Minimizing the variance of a mathematical expectation estimate for a diffusion process functional based on a parametric transformation of a parabolic boundary value problem

This paper deals with finding ways of reducing the variance of a mathematical expectation estimate for the functional of a diffusion process moving in a domain with an absorbing boundary. The estimate of mathematical expectation of the functional is obtained based on a numerical solution of stochastic differential equations (SDEs) by using the Euler method. A formula of the limiting variance is derived with decreasing integration step in the Euler method. A method of reducing the variance value of the estimate based on transformation of the parabolic boundary value problem corresponding to the diffusion process is proposed. Some numerical results are presented.     

A singular-perturbed two-phase Stefan problem

The asymptotic behavior of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit, the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.     

Estimation of derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary

A statistical method is proposed for estimating derivatives with respect to parameters of a functional of a diffusion process moving in a domain with absorbing boundary. The functional considered defines the probability representation of the solution of a corresponding parabolic first boundary-value problem. The problem posed is tackled by numerically solving stochastic differential equations (SDE) using the Euler method. An error of the proposed method is evaluated, and estimates of the variance of the resultant parametric derivatives are given. Some numerical results are presented.     

Moving collocation method for a reaction‐diffusion equation with a traveling heat source

We consider a reaction‐diffusion equation with a traveling heat source on an unbounded domain. The numerical simulation of the problem is difficult because of the moving singularity, the blow‐up phenomenon, and the delta function in the equation. Because we are only interested in the solution behavior near the heat source, we choose a bounded moving domain which contains the heat source and has the same speed as the source. Local absorbing boundary conditions are constructed on the boundaries of the moving domain. Then, we transform the moving domain to a fixed one. At last, a special moving collocation method is adopted. The new method is much simpler than the existing moving finite difference methods. Moreover, numerical experiments illustrate the accuracy and efficiency of our moving collocation method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Application of the spherical metric tensor to grid adaptation and the solution of applied problems

New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solution of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three-dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.     

药物从柱型高聚物基体内扩散释放的活动边界问题

采用修正积分法,得到了药物从柱形高聚物基体内扩散释放的活动边界问题的近似解析解,给出了扩散边界和药物释放分数的计算公式及其在不同初装浓度下的计算结果。计算结果与实验结果是一致的。进而,对药物从片形、柱形、球形等不同几何形状的基体内的释放问题进行了对照,分析了基体的不同几何形状对药物释放的影响。还给出了药物从柱形基体内扩散释放的有效时间的近似计算公式。这对临床实验具有一定的指导意义,也为进一步研究药物从柱型高聚物基体内扩散释放问题以及缓释制剂的设计提供了理论工具。     

A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate

In this article, we discuss a space-fractional diffusion logistic population model with Caputo fractional derivative and density-dependent dispersal rate. The numerical solution of the problem is obtained by using a finite difference scheme. The consistency and stability of the scheme for our solution to the problem are also discussed. The effect of the density-dependent dispersal rate and order of the space-fractional derivative are analyzed for the population density and expanding front (moving boundary).     

Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

Abstract

We study the limit at zero of the first-passage time density of a one-dimensional diffusion process over a moving boundary and we also deal with the inverse first-passage time problem, which consists of determining the boundary shape when the first-passage density is known. Our results generalize the analogous ones already known for Brownian motion. We illustrate some examples for which the results are obtained analytically and by a numerical procedure.     

An adaptive algorithm for the Thomas–Fermi equation

A free boundary value problem is introduced to approximate the original Thomas–Fermi equation. The unknown truncated free boundary is determined iteratively. We transform the free boundary value problem to a nonlinear boundary value problem defined on [0,1]. We present an adaptive algorithm to solve the problem by means of the moving mesh finite element method. Comparison of our numerical results with those obtained by other approaches shows high accuracy of our method.     

Radial basis function Hermite collocation approach for the numerical simulation of the effect of precipitation inhibitor on the crystallization process of an over‐saturated solution

This work is concerned with the analysis of the effect of precipitation inhibitors on the growth of crystals from over‐saturated solutions, by the numerical simulation of the fundamental mechanisms of such crystallization process. The complete crystallization process in the presence of precipitation inhibitor is defined by a set of coupled partial differential equations that needs to be solved in a recursive manner, due to the inhibitor modification of the molar flux of the mineral at the crystal interface. This set of governing equations needs to satisfy the corresponding initial and boundary conditions of the problem where it is necessary to consider the additional unknown of a moving interface, i.e., the growing crystal surface. For the numerical solution of the proposed problem, we used a truly meshless numerical scheme based upon Hermite interpolation property of the radial basis functions. The use of a Hermitian meshless collocation numerical approach was selected in this work due to its flexibility on dealing with moving boundary problems and their high accuracy on predicting surface fluxes, which is a crucial part of the diffusion controlled crystallization process considered here. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

On a Variational Inequality Formulation of an Electrochemical Machining Moving Boundary Problem and its Approximation by the Finite Element Method

An electrochemical machining moving boundary problem is formulated,after a change of variable, as an elliptic variational inequality.The unknown anode surface may now be found by solving just oneelliptic free boundary problem. The variational inequality isapproximated by the finite element method and numerical resultsare presented.     

Mathematical Modelling of Heat Flow in Czochralski Crystal Pulling

A mathematical model of the heat flow in the holm region incrystal pulling by the Czochralski technique is developed. Thisis a moving boundary problem with two moving boundaries, thephase change surface and the air—liquid meniscus. Usingthe enthalpy method and co-ordinate transformation techniques,the problem is cast into a form suitable for numerical solution.A numerical scheme is outlined, and some results for the growthof germanium crystals are shown.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

A numerical method based on integro‐differential formulation for solving a one‐dimensional Stefan problem

A numerical method based on an integro‐differential formulation is proposed for solving a one‐dimensional moving boundary Stefan problem involving heat conduction in a solid with phase change. Some specific test problems are solved using the proposed method. The numerical results obtained indicate that it can give accurate solutions and may offer an interesting and viable alternative to existing numerical methods for solving the Stefan problem. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A Mildly Non-linear System of Parabolic Differential Equations Arising from Mass Transfer with Chemical Reaction in Electrochemistry

A system of three connected parabolic equations is studied.The first equation is the straight forward diffusion equationin one space dimension and the solution can be written down.The remaining two cannot be solved analytically but it is interestingto observe that a solution does exist for their difference.By considering the problem as a moving boundary value probleman approximate solution is obtained by a finite difference technique.An analysis of stability is performed and numerical resultsfor a specific chemical reaction are presented.     

Laguerre reproducing kernel method in Hilbert spaces for unsteady stagnation point flow over a stretching/shrinking sheet

This paper investigates the nonlinear boundary value problem resulting from the exact reduction of the Navier-Stokes equations for unsteady magnetohydrodynamic boundary layer flow over the stretching/shrinking permeable sheet submerged in a moving fluid. To solve this equation, a numerical method is proposed based on a Laguerre functions with reproducing kernel Hilbert space method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate.     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.     

Dynamic analysis of a non-conservative band–wheel system with a moving boundary part 2: Numerical results

The dynamic formulations of a non-conservative band/wheel system with a moving boundary are derived in the companion paper by use of Hamilton’s principle and calculus of variation. In this paper, numerical simulations for the system are obtained in transient amplitudes of the string and positions of the moving boundary by a modified finite difference method (FDM). Since the moving boundary position may not locate exactly at a grid point for any computational time, a special technique of the FDM is proposed to approximate both the transversality condition of a moving boundary and the partial differential equation of the neighboring grid points. The effects of parameters such as radius of the wheel, tension of the string, propagation speeds of the longitudinal and transverse wave and various initial conditions on the transient responses are investigated and compared with those of the fixed boundary problem.     

On a Moving Boundary Problem from Biomechanics

A moving boundary problem arising in biomechanical diffusiontheory which has previously been investigated by Crank &Gupta (1972a, b) is studied using a different method of solution.The method is based on an integral equation for the functiondefining the position of the moving boundary and an integralformula for the concentration. The integral equation is solvedasymptotically for small times and numerically during the entiremotion of the boundary. The concentration is estimated asymptoticallyfor small times and computed by numerical quadrature at laterinstants. The results are compared with those of Crank &Gupta. In most cases the agreement is fair.