Stable numerical solution of a fractional-diffusion inverse heat conduction problem

The ill-posed problem of attempting to recover the boundary temperature and the heat flux functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques is introduced for the numerical solution of the discrete problem. Stability bounds, error estimates and numerical examples of interest are also presented.     

多目标应急设施选址问题的模拟退火算法

考虑应急设施选址时的成本和应急时间因素,给出了多目标应急设施选址问题的模型,通过设置罚函数将该多约束问题转化成易于计算机求解的简单约束模型,进而在初始解的选取、温度参数的控制、可行解的迭代策略和算法终止条件等方面为之设计了模拟退火算法,并通过仿真证明了该算法的有效性。     

Numerical treatment of a quasilinear initial value problem with boundary layer

The paper deals with the singularly perturbed quasilinear initial value problem exhibiting initial layer. First the nature of solution of differential problem before presenting method for its numerical solution is discussed. The numerical solution of the problem is performed with the use of a finite-fitted difference scheme on an appropriate piecewise uniform mesh (Shishkin-type mesh). An error analysis shows that the method is first-order convergent except for a logarithmic factor, in the discrete maximum norm, independently of the perturbation parameter. Finally, numerical results supporting the theory are presented.     

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

We introduce a continuous (downscaling) data assimilation algorithm for the 2D Bénard convection problem using vorticity or local circulation measurements only. In this algorithm, a nudging term is added to the vorticity equation to constrain the model. Our numerical results indicate that the approximate solution of the algorithm is converging to the unknown reference solution (vorticity and temperature) corresponding to the measurements of the 2D Bénard convection problem when only spatial coarse-grain measurements of vorticity are assimilated. Moreover, this convergence is realized using data which is much more coarse than the resolution needed to satisfy rigorous analytical estimates.     

一类周期多孔固体材料热传导性能计算的孔洞填充方法

对于一类周期多孔固体材料,提出了一种孔洞填充方法,用一种极低热导率的材料填充孔洞,将原本几何复杂的单相多孔区域的热传导边值问题转化为几何简单的多相无孔区域上的问题,借助于延拓定理给出了填充前后材料的热传导问题解和均匀化热导率的误差估计,对结果的分析表明可以用填充后材料热传导问题的双尺度解近似原孔洞问题的解,在最后的数值算例中,讨论了具有对称和非对称单胞构造的周期多孔固体的热传导边值问题,分别比较了多孔固体材料填充前后的均匀化热导率、温度和温度梯度解,结果表明孔洞填充方法的确可行。     

A semi-discrete tailored finite point method for a class of anisotropic diffusion problems

This work proposes a tailored finite point method (TFPM) for the numerical solution of an anisotropic diffusion problem, which has much smaller diffusion coefficient along one direction than the other on a rectangular domain. The paper includes analysis on the differentiability of the solution to the given problem under some compatibility conditions. It has detailed derivation for a semi-discrete TFPM for the given problem. This work also proves a uniform error estimate on the approximate solution. Numerical results show that the TFPM is accurate as well as efficient for the strongly anisotropic diffusion problem. Examples include those that do not satisfy compatibility and regularity conditions. For the incompatible problems, numerical experiments indicate that the method proposed can still offer good numerical approximations.     

Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation

In this paper we consider the Monte Carlo solution of the Cauchy problem for a nonlinear parabolic equation. Using the fundamental solution of the heat equation, we obtain a nonlinear integral equation with solution the same as the original partial differential equation. On the basis of this integral representation, we construct a probabilistic representation of the solution to our original Cauchy problem. This representation is based on a branching stochastic process that allows one to directly sample the solution to the full nonlinear problem. Along a trajectory of these branching stochastic processes we build an unbiased estimator for the solution of original Cauchy problem. We then provide results of numerical experiments to validate the numerical method and the underlying stochastic representation.     

Acceleration-constrained time-optimal control inndimensions

The minimum-time acceleration of a generalized point mass from an initial position and velocity to the origin inndimensions is solved by transforming the problem to an equivalent problem in two dimensions and analytically integrating the system differential equations. Computation of the optimal control is thereby reduced to the solution of five simultaneous nonlinear equations. A numerical continuation method is presented for solving these equations by starting at the known solution of a related single-dimensional problem and progressing incrementally to the desired solution. The problem and solution method are illustrated by a numerical example.     

全变差正则化在抛物型方程初始条件反问题的应用

当反问题反演的函数不连续时,一般的正则化算法反演效果不令人满意,用全变差正则化方法对抛物型方程初始条件反问题进行求解,并进行了数值分析和数值模拟,结果显示数值解与真解吻合较好,表明该方法对于不连续函数求解具有高效、稳定等优点.     

On the problem of spacing of oil wells and control of their production rates

A statement and a numerical solution of the optimization problem for sites of spacing of wells (well clusters) and modes of their operation in an oil field are presented. The mathematical problem represents a parametric problem of optimal control of the distributed system with concentrated sources, which is described by partial differential equations. Similar problems arise in the development and control of technological processes of various assignments and the design of control systems of technical objects. For its numerical solution, the initial problem is reduced to a finite-dimensional optimization problem with constraints of special features. Formulas of a gradient of the functional of the reduced problem are derived. Results of the solution of control problems are given.     

Asymptotic behaviour of a mathematical model of hematopoietic stem cell dynamics

We deeply researched into the asymptotic behaviour of a numerical method adapted for the solution of mathematical model of hematopoiesis which describes the dynamics of a stem cell population. We investigated the stationary solutions of the original model by their numerical approximation: we proved the existence of a numerical stationary solution that provides a good approximation to the nontrivial equilibrium solution of the problem. Also, we presented a numerical simulation which confirms this behaviour.     

一类自治脉冲微分方程的动力学研究

对一类自治脉冲微分方程的动力学性质进行了研究,给出了半平凡周期解的存在与稳定的充分条件,建立Poincare映射将周期解问题转化为不动点问题．理论分析及数值模拟表明,半平凡周期解通过跨临界分岔获得稳定的正周期-1解．数值模拟显示,随着控制参数的变化,正周期-1解通过倍周期分岔出正周期-2解,再通过一系列倍周期分岔通向混沌．     

Numerical procedure for the determination of an unknown parameter in a parabolic equation

A numerical procedure for an inverse problem of determining unknown source parameter of one-dimensional parabolic equation subject to the specification of the solution at internal point along with the usual initial boundary conditions is considered. By using some transformation the problem is reformulated to a nonlocal parabolic problem. Some numerical examples using the proposed numerical procedure are presented.     

A truncated Legendre spectral method for solving numerical differentiation

A numerical differentiation problem for a given function with noisy data is discussed in this paper. A truncated spectral method has been introduced to deal with the ill-posedness of the problem. The theoretical analysis shows that the smoother the genuine solution, the higher the convergence rate of the numerical solution by our method. Numerical examples are also given to show the efficiency of the method.     

由两个特征对构造正定Jacobi矩阵

132 数值计算与计算机应用2002 it巨1.引言具有如下形状的实对称矩阵:ldl hi \Ibid。。6,l人=I”.”·.’·.I 门)16、_,tL、_fo、_11\ b、;_IG、l称为n阶实对称三对角矩阵.若人还满足:(a)b;>     

A numerical algorithm for singular perturbation problems exhibiting weak boundary layers

A class of singularly perturbed two-point boundary-value problems (BVP) for second-order ordinary differential equations is considered here. To avoid the numerical difficulties in the solution to these problems, we divide the domain into two subdomains. The first BVP is a layer domain problem and the second BVP is a regular domain problem. Error estimates are derived for the numerical solution. Numerical examples are provided in support of the proposed method.     

Optimal control of the elastic development drive of an oil pool

In the present article a problem of optimal control with respect to performance to determine the optimal period of development of an oil pool is posed and a numerical method for the solution of the problem is proposed. Results of a numerical experiment are presented.     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

Flow rate boundary problems for an incompressible fluid in deformable domains: Formulations and solution methods

In this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain. As well known, in this case there is the problem of prescribing realistic boundary data on the artificial sections, when only partial data are available. This problem has been investigated extensively for the rigid case. In this work we start considering the compliant case, by focusing on the flow rate conditions for the fluid. We propose three formulations of this problem, different algorithms for its numerical solution and carry out several 2D numerical simulations with the aim of comparing the performances of the different algorithms.     

A new numerical approach to a non-steady filtration problem

A free-boundary transient problem of seepage flow is studied from a numerical standpoint. From a suitable formulation of the problem in terms of variational inequality we introduce a new numerical approach of the implicit type and based on the finite element method. In this approach the problem is solved on a fixed region and the position of the free boundary is automatically found as part of the solution of the problem; so it is not necessary to solve a succession of problems with different positions of the free boundary. We prove stability and convergence for the approximate solution and we give several numerical results. Work supported by C. N. R. of Italy through the Laboratorio di Analisi Numerica of Pavia.