具有自由边界的二维渗流问题

渗流的自由边界问题是工程上很受关注的问题．现有的数值分析方法需事先估计边界形态，逐次逼近．本文采用“变分不等式”的模式，结合有限元方法研究有自由边界的渗流问题，在整个结构区域内作有限元剖分，避免了传统的有限元分析中估计自由边界、反复修正计算区域的迭代过程．本文方法为简单而快速地分析渗流自由边界问题提供了一条有效的途径．     

一个反应扩散方程的自由边界问题

该文研究了一个描述原细胞生长的反应扩散方程的自由边界问题.利用非线性分析中的线性化思想和偏微分方程的估计理论,证明了该自由边界问题局部古典解的存在唯一性.     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．     

具有抛物线边界的二维弹性介质的Green函数

文章求解了具有抛物线边界的二维弹性介质的两种Ｇｒｅｅｎ函数，一种是自由边界问题，另一种是刚性边界问题。我们还求得了当抛物线边界退化成半无限裂纹或半无限刚性裂纹时裂纹尖端的奇异场，得到了集中力作用于边界的基本解，这个基本解使得我们可以通过沿边界积分确定任意分布荷载的弹性解．     

拉单晶问题差分解的收敛性

本文考虑一源于拉单晶热传导数学模型的三角形区域上的自由边界问题的数值解，给出差分格式，并证明其数值解的收敛性。     

非稳定井流的自由边界问题

<正> 近年来 C.Baiocchi 在研究水坝的浸润面时,提出了一种 Baiocchi 变换,它成为研究一类稳定自由边界问题的有力工具.A.Torelli 在处理非稳定问题时发展了 Baioc-chi 变换.基于 Baiocchi 等人的工作,C.W.Cryer 和 H.Fetter 研究了稳定井流的自由边界问题.本文进一步讨论非稳定可压缩井流的自由边界问题.我们通过 Torelli 变换把它归结为一个发展型变分不等方程,证明了该方程的解存在唯一.     

一类周期自由边界问题解的渐近性态

本文证明了化学反应扩散过程中的一类周期自由边界问题古典解的存在唯一性，并且讨论了潜热L→0时周期Stefan问题古典解的渐近性态及误差估计．     

一个肿瘤生长自由边界问题解的渐近性态

该文研究根据Byrne和Chaplain的思想建立的一个描述抑制物作用下无坏死核肿瘤生长的数学模型, 这个模型是一个非线性反应扩散方程组的自由边界问题. 作者运用反应扩散方程理论中的上下解方法结合自由边界问题的迭代技巧, 研究了解的渐近性态, 在营养物消耗函数f、抑制物消耗函数g和肿瘤细胞繁衍函数S的一些一般条件下,证明当常数c₁,c₂(肿瘤细胞分裂速率和营养物、抑制物扩散速率的比值)都非常小时,在一定的初边值条件下肿瘤趋于消失,在另外一些初边值条件下肿瘤半径趋于一个常数,进而时变解将趋于一个稳态解.     

拟线性双曲型方程组典型自由边界问题的整体经典解

本文对拟线性双曲组的某些典型边值问题及典型自由边界问题,证明了整体经典解的存在唯一性。     

一类永久博弈期权的定价

博弈期权是由Kifer引进的,本质上是美式期权的一种,它使买卖双方都有权在到期日前的任何时刻中止合约来维护自己的权益。在股票波动率非常数时,对一类特殊类型的博弈期权进行了研究,通过解一个自由边界问题,得到了其价格的闭式解。     

轴对称渗流中自由边界问题的近似解法

１．问题的提法 本文讨论各向异性非均匀介质的轴对称稳定渗流问题,我们延用［７］的记号。设有一水井（或油井）,其截面如图１所示,ｚ轴为对称轴,Ｄ为渗流区域,其边界为ＡＢＣＥＦＡ,Ｋ＝｛ｋｉｊ（ｒ,ｚ,ｈ,ｑ）｝为对称渗流张量,它依赖于柱坐标中的ｒ,ｚ,压头ｈ＝ｚ＋ｐ／ρｇ及渗流速率其中ｐ为点（ｒ,ｚ）处的压力,ρ为流体密度,ｇ为重力加速度．ｒ０为井的半径,Ｈ１为液面的高度,同时假设当ｒ≥Ｒ时,其渗流速度Ｖ＝０． 由渗流理论,有引入热函数,流函数 Ｄ中一点）,则满足下列一阶非线性方程组其中,若（ｉ＝ １…     

Study of a boundary value transmission problem for two-dimensional flows in a piecewise anisotropic inhomogeneous porous layer

We consider a boundary value transmission problem for two-dimensional filtration flows in an anisotropic porous layer consisting of adjacent domains in which the media have essentially different conductivities (permeability and thickness). In general, the layer conductivity is specified by a nonsymmetric second rank tensor whose components are modeled by continuously differentiable functions of coordinates. To study the problem, we use two complex planes, the physical plane and an auxiliary plane, which are related by a homeomorphic (one-to-one and continuous) transformation satisfying an equation of the Beltrami type. On the physical plane, we pose a transmission problem for a rather complicated elliptic system of equations. This problem is reduced on the auxiliary plane to canonical form, which dramatically simplifies the analysis of the problem. Then the problem is reduced to a system of boundary singular integral equations with generalized kernels of the Cauchy type, which are expressed via the fundamental solutions of the main equations. The boundary value transmission problem studied here can be used as a mathematical model of processes arising in the recovery of fluids (water and oil) from natural soil formations of complicated geological structure.     

A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

On a price formation free boundary model by Lasry and Lions

We discuss global existence and asymptotic behaviour of a price formation free boundary model introduced by Lasry and Lions in 2007. Our results are based on a construction which transforms the problem into the heat equation with specially prepared initial datum. The key point is that the free boundary present in the original problem becomes the zero level set of this solution. Using the properties of the heat operator we can show global existence, regularity and asymptotic results of the free boundary.     

Gas emission into the excavated space during the mining of coal deposits

We consider the problem of nonsteady filtration of a gas in a bounded domain with an unknown moving boundary. We obtain a closed-form solution both for the case when the moving boundary is smaller than the boundary of the filtration region and in the case when the two boundaries coincide. We study the regularities of variation of gas emission, pressure distribution, and advance of the free boundary as functions of time and the filtration properties of the coal vein. Four figures. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 81–85, 1991.     

Proprieta’ della frontiera libera in un problema di filtrazione

One describes the behavior of the free boundary in the problem of filtration of a fluid through a porous medium.     

Liquid Flow in Partially Saturated Porous Media

We consider a picture for the filtration of a liquid in a partiallysaturated porous medium, leading to a two-phase one-dimensionalfree boundary problem of the following type: The liquid pressuresatisfies an elliptic equation in the saturated region and anon-linear parabolic equation in the unsaturated region, whilepressure and velocity are continuous across the interface. This scheme reduces to the study of the non-linear parabolicfree boundary problem in the unsaturated phase with cauchy dataprescribed on the free boundary, for such a problem existence,uniqueness and continuous dependence theorems are proved.     

On a multiple credit rating migration model with stochastic interest rate

In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.     

On a price formation free boundary model by Lasry and Lions: The Neumann problem

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry and Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.     

Convergence analysis of trial free boundary methods for the two-dimensional filtration problem

Summary In this paper, we present a scheme of convergence analysis of trial free boundary methods for the two-dimensional filtration (or dam) problem. For the purpose we present a new variational principle of the filtration problem. This variational principle is defined on the set of admissible domains (candidates of the solution) in the dam. Under mild assumptions on the configuration of the dam, we may assume that all admissible domains are mapped from the unit disk by conformal mappings. Thus, proving convergence of trial free boundaries is reduced to proving convergence of the conformal mappings on the unit disk, and it is done using a method in the theory of minimal surfaces. Numerical examples are given.