems for parabolic equations in Sobolev spaces with mixed-norms

We establish \(L_{q,p}\)-estimates and solvability for mixed Dirichlet–conormal problems for parabolic equations in a cylindrical Reifenberg-flat domain with a rough time-dependent separation.

Chernoff approximations of Feller semigroups in Riemannian manifolds

AbstractChernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non‐compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non‐compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.

Boundary Value Problems for Parabolic Equations in Noncylindrical Domains

Boundary Value Problems for Parabolic Equations in Noncylindrical Domains

INITIAL-BOUNDARY VALUE PROBLEM FOR HIGHER-ORDERS NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENTS OF THE NONLINEARITY IN UNBOUNDED DOMAINS WITHOUT CONDITIONS AT INFINITY

Initial-boundary value problems for parabolic equations in unbounded domains with respect to the spatial variables were studied by many authors. As is well known, to guarantee the uniqueness of the solution of the initial-boundary value problems for linear and some nonlinear parabolic equations in unbounded domains we need some restrictions on solution's behavior as $|x|\to +\infty$ (for example, solution's growth restriction as $|x|\to +\infty$, or belonging of solution to some functional spaces). Note that we need some restrictions on the data-in behavior as $|x|\to +\infty$ to solvability of the initial-boundary value problems for parabolic equations considered above. However, there are nonlinear parabolic equations for which the corresponding initial-boundary value problems are unique solvable without any conditions at infinity. Nonlinear differential equations with variable exponents of the nonlinearity appear as mathematical models in various physical processes. In particular, these equations describe electroreological substance flows, image recovering processes, electric current in the conductor with changing temperature field. Nonlinear differential equations with variable exponents of the nonlinearity were intensively studied in many works. The corresponding generalizations of Lebesgue and Sobolev spaces were used in these investigations. In this paper we prove the unique solvability of the initial--boundary value problem without conditions at infinity for some of the higher-orders anisotropic parabolic equations with variable exponents of the nonlinearity. An a priori estimate of the generalized solutions of this problem was also obtained.

Modeling of filtration processes in periodic porous media

Modeling of dynamic processes of diffusion and filtration of liquids in porous media are discussed. The media are formed by a large number of blocks with low permeability, and separated by a connected system of faults with high permeability. The modeling is based on solving initial boundary value problems for parabolic equations of diffusion and filtration in porous media. The structure of the media leads to the dependence of the equations on a small parameter. Assertions on the solvability and regularity of such problems and the corresponding homogenized convolution problems are considered. The statements are actual for the numerical solution of this problem with guaranteed accuracy that is necessary to model the considered processes.

Final Value Problem for Parabolic Equation with Fractional Laplacian and Kirchhoff’s Term

In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero whenδtends to zero.

Initial-boundary value problem for parabolic equations containing an elliptic operator of arbitrary order

Initial-boundary value problem for parabolic equations containing an elliptic operator of arbitrary order

On a nonlinear parabolic equation with fractional Laplacian and integral conditions

The paper deals with non-classical initial-boundary value problems for parabolic equations with a fractional Laplacian. We study the existence and uniqueness of a mild solution to our problem. The continuous dependence of the solution on the given data is shown and the ill-posedness of the mild solution at t = 0 is also considered. In order to avoid such ill-posedness, we construct a regularized solution using the Fourier truncation method. An error estimate and the convergence rate between the regularized solution and the exact solution are obtained.

A convergence result on the second boundary value problem for parabolic equations

We establish a Schn$\ddot{\text{u}}$rer's convergence result and then apply it to obtain the existence of solutions on the second boundary value problem for a family of special Lagrangian equations

On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation

On the problem of recovering boundary conditions in the third boundary value problem for parabolic equation

Global and local optimization in identification of parabolic systems

Abstract The problem of identification of coefficients and initial conditions for a boundary value problem for parabolic equations that reduces to a minimization problem of a misfit function is investigated. Firstly, the tensor train decomposition approach is presented as a global convergence algorithm. The idea of the proposed method is to extract the tensor structure of the optimized functional and use it for multidimensional optimization problems. Secondly, for the refinement of the unknown parameters, three local optimization approaches are implemented and compared: Nelder–Mead simplex method, gradient method of minimum errors, adaptive gradient method. For gradient methods, the evident formula for the continuous gradient of the misfit function is obtained. The identification problem for the diffusive logistic mathematical model which can be applied to social sciences (online social networks), economy (spatial Solow model) and epidemiology (coronavirus COVID-19, HIV, etc.) is considered. The numerical results for information propagation in online social network are presented and discussed.

On a final value problem for parabolic equation on the sphere with linear and nonlinear source

Parabolic equation on the unit sphere arise naturally in geophysics and oceanography when we model a physical quantity on large scales. In this paper, we consider a problem of finding the initial state for backward parabolic problem on the sphere. This backward parabolic problem is ill-posed in the sense of Hadamard. The solutions may be not exists and if they exists then the solution does not continuous depends on the given observation. The backward problem for homogeneous parabolic problem was recently considered in the paper Q.T. L. Gia, N.H. Tuan, T. Tran. However, there are very few results on the backward problem of nonlinear parabolic equation on the sphere. In this paper, we do not consider the its existence, we only study the stability of the solution if it exists. By applying some regularized method and some techniques on the spherical harmonics, we approximate the problem and then obtain the convalescence rate between the regularized solution and the exact solution.

Energy‐dissipation in a coupled system of Allen–Cahn‐type equation and Kobayashi–Warren–Carter‐type model of grain boundary motion

In this paper, we consider a system of initial‐boundary value problems for parabolic equations, as a generalized version of the “φ‐η‐θ model” of grain boundary motion, proposed by Kobayashi (2001). The system is a coupled system of an Allen–Cahn‐type equation with a given temperature source and a phase‐field model of grain boundary motion, known as “Kobayashi–Warren–Carter‐type model.” The focus of the study is on a special kind of solution, called energy‐dissipative solution, which is to reproduce the energy‐dissipation of the governing energy in time. Under suitable assumptions, two Main Theorems, concerned with the existence of energy‐dissipative solution and and the large‐time behavior, will be demonstrated as the results of this paper.

Rotationally-axisymmetric Motion of a Binary Mixture with a Flat Free Boundary at Small Marangoni Numbers

Rotationally-axisymmetric motion of a binary mixture with a flat free boundary at small Marangoni numbers is investigated. The problem is reduced to the inverse linear initial-boundary value problem for parabolic equations. Using Laplace transformation properties the exact analytical solution is obtained. It is shown that a stationary solution is the limiting one with the growth of time if there is a certain relationship between the temperature of the solid wall and the external temperature of the gas. If there is no connection, the convergence to the stationary solution is broken. Some examples of numerical reconstruction of the temperature, concentration and velocity fields are given, which confirm the theoretical conclusions

Исследование некоторых математических моделей нестационарных фильтрационных процессов

Рассматриваются математические модели, связанные с изучением нестационарных процессов фильтрации в подземной гидродинамике. Они представляют собой нелинейные задачи для параболических уравнений с неизвестной функцией источника в правой части. Одна из постановок является системой, которая состоит из краевой задачи с граничными условиями первого рода и из уравнения, задающего закон изменения по времени искомой функции источника. В другой постановке соответствующая система включает в себя краевую задачу с граничными условиями второго рода. Указанные постановки существенно отличаются от обычных краевых задач для параболических уравнений. Цель исследования - установить для этих нелинейных параболических задач условия однозначной разрешимости в классе гладких функций на основе априорных оценок метода Ротэ. We consider some mathematical models connected with the study of nonstationary filtration processes in underground hydrodynamics. These models involve nonlinear problems for parabolic equations with unknown source functions. One of the problems is a system consisting of a boundary value problem of the first kind and an equation describing a time dependence of the sought source function. In the other problem, the corresponding system is distinguished from the first one by boundary conditions of the second kind. These problems essentially differ from usual boundary value problems for parabolic equations. The aim of our study is to establish conditions of unique solvability in a class of smooth functions for the considered nonlinear parabolic problems. The proposed approach involves the proof of a priori estimates for the Rothe method.