scholarly journals A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1907
Author(s):  
Jorge E. Macías-Díaz
Keyword(s):  
Stochastic Model ◽  
Moving Boundary ◽  
Numerical Study ◽  
Random Variable ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Structural Features ◽  
Stability And Convergence ◽  
General Stochastic Model ◽  
Initial Boundary Value

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

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.

Methods of Moving Boundary Based on Artificial Boundary in Heat Conduction Direct Problem

2012 ◽  
Vol 490-495 ◽  
pp. 2282-2285
Author(s):  
Xue Yan Zhang ◽  
Qian Li ◽  
Da Quan Gu ◽  
Tai Ping Hou
Keyword(s):  
Neumann Boundary Condition ◽  
Direct Problem ◽  
Moving Boundary ◽  
Classical Problem ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moving Boundary Problem ◽  
Artificial Boundary ◽  
Initial Boundary Value

The initial boundary value problem for parabolic equation with Neumann boundary condition is a kind of classical problem in partial differential equations. In this paper we use the artificial boundary to solve the moving boundary problem. Potential theory and difference method are discussed. Numerical results are given to support the proposed schemes and to give the compare of the two methods.

scholarly journals A Unified Analytical Approach to Fixed and Moving Boundary Problems for the Heat Equation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 749
Author(s):  
Marianito R. Rodrigo ◽  
Ngamta Thamwattana
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Moving Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moving Boundary Problems ◽  
Initial Value ◽  
Boundary Problems ◽  
Fixed Boundary Problems ◽  
Initial Boundary Value

Fixed and moving boundary problems for the one-dimensional heat equation are considered. A unified approach to solving such problems is proposed by embedding a given initial-boundary value problem into an appropriate initial value problem on the real line with arbitrary but given functions, whose solution is known. These arbitrary functions are determined by imposing that the solution of the initial value problem satisfies the given boundary conditions. Exact analytical solutions of some moving boundary problems that have not been previously obtained are provided. Moreover, examples of fixed boundary problems over semi-infinite and bounded intervals are given, thus providing an alternative approach to the usual methods of solution.

scholarly journals Factorized difference scheme for two-dimensional subdiffusion equation in nonhomogeneous media

2016 ◽  
Vol 99 (113) ◽  
pp. 1-13 ◽  
Author(s):  
Aleksandra Delic ◽  
Sandra Hodzic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Nonhomogeneous Media ◽  
Subdiffusion Equation ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional subdiffusion equation in nonhomogeneous media is proposed. Its stability and convergence are investigated. The corresponding error bounds are obtained.

HIGH-ACCURACY DIFFERENCE SCHEMES FOR THE NONLINEAR TRANSFER EQUATION

2007 ◽  
Vol 12 (4) ◽  
pp. 469-482 ◽  
Author(s):  
Agnieszka Paradzinska ◽  
Piotr Matus
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Nonlinear Transport ◽  
Basic Principles ◽  
Order Of Accuracy ◽  
Accuracy Difference ◽  
Initial Boundary Value ◽  
The Right

In the present paper, for the initial boundary value problem for the non‐homogeneous nonlinear transport equationthe basic principles for constructing difference schemes of any order of accuracy O(#GTM), M ≥ 1, on characteristic grids with the minimal stencil were introduced. To construct a difference scheme the Steklov averaging idea for the right‐hand sidewas used. The case of f(u) = λu2 was investigated in detail. A strict analysis of the order of approximation, stability, and convergence in nonlinear case was made. The performed numerical experiments justify theoretical results.

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