伴有边界摄动的Volterra型积分微分方程组的奇摄动

讨论了伴有边界摄动的二阶非线性Volterra型积分微分方程组的奇摄动.在适当的条件下,利用对角化技巧证明了解的存在性,构造出解的渐近展开式并给出余项的一致有效估计.     

Higher-order Lidstone boundary value problems for elliptic partial differential equations

The aim of this paper is to show the existence and uniqueness of a solution for a class of 2nth-order elliptic Lidstone boundary value problems where the nonlinear functions depend on the higher-order derivatives. Sufficient conditions are given for the existence and uniqueness of a solution. It is also shown that there exist two sequences which converge monotonically from above and below, respectively, to the unique solution. The approach to the problem is by the method of upper and lower solutions together with monotone iterative technique for nonquasimonotone functions. All the results are directly applicable to 2nth-order two-point Lidstone boundary value problems.     

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

     

Singular perturbation problems for linear elliptic differential operators of arbitrary order. I. Degeneration to elliptic operators

Dirichlet problems of singular perturbation type for linear elliptic differential operators of arbitrary order are studied. The asymptotic validity of approximations constructed by the boundary layer method is demonstrated in the maximum norm by means of a priori estimates.     

Hamiltonian identities for elliptic partial differential equations

New identities for elliptic partial differential equations are obtained. Several applications are discussed. In particular, Young's law for the contact angles in triple junction formation is proven rigorously. Structure of level curves of saddle solutions to Allen-Cahn equation are also carefully analyzed.     

Hopscotch methods for elliptic partial differential equations

This paper analyses hopscotch algorithms when used to solve elliptic partial differential equations. A comparison with standard methods is made for the model problem.     

Nonlocal boundary-value problems for systems on linear partial differential equations

We study the classical well-posedness of problems with nonlocal two-point conditions for typeless systems of linear partial differential equations with variable coefficients in a cylindrical domain. We prove metric theorems on lower bounds for small denominators that appear in the construction of solutions of such problems.     

On numerical solution of stochastic partial differential equations of elliptic type

We study lattice approximations of stochastic PDEs of elliptic type, driven by a white noise on a bounded domain in ? ^d , for d = 1, 2, 3. We obtain estimates for the rate of convergence of the approximations.     

Nonclassical boundary value problems for some systems of partial differential equations

We consider nonclassical boundary value problems with discontinuous conditions for some systems of partial differential equations of the first and second order.     

Fredholm property of elliptic boundary value problems for partial differential and differential- operator equations

In this paper we find conditions on boundary value problems for elliptic differential-operator equations of the 4-th order in an interval to be fredholm. Apparently, this is the first publication for elliptic differential-operator equations of the 4-th order, when the principal part of the equation has the form u′?n(t) + Au″(t) + Bu(t), where AB^-1/2 is a bounded operator and is not compact. As an application we find some algebraic conditions on boundary value problems for elliptic partial equations of the 4-th order in cylindrical domains to be fredholm. Note that a new method has actually been suggested here for investigation of boundary value problems for elliptic partial equations of the 4-th order.     

The method of quasi-reversibility to solve the Cauchy problems for elliptic partial differential equations

We use the method of quasi-reversibility, first introduced in [1], to solve the ill-posed Cauchy problems for an elliptic operator such as P = –Δ. In particular, a non-conforming method is implemented, some a priori error estimates are derived, and a few numerical computations are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular boundary-value problems for ordinary second-order differential equations

This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.)Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105–201, 1987.     

Quadratic spline collocation methods for elliptic partial differential equations

We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard formulation of these methods leads to non-optimal approximations. In order to derive optimal QSC approximations, high order perturbations of the PDE problem are generated. These perturbations can be applied either to the PDE problem operators or to the right sides, thus leading to two different formulations of optimal QSC methods. The convergence properties of the QSC methods are studied. OptimalO(h ^3–j ) global error estimates for thejth partial derivative are obtained for a certain class of problems. Moreover,O(h ^4–j ) error bounds for thejth partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of the QSC methods. Performance results also show that the QSC methods are very effective from the computational point of view. They have been implemented efficiently on parallel machines.This research was supported in part by David Ross Foundation (U.S.A) and NSERC (Natural Sciences and Engineering Research Council of Canada).     

A posteriori estimators for nonlinear elliptic partial differential equations

Many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations. For each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible. Usually, the available estimators may be classified into two types, i.e., the implicit estimators (based on the solution of a local problem) and the explicit estimators (based on some suitable norm of the residual in a dual space). Regarding the performance, an estimator is called asymptotically exact if it is a higher-order perturbation of a norm of the exact error. Nowadays, one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems. The situation is even worse when the nonlinearities involve the highest derivatives occurring in the PDE being considered (strongly nonlinear PDEs). In this work we establish conditions under which those estimators, originally developed for linear problems, may be used for strongly nonlinear problems, and how that could be done. We also show that, under some suitable hypothesis, the estimators will be asymptotically exact, whenever they are asymptotically exact for linear problems. Those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems.