一个二次冻胀的数学模型

In this paper a new mathematical model of secondary frost heave is presented. It isexpected that the problem considered under some assumptions is well-posed.     

A novel self-adaptive control strategy of frost prevention and retardation for air source heat pumps in winter conditions

As frost accumulates on the heat exchanger surface with time, system operating performance will be dramatically degraded, and limit its use in climates susceptible to frost formation. A novel self-adaptive control strategy of frost prevention and retardation for air source heat pumps (ASHP) is introduced in this paper. The control strategy relies on a new thermodynamic model, which involves a Dimensionless Artificial Neural Network (DANN) correlation model describing frost accumulation for ASHP on the air-side of the fin-and-tube heat exchanger. The dimensionless parameters of this DANN model, including the ambient conditions, 6 commonly used refrigerants, and the geometric parameters of the heat exchanger, are considered in the model. To enhance the reliability of DANN, we develop a self-adaptive algorithm, including determining the optimal transfer algorithm and selecting the number of neurons in the hidden layer, for the DANN model. Results show a limited relative error (7.55%) between calculated values and experimental data, which help researchers and manufacturers analyze the complicated frosting process and design the new ASHPs more reasonably in different regions with different ambient conditions.     

A moving boundary problem derived from heat and water transfer processes in frozen and thawed soils and its numerical simulation

The seasonal change in depths of the frozen and thawed soils within their active layer is reduced to a moving boundary problem,which describes the dynamics of the total ice content using an independent mass balance equation and treats the soil frost/thaw depths as moving(sharp)interfaces governed by some Stefan-type moving boundary conditions,and hence simultaneously describes the liquid water and solid ice states as well as the positions of the frost/thaw depths in soil.An adaptive mesh method for the moving boundary problem is adopted to solve the relevant equations and to determine frost/thaw depths,water content and temperature distribution.A series of sensitivity experiments by the numerical model under the periodic sinusoidal upper boundary condition for temperature are conducted to validate the model,and to investigate the effiects of the model soil thickness,ground surface temperature,annual amplitude of ground surface temperature and thermal conductivity on frost/thaw depths and soil temperature.The simulated frost/thaw depths by the model with a periodical change of the upper boundary condition have the same period as that of the upper boundary condition,which shows that it can simulate the frost/thaw depths reasonably for a periodical forcing.     

A class of free boundary problems to quasi-linear parabolic equations

1.IntroductionStefan-likeproblemswithakineticconditiononthefreeboundarytolinearparabolicequationshavebeenconsideredbyseveralauthors(see[1-4]andreferencestherein),buttoquasi-linearonesproblemssimilar,whichareconsideredhereandmoredifficultthanpreviouso...     

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.     

An Interactive Algorithm for Vehicle Routeing for Winter — Gritting

In winter, when roads may become dangerously slippery due to frost, ice or snow, a de-icing agent (usually salt) is spread on them by a local authority for safety reasons. A gritter only needs to travel once down all those roads requiring treatment, as it can spread the salt onto both sides of the carriageway. The problem studied is how to design routes for gritters which will minimise costs. This problem is a type of Capacitated Arc Routeing Problem including consideration of multiple depot locations, limited vehicle capacities, time constraints on when roads must be gritted, roads with different priorities for gritting, the existence of one-way roads and salt-refilling locations. The objective function to be optimised depends on both the total distance travelled and the number and capacity of the gritters. A heuristic algorithm is devised with a computer program which allows user-interaction, and provides a practical tool for planning gritter routes. The model is linked to a GIS containing information on the road network for the County of Lancashire. Test results from the interactive algorithm are found to outperform another existing approach which solves the same problem.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

Some algorithmic problems for relatively free groups

The paper contains the proof of the algorithmic unrecognizability for two properties of some relatively free soluble groups, as well as the proof of the lack in a certain sense of the recursive enumerability for one more property of such groups. It is also proved that the indecomposability in the class of soluble finitely based varieties of groups is unrecognizable. An example is given of a relatively free soluble group with solvable word problem and with unsolvable identity problem. A result is obtained involving truth-table degrees of the word problem for relatively free soluble groups. Bibliography: 22 titles.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

A free boundary problem for two-species competitive model in ecology

This paper deals with a free boundary problem which is used to describe the two-species competitive model in ecology. The existence and uniqueness of a global classical solution are given by invoking the Schauder fixed point theorem. We study the evolution of the free boundary problem and show that the free boundary problem is well posed.     

On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

Free products and the word problem

An example is given to show that even in a variety whose free word problem (identity problem) is solvable, the free product of two recursive presentations with solvable word problems may have unsolvable word problem. Under some extra conditions on the syntax of the identities defining the variety however, the free product is shown to preserve solvability of the word problem for recursive presentations. The conditions can be checked mechanically, and common varieties such as semigroups and groups satisfy them, as well as many less familiar varieties. The results are obtained by using rewrite-completion techniques.Presented by Ralph Freese.     

Numerical solution of the Falkner-Skan equation based on quasilinearization

We present two iterative methods for solving the Falkner-Skan equation based on the quasilinearization method. We formulate the original problem as a new free boundary value problem. The truncated boundary depending on a small parameter is an unknown free boundary and has to be determined as part of solution. Using a change of variables, the free boundary value problem is transformed to a problem defined on [0, 1]. We apply the quasilinearization method to solve the resulting nonlinear problem. Then we propose two different iterative algorithms by means of a cubic spline solver. Numerical results for various instances are compared with those reported previously in the literature. The comparisons show the accuracy, robustness and efficiency of the presented methodology.     

具有两种流体的轴对称渗流的一个自由边界问题

本文处理带有两种流体的轴对称的一个自由边界问题，其中在渗流区域的上部都是油，下部是水，这是同时取油注水的一个数学模型。下面，我们将用复分析方法求出此自由边界问题的一个解，并证明其解的唯一性。     

A nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems

The article proposes a nonoverlapping domain decomposition method for variational inequalities derived from free boundary problems. The free boundary value problem is broken up into two problems on nonoverlapping regions. In one region the problem is treated as a partial differential equation, while in the second region that contains the free boundary part, a variational inequality is considered. By solving these two related problems successively, we have shown that the successive solutions converge to the solution of the original problem. Application to a free surface seepage problem is given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

美式期权自由边界的计算方法

美式期权的自由边界问题在金融工程文献中已经引起了广泛的关注,然而,它的数值计算方法一直是一个难点.基于差分技巧,给出了满足具有有限到期日的美式期权自由边界的两种计算方法,即,根据股票期权价格和相应的偏导数来确定自由边界条件.数值结果表明了上述两种方法下自由边界是一致性的.此外研究结果对自由边界的计算提供很好的科学依据.