scholarly journals On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Variable Exponent ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Medium Equation ◽  
Partial Boundary Value Condition ◽  
Initial Boundary Value

The initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient ax,t and the variable exponent px,t depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of ∂Ω×0,T. By this innovation, the stability of weak solutions is proved.

scholarly journals On the Solutions of a Porous Medium Equation with Exponent Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

scholarly journals The Stability of the Solutions for a Porous Medium Equation with a Convection Term

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Huashui Zhan ◽  
Miao Ouyang
Keyword(s):  
Porous Medium ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convection Term ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

This paper studies the initial-boundary value problem of a porous medium equation with a convection term. If the equation is degenerate on the boundary, then only a partial boundary condition is needed generally. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of the solutions is studied. In some special cases, the stability can be proved without any boundary value condition.

scholarly journals Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium

2020 ◽  
pp. 72-76
Author(s):  
R.A. Virts ◽  
A.A. Papin ◽  
W.A. Weigant
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Second Order ◽  
Initial System ◽  
Effective Pressure ◽  
Initial Boundary Value

The paper considers a model for filtering a viscous incompressible fluid in a deformable porous medium. The filtration process can be described by a system consisting of mass conservation equations for liquid and solid phases, Darcy's law, rheological relation for a porous medium, and the law of conservation of balance of forces. This paper assumes that the poroelastic medium has both viscous and elastic properties. In the one-dimensional case, the transition to Lagrange variables allows us to reduce the initial system of governing equations to a system of two equations for effective pressure and porosity, respectively. The aim of the work is a numerical study of the emerging initial-boundary value problem. Paragraph 1 gives the statement of the problem and a brief review of the literature on works close to this topic. In paragraph 2, the initial system of equations is transformed, as a result of which a second-order equation for effective pressure and the first-order equation for porosity arise. Paragraph 3 proposes an algorithm to solve the initial-boundary value problem numerically. A difference scheme for the heat equation with the righthand side and a Runge–Kutta second-order approximation scheme are used for numerical implementation.

scholarly journals On a degenerate parabolic equation from double phase convection

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Double Phase ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$ a ( x ) and $b(x)$ b ( x ) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and $u_{t}\in L^{2}(Q_{T})$ u t ∈ L 2 ( Q T ) is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$ a ( x ) and $b(x)$ b ( x ) . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by $a(x)b(x)|_{x\in \partial \Omega }=0$ a ( x ) b ( x ) | x ∈ ∂ Ω = 0 is found for the first time.

scholarly journals Numerical Solution of One Problem of Carbon Dioxide Injection into the Rock

2021 ◽  
pp. 81-85
Author(s):  
R.A . Virts
Keyword(s):  
Carbon Dioxide ◽  
Porous Medium ◽  
Boundary Value Problem ◽  
Numerical Study ◽  
Boundary Value ◽  
Original System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Carbon Dioxide Injection ◽  
Initial Boundary Value

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible liquid or gas in a porous medium. A unique feature of the model under consideration is the incorporation of poroelastic properties of the solid skeleton. From a mathematical point of view, the equations of mass conservation for liquid / gaseous and solid phases, Darcy's law, the rheological ratio for a porous medium, and the conservation law of the balance of forces are considered. The work is aimed at numerical study of the model initial-boundary value problem of carbon dioxide injection into the rock with minimum initial porosity. Also, it is necessary to find out the parameters at which the porosity will increase in the upper layers of the rock and, as a result, the gas will come to the surface. Section 1 contains a statement of the problem and a brief review of scientific papers related to this topic. In Section 2, the original system of constitutive equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and a first-order equation for porosity. Section 3 presents the results and conclusions of a numerical study of the initial-boundary value problem.

scholarly journals Numerical Solution of a Two-Dimensional Problem of Fluid Filtration in a Deformable Porous Medium

2021 ◽  
pp. 88-92
Author(s):  
R.A. Virts
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Deformable Porous Medium ◽  
Balance Of Forces ◽  
Initial Boundary Value

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible fluid in a deformable porous medium. The model is based on the equations of conservation of mass for liquid and solid phases, Darcy’s law, the rheological relationship for a porous medium, and the law of conservation of the balance of forces. In this article, the equation of the balance of forces is taken in full form, i.e. the viscous and elastic properties of the medium are taken into account. The aim of the work is a numerical study of a model initial-boundary value problem. Section 1 gives a statement of the problem and a brief review of the literature on works related to this topic. In item 2, the original system of equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and the first-order equation for porosity. Section 3 proposes an algorithm for the numerical solution of the resulting initial-boundary value problem. For the numerical implementation, a variable direction scheme for the heat equation with variable coefficients is used, as well as the Runge — Kutta scheme of the fourth order of approximation.

COMPLETENESS OF ROOT FUNCTIONS AND ELEMENTARY SOLUTIONS OF THE THERMOELASTICITY SYSTEM

1995 ◽  
Vol 05 (05) ◽  
pp. 587-598
Author(s):  
YAKOV YAKUBOV
Keyword(s):  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Root Functions ◽  
Associated Functions ◽  
Initial Boundary Value

In this paper we prove the completeness of the root functions (eigenfunctions and associated functions) of an elliptic system (in the sense of Douglis-Nirenberg) corresponding to the thermoelasticity system with the Dirichlet boundary value condition. The problem is considered in a domain with a non-smooth boundary. Then an initial boundary value problem corresponding to the thermoelasticity system with the Dirichlet boundary value condition is considered. We find sufficient conditions that guarantee an approximation of a solution to the initial boundary value problem by linear combinations of some “elementary solutions” to the thermoelasticity system.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

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