One-parameter imbedding techniques for the solution of nonlinear boundary-value problems

A powerful numerical tool for the solution of nonlinear boundary-value problems —the one-parameter imbedding technique—is suggested. The basic principle is more than twenty years old; however, its numerical utilization has had only a few restricted applications up until recent times. The methods are divided into two categories: one- and multi-loop techniques. It is shown that the multi-loop techniques are of correct and incorrect type. Based on correct procedures new iteration techniques may be developed. Numerical solutions of differential equations arising for the one-parameter imbedding methods are presented along with the corresponding iteration techniques. Some typical imbedding procedures are discussed, and practical application of the method is demonstrated on calculated examples.     

Solutions explicites de problèmes de plongement

We construct the solutions of a class of solvable imbedding problems. Explicit examples are provided.     

Invariant imbedding for block tridiagonal systems

The invariant imbedding technique is applied to the block tridiagonal systems in a general setting using, when necessary, the generalized Moore-Penrose inverse of a matrix. Three examples are given, and the existence and the stability of solutions are discussed.     

泛函极小与非线性椭圆方程解的局部正则性

利用Giaquinta和Giusti的嵌入不等式和Sobolev空间方法,证明了在更一般条件下的变分问题中的泛函极小与非线性椭圆方程弱解的局部正则性.     

轴对称渗流中自由边界问题的近似解法

１．问题的提法 本文讨论各向异性非均匀介质的轴对称稳定渗流问题,我们延用［７］的记号。设有一水井（或油井）,其截面如图１所示,ｚ轴为对称轴,Ｄ为渗流区域,其边界为ＡＢＣＥＦＡ,Ｋ＝｛ｋｉｊ（ｒ,ｚ,ｈ,ｑ）｝为对称渗流张量,它依赖于柱坐标中的ｒ,ｚ,压头ｈ＝ｚ＋ｐ／ρｇ及渗流速率其中ｐ为点（ｒ,ｚ）处的压力,ρ为流体密度,ｇ为重力加速度．ｒ０为井的半径,Ｈ１为液面的高度,同时假设当ｒ≥Ｒ时,其渗流速度Ｖ＝０． 由渗流理论,有引入热函数,流函数 Ｄ中一点）,则满足下列一阶非线性方程组其中,若（ｉ＝ １…     

Existence and uniqueness of Sobolev type integrodifferential equations

An imbedding method for nonlinear Fredholm integral equations gives rise to a Sobolev type integrodifferential equation. For such equations, sufficient conditions are given to guarantee local existence and uniqueness of solutions. A Picard type theorem utilizing a Lipschitz condition is obtained.     

一类强耦合抛物型方程组初边值问题解的整体存在性

In this paper, we consider a strongly-coupled parabolic system with initial boundary values. Under the appropriate conditions, using Gagliard-Nirenberg inequality, Poincare inequality, Gronwall inequality and imbedding theorem, we obtain the global existence of solutions.     

Invariant imbedding and Riccati transformations

From its inception, the theory of invariant imbedding has been concerned with the study of various relations between the inputs and outputs of various physical processes. Where the processes could be modelled by differential or integro-differential equations, these ideas have led to the heuristic development of various functional relationships for the solutions of these equations. In this work, we show that for a general class of two point boundary value problems these relations can be obtained from mathematical arguments rather than physical ones. The principal result is the establishment of the equivalence of solving a family of two point boundary value problems and that of determining the existence of two transformations on the set of solutions of the given differential equations. We refer to these transformations as Riccati transformations. They are shown to be determined by a set of initial value problems which generalize the invariant imbedding equations obtained by previous authors. We work in the coordinate free setting of a Banach space. The usefulness of this approach is shown as we are able to readily extend our results to nonlocal and multipoint boundary conditions. An indication is made of how a similar theory applies to a class of problems for difference equations.     

常微分方程组的整体解及两点边值问题——Ⅰ.整体解方法

众所周知,n维向量函数u(x)的一阶常微分方程组,如在某点上只给出n_1相似文献   

一类更广泛的Захаров方程组的有限元分析

其中n=n(x,i)为离子的扰动量(实函数,ε为场量(复函数)。该方程组具有一系列重要性质,如具有一维孤立子解,即Langmuir孤立子,它的形成、发展和相互作用不同于KDV方程的孤立子,因而引起人们的兴趣和关注.[2]研究了这个方程组的周期初值问     

Isoperimetric Inequalities and Imbedding Theorems in Irregular Domains

The paper proves several weighted imbedding theorems for domainswith fractal boundaries. The weights considered are distancesto the boundary to certain powers, and the domains are so-calleds-John domains. The paper also proves, in the general setting,that the existence of an imbedding implies compactness of theimbedding for lower exponents. Moreover, following Maz'ya, thepaper reformulates the imbedding theorems in the language oflocal isoperimetric and capacity estimates.     

Existence of solutions to Schrödinger‐Poisson systems with critical and supercritical nonlinear terms

In this paper, the existence of nontrivial solutions to a class of Schrödinger‐Poisson systems with critical and supercritical nonlinear terms is obtained via variational methods. By using the potential function, a compactness imbedding result is obtained. The properties of potential function play an important role for insuring variational setting.     

Storage problems in continuous time with random inputs,random outputs,and deterministic release

The subject of study here is the model of a dam, with random inputs and outputs along with a deterministic release. The amounts of the Poisson jumps, either up or down, are independently and identically distributed. Closed form solutions are obtained for the Laplace transforms of first passage densities to different situations of overflow or emptiness. These results can throw insights regarding different threshold studies in storage, inventory, biological, and environmental problems. The closed form solutions are obtained by applying imbedding methods for different types of densities conceptualized in novel ways.     

Invariant imbedding and multiple shooting for the solution of unstable linear boundary value problems

Invariant imbedding has been used to solve unstable linear boundary value problems for a few years. First this method is derived using the theory of characteristics; there the boundary value problem has to be imbedded in a problem of double dimension. If the corresponding Riccati equation has a critical length, one has to repeat the algorithm. A relation between this repeated invariant imbedding and multiple shooting is shown. In examples invariant imbedding, repeated invariant imbedding, multiple shooting and the superposition principle are compared.     

不分明嵌入理论及其应用

本文讨论一类格上拓扑学中嵌入问题,确切说是讨论值域为fuzzy格的L不分明拓扑空间中嵌入理论及其应用.首先概述若干诸如不分明单位区间、重域构造以及格上保并映射类的代数运算等基础性成果.其次给出不分明完全正则的点式刻划与关于一致结构的著名Weil定理的不分明推广并从而建立了在不分明单位方体中一般性的嵌入定理.最后作为嵌入定理的应用,得到了不分明Urysohn度量化定理并完成了不分明Stone-Cech紧化的一般理论。     

Application of invariant imbedding to linear multipoint boundary value problems with general boundary conditions

Two extensions of the usual application of invariant imbedding to the solution of linear boundary value problems are presented. The invariant imbedding formulation of a linear two point boundary value problem in which functional relationships are given between the variables at either one or both of the boundary points is presented. Also, extension of invariant imbedding to linear multipoint boundary value problems is given. Using these extensions singly or in combination, a general multipoint boundary value of linear ordinary differential equations can be solved. In addition, the problems of infinite initial conditions and / or indeterminate initial derivatives are resolved. Numerical examples demonstrate the feasibility and accuracy of the method.     

Banach空间中的相补问题

本文在Ｂａｎａｃｈ空间中研究了三类相补问题解的存在性。所得结果是［４，５，６，９，１１－１４］中相应结果的深入和发展。     

On genus imbeddings of the tensor product of graphs

In this article new genus results for the tensor product H ⊗ G are presented. The second factor G in H ⊗ G is a Cayley graph. The imbedding technique used to establish these results combines surgery and voltage graph theory. This technique was first used by A. T. White [17]. This imbedding technique starts with a suitable imbedding of H on some surface and proceeds by modifying H according to the structure of G to give H*. The resulting pseudograph H* is a voltage graph whose covering graph is the tensor product H ⊗ G. Using our knowledge of the order, size, and the number of regions in the imbedding of H*, together with the theory of voltage graphs, we are able to find the minimum genus of the imbedding surface for several families of the product H ⊗ G. © 1996 John Wiley & Sons, Inc.     

Mechanisms for local search

Local search methods for combinatorial optimization make a series of steps, at each stage improving the current solution by moving to a neighbouring solution. This is usually done by considering the neighbouring solutions one at a time and moving to the first one which gives an improvement (a first-improving method). In this paper we consider whether there are circumstances in which some other strategy will have better performance. In exploring this question we begin by giving a theoretical treatment of a simple model with random objective values at each solution point. We carry out some experiments on the Travelling Salesman Problem and the Quadratic Assignment Problem using varying values of a spread parameter, k. The value of k corresponds to the number of improving solutions looked at before making a move. We also make some conjectures relating the overall performance of the local search method to the average number of solutions which are evaluated before a local minimum is reached.