Implicit difference methods for evolution functional differential equations

A general theory of implicit difference schemes for nonlinear functional differential equations with initial boundary conditions is presented. A theorem on error estimates of approximate solutions for implicit functional difference equations of the Volterra type with an unknown function of several variables is given. This general result is employed to investigate the stability of implicit difference schemes generated by first-order partial differential functional equations and by parabolic problems. A comparison technique with nonlinear estimates of the Perron type for given functions with respect to the functional variable is used.     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.     

二维非饱和水流的全离散广义差分格式分析及其数值模拟

建立二维非饱和水流问题的全离散广义差分格式,讨论了全离散广义差分解的存在唯一性,并给出最优误差估计的证明．最后给出数值算例,验证方法的有效性.     

一维线性Sobolev方程广义差分法

1引言Sobolev方程在流体穿过裂缝岩石的渗透理论,土壤中湿气迁移问题,不同介质中的热传导等许多数学物理问题中有着广泛的应用．Sobolev方程解的存在唯一性以及标准有限元和混合有限元已得到,但因为广义差分法是近几十年发展起来的一种计算方法,而且建立误差估计非常困难,所以有关广义差分法的理论并不完善．关于广义差分法的W~1,p和L~P模估计可参考文献[12]．本文将给出一维线性Sobolev方程的广义差分格式,通过     

Some results on numerical divided difference formulas

The remainder estimates of numerical divided difference formula are given for the functions of lower and higher smoothness, respectively. Then several divided difference formulas with super-convergence are derived with their remainder expressions.     

Adequacy of finite difference schemes for convection‐diffusion equations

Finite difference schemes for the numerical solution of singularly perturbed convection problems on uniform grids are studied in the limit case where the viscosity and the meshsize approach zero at the same time. The present error estimates are given in terms of order of magnitude in the above limit process and are useful in a priori choosing adequate schemes and meshsizes for boundary‐layer problems and problems with closed characteristics. Published 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 280–295, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10007     

Discrete Approximations of Controlled Stochastic Systems with Memory: A Survey

This survey article considers discrete approximations of an optimal control problem in which the controlled state equation is described by a general class of stochastic functional differential equations with a bounded memory. Specifically, three different approximation methods, namely (i) semidiscretization scheme; (ii) Markov chain approximation; and (iii) finite difference approximation, are investigated. The convergence results as well as error estimates are established for each of the approximation methods.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

方程u_(tt)=u_(xxt)+f(u_x)_x初边值问题的差分法

The finite difference method is considered for the followinginitial-boundary-value problem: arrayllutt=uxxt+f(ux)x, & (x,t) QT, u(x,0) =(x), & x ［0,1］, ut(x,0) = (x), & x ［0,1］, u(0,t) =u(1,t) =0, & t ［0,T］,array. where f(s),(x) and (x) are given functions;QT=［0,1］ ［0,T］. The convergence of the finite difference schemesis verified by discrete functional analysis methods and prior estimationtechniques.     

The hyperbolic–elliptic equation with the nonlocal condition

The nonlocal boundary value problem for a hyperbolic–elliptic equation in a Hilbert space is considered. The stability estimate for the solution of the given problem is obtained. The first and second orders of difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes are supported by the results of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Gidas-Ni-Nirenberg results for finite difference equations: Estimates of approximate symmetry

Are positive solutions of finite difference boundary value problems Δhu=f(u) in Ωh, u=0 on ∂Ωh as symmetric as the domain? To answer this question we first show by examples that almost arbitrary non-symmetric solutions can be constructed. This is in striking difference to the continuous case, where by the famous Gidas-Ni-Nirenberg theorem [B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related problems via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243] positive solutions inherit the symmetry of the underlying domain. Then we prove approximate symmetry theorems for solutions on equidistantly meshed n-dimensional cubes: explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer. The quality of the estimates depends on whether or not f(0)<0. The one-dimensional case stands out in two ways: the proofs are elementary and the estimates for the defect of symmetry are O(h) compared to O(1/|log(h)|) in the higher-dimensional case.     

Delay difference equations in Banach spaces

We derive explicit stability conditions for semilinear delay difference equations in a Banach space. It is assumed that the nonlinearities of the considered equations satisfy the local Lipschitz condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.     

Suboptimal Solutions to Dynamic Optimization Problems via Approximations of the Policy Functions

The approximation of the optimal policy functions is investigated for dynamic optimization problems with an objective that is additive over a finite number of stages. The distance between optimal and suboptimal values of the objective functional is estimated, in terms of the errors in approximating the optimal policy functions at the various stages. Smoothness properties are derived for such functions and exploited to choose the approximating families. The approximation error is measured in the supremum norm, in such a way to control the error propagation from stage to stage. Nonlinear approximators corresponding to Gaussian radial-basis-function networks with adjustable centers and widths are considered. Conditions are defined, guaranteeing that the number of Gaussians (hence, the number of parameters to be adjusted) does not grow “too fast” with the dimension of the state vector. The results help to mitigate the curse of dimensionality in dynamic optimization. An example of application is given and the use of the estimates is illustrated via a numerical simulation.     

Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

On the convergence of the method for indefinite integration of oscillatory and singular functions

We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller [8]. To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates.     

Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

On some optimal control problems and their finite difference approximations and regularization for quasilinear elliptic equations with controls in the coefficients

Mathematical statements of the optimal control problems for quasilinear elliptic equations with the controls in the variable coefficients of the equation of state are considered. Both local and integral constraints on the controls are considered. The objective functionals correspond to the optimization with respect to a certain number of quality indexes. Finite difference approximations of optimization problems are constructed, and estimates of the approximation error with respect to the state and to the objective functional are established. The weak convergence in control is proved. The approximations are regularized after Tikhonov. Interesting examples of some applied optimization problems that naturally lead to the nonlinear optimal control problems examined in this paper are considered.     

非线性对流扩散方程的UNO-MMOCAA差分方法

本文把MMOCAA差分方法与UNO插值相结合，提出了求解对流占优扩散问题的UN0—MMOCAA差分方法，它避免了基于高次(≥2)Lagrange插值的MMOCAA差分方法在方程解的陡峭前沿附近产生的振荡．本文通过引入辅助插值算于等方法，给出了非线性UNO—MMOCAA差分格式的误差分析．数值例子表明新格式无振荡。     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

带五次项的非线性Schrödinger方程的守恒差分格式

本文研究带有五次项的非线性Schrödinger方程初边值问题的有限差分法,其中方程中二阶偏导数项的系数、五次项的系数及初值满足下面的条件（1.6）.针对此问题,我们研究了一个守恒差分格式,在条件（1.6）下,差分解的$L^{\infty}$模先验估计被得到.在此基础上,我们得到了差分解最优$L^2$模的误差估计.