Showing papers in "Siam Review in 1976"
TL;DR: A survey of nonlinear functional analysis in ordered Banach spaces can be found in this paper, where some of the most important methods and results of non-linear functional analyses are discussed.
Abstract: This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered Banach spaces. By means of iterative techniques and by using topological to...
1,804 citations
TL;DR: A survey of results for the Korteweg-deVries equation can be found in this paper, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method.
Abstract: The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...
623 citations
507 citations
TL;DR: A survey of theoretical results and solution methods for the set partitioning problem can be found in this article, and while we have tried not to omit anything important, we have no claim to completeness. Critical comments pointing out possible omissions or misstatements are welcome.
Abstract: This paper discusses the set partitioning or equality-constrained set covering problem. It is a survey of theoretical results and solution methods for this problem, and while we have tried not to omit anything important, we have no claim to completeness. Critical comments pointing out possible omissions or misstatements will be welcome.Part 1 gives some background material. It starts by discussing the uses of the set partitioning model; then it introduces the concepts to be used throughout the paper, and connects our problem to its close and distant relatives which play or may play a role in dealing with it: set packing and set covering, edge matching and edge covering, node packing and node covering, clique covering. The crucial equivalence between set packing/partitioning and node packing problems is introduced.Part 2 deals with structural properties of the set packing and set partitioning polytopes. We discuss necessary and sufficient conditions for all vertices of the set packing polytope to be intege...
453 citations
229 citations
TL;DR: In this article, the characteristics and capabilities of the best codes for solving the initial value problem for ordinary differential equations are studied. Only codes which are readily available, portable, and v...
Abstract: The characteristics and capabilities of the best codes for solving the initial value problem for ordinary differential equations are studied. Only codes which are readily available, portable, and v...
227 citations
TL;DR: The fractional derivative operator as mentioned in this paper is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representation
Abstract: The fractional derivative operator is an extension of the familiar derivative operator $D^n $ to arbitrary (integer, rational, irrational, or complex) values of n. The most important representation...
178 citations
TL;DR: In this paper, the authors present a new proof based on a recursion relation of Lenart, for the existence of an infinite sequence of conserved functionals, which are invariant under the KdV flow.
Abstract: In this talk we discuss the almost periodic behavior in time of space periodic solutions of the KdV equation \[ u_t + uu_x + u_{xxx} = 0.\] We present a new proof, based on a recursion relation of Lenart, for the existence of an infinite sequence of conserved functionals $F_n (u)$ of form$\int {P_n (u)dx} $, $P_n $ a polynomial in u and its derivatives; the existence of such functionals is due to Kruskal, Zabusky, Miura and Gardner. We review and extend the following result of the speaker: the functions u minimizing $F_{N + 1} (u)$ subject to the constraints $F_j (u) = A_j $,$j = 0, \cdots ,N,$ form N-dimensional tori which are invariant under the KdV flow. The extension consists of showing that for certain ranges of the constraining parameters $A_j $ the functional $F_{N + 1} (u)$ has minimax stationary points; these too form invariant N-tori. The Hamiltonian structure of the KdV equation, discovered by Gardner and also by Faddeev and Zakharov, which is used in these studies, is described briefly. In an ...
171 citations
147 citations
TL;DR: In this article, the Lagrange multipliers are used with finite elements to achieve desirable properties in the underlying approximation for elliptic boundary value problems, in which all boundary conditions are natural.
Abstract: The purpose of this paper is to show how Lagrange multipliers can be used with finite elements to achieve a number of desirable properties in the underlying approximation. For elliptic boundary value problems, variational principles can be developed in which all boundary conditions are natural. In fluid flow problems, one can endow the approximations with physically essential conservation laws.
112 citations
TL;DR: In this paper, the title problem is posed as a linear heat equation in one space dimension and time, with a nonlinear radiative-type boundary condition on the surface, and the authors present a solution to the problem.
Abstract: The title problem is posed as a linear heat equation in one space dimension $(x > 0)$ and time $(t > 0)$, with a nonlinear radiative-type boundary condition on the surface $(x = 0)$. Existence and ...
TL;DR: In this article, various algorithms for obtaining minimal $l_2 $-solutions to consistent linear systems of equations $Ax = y$ are discussed, which are based upon factorizations of matrices, A, with full...
Abstract: Various algorithms for obtaining minimal $l_2 $-solutions to consistent linear systems of equations $Ax = y$ are discussed. These algorithms are based upon factorizations of matrices, A, with full ...
TL;DR: In this article, the problem of generating approximations to functions, specified by asymptotic expansions about one or more points, is considered, and the development is limited to expansions of the power series type.
Abstract: The problem of generating approximations to functions, specified by asymptotic expansions about one or more points, is considered. The development is limited to expansions of the power series type ...
TL;DR: Finite difference methods for degenerate linear and (certain) nonlinear elliptic and parabolic equations are analyzed from a probabilistic point of view, and the techniques are used to show convergence to the correct weak or strong sense solution as discussed by the authors.
Abstract: Finite difference methods for degenerate linear and (certain) nonlinear elliptic and parabolic equations are analyzed from a probabilistic point of view, and probabilistic methods are used to show convergence to the correct weak or strong sense solution. The techniques generalize results in numerical analysis, and the probabilistic approach allows some additional physical insight to be brought to bear on the problem of selecting suitable approximations and methods of solution. The equations which are discussed all have probabilistic interpretations. In each case, the finite difference equations (with appropriately chosen finite difference approximations) are also equations which are satisfied by certain functionals of certain Markov chains, whose transition functions are given directly by the coefficients in the finite difference equation. Suitable continuous time interpolations of the chains converge to various diffusion processes which are connected with the original partial differential operators, and ...
TL;DR: Two stochastic simulation models for the study of immunization strategies are presented and it is shown that immunization can have a major impact on epidemic size even under such adverse conditions.
Abstract: Two stochastic simulation models for the study of immunization strategies are presented. The first is for influenza in a community of one thousand persons in four age groups and allows for clustering in families, neighborhoods, schools and play groups. The lengths of the latent and infectivity periods are random variables. Variations in susceptibility, illness and withdrawal to the home are considered. Results are given for a number of immunization schedules.The second model is for poliomyelitis in a community in which environmental contamination with competing and interfering enteric agents results in a $50\% $ vaccination failure rate. This is a competitive risk model in which the live virus Sabin vaccine and the wild agent compete with each other and with the contaminating agents. It is shown that immunization can have a major impact on epidemic size even under such adverse conditions.
TL;DR: In this article, a simple change of variable formula is used to extend existing results on boundedness and stability of solutions to ordinary differential equations of second order to ODEs, which can also be used on oscillation criteria and other qualitative properties.
Abstract: This paper illustrates the systematic use of a simple change of variable formula to extend existing results on boundedness and stability of solutions to ordinary differential equations of second order. This approach can also be used on oscillation criteria and other qualitative properties.
TL;DR: In this article, the main result of the present note is the following partial converse of this statement: if a power of T is a contraction, then T is also a contraction with respect to a new metric which is easily defined in terms of the old one.
Abstract: If T is a contractive mapping in a metric space, then powers of T are also contractive. The main result of the present note is the following partial converse of this statement: If a power of T is a contraction, then T is a contraction with respect to a new metric which is easily defined in terms of the old one. Using this result, any proposition regarding a contractive mapping carries over to the case where a power of T is a contraction. Several examples are given yielding old and new results.