Representation of the solution to a boundary-value problem as a continuum integral

A Feynman integral representation is obtained for the solution of the Dirichlet and Neumann problems for a second-order parabolic equation in a Riemannian space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 6–11, 1974.In conclusion, the author wishes to thank V. S. Buslaev for his attention to this work.     

Toward a solution to the high Weissenberg number problem

We present results of a numerical study of high Weissenberg number flows using the method of Trebotich, Colella and Miller, J. Comput. Phys. vol. 205 (2005). We consider benchmark viscoelastic flow of an Oldroyd-B fluid in four to one sudden contractions for We = 1, 10, Re = 1. We conclude that (i) underlying wave behavior in the solution for flows at critical and supercritical elastic Mach number prevents the solution from reaching a steady-state; and (ii) high resolution is needed to resolve structure in the solution downstream from the re-entrant corner in order to determine convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the solution and complexity of a generalized linear complementarity problem

We introduce some sufficient conditions under which a generalized linear complementarity problem (GLCP) can be solved as a pure linear complementarity problem. We also establish that the GLCP is in general a NP-Hard problem.Support of this work has been provided by the Instituto Nacional de Investigação Cientifica de Portugal (INIC) under contract 89/EXA/5.     

The generalized solution of ill-posed boundary problem

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk,p0(Ω,∑). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.     

Numerical solution of the generalized Stokes problem for the flow in a channel

In this article we introduce the separation of variables in the two-dimensional generalized Stokes problem. −νΔu + αu + inverted delta p = f, for the flow in a channel. Also for the first time, we discuss the implementation of the Incremental Unknowns Method with a data structure of Compressed Column Storage. Two examples of application of the Incremental Unknowns method for this problem are presented in which we compare the CPU times of three methods: Conjugate Gradient (CG), Incremental Unknowns (IU), and Uzawa Algorithm (Uzawa). © 1997 John Wiley & Sons, Inc.     

Approximate method for the solution of the generalized Dirichlet problem

We extend the method for approximate solution of classical boundary-value problems for the Laplace equation suggested in [1–3] to the case of the Poisson equation with generalized functions on the right-hand side of the equation and in the boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.46, No. 10, pp. 1417–1420, October, 1994.     

Existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem

In this paper two theorems are proved on the existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem, from which the solvability of this problem in the class of continuous functions follows.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 447–452, October, 1971.     

Partial regularity of solution to generalized Navier-Stokes problem

In the presented work, we study the regularity of solutions to the generalized Navier-Stokes problem up to a C ² boundary in dimensions two and three. The point of our generalization is an assumption that a deviatoric part of a stress tensor depends on a shear rate and on a pressure. We focus on estimates of the Hausdorff measure of a singular set which is defined as a complement of a set where a solution is Hölder continuous. We use so-called indirect approach to show partial regularity, for dimension 2 we get even an empty set of singular points.     

Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem

This paper deals with the stability for a parametric generalized strong vector equilibrium problem. Under new assumptions, which do not contain any information about the solution set and monotonicity, we establish the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result. These results are improve the corresponding ones in recent literature. Some examples are given to illustrate our results.     

Iterative solution of the generalized Love's problem in anisotropic media

An integral equation method is used to show the well posedness of the generalized Love's problem for an elastic anisotropic layer superimposed to a homogeneous substrate. The solution is derived using an iterative technique and the dispersion equation is obtained by imposing the pertinent boundary conditions. Inhomogeneous layers with different profiles of the material parameters are considered as examples and numerical results are given. The corresponding generalized problem is discussed for Lamb's modes.     

On an approach to the construction of a solution of the generalized coupling problem

We propose a method of solving generalized coupling problems by reducing them to an inhomogeneous differential equation in the space of generalized functions that can be solved using a normalized fundamental function or a Green's function.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 198–203.     

An iterative solution to the generalized Towers of Hanoi problem

An iterative solution to the generalized Towers of Hanoi problem, and its derivation are presented. In this generalization, one or more towers, consisting of a total ofn discs, are given as an initial legal configuration, and the task is to move them to a specified peg under the usual restrictions. An analysis of the iterative algorithm is also discussed.     

Exact solution of the generalized routing problem through graph transformations

In the general routing problem, the aim is to determine a least cost traversal of a subset of edges, arcs and vertices of a graph. The problem can be transformed into an equivalent traveling salesman problem or rural postman problem and solved optimally. Computational results are reported.     

Boundary element solution of a scattering problem involving a generalized impedance boundary condition

A boundary element method is introduced to approximate the solution of a scattering problem for the Helmholtz equation with a generalized Fourier–Robin‐type boundary condition given by a second‐order elliptic differential operator. The formulation involves three unknown fields, but is free from any hypersingular integral. Existence and uniqueness of the solution are established using a Babuška inf–sup condition. When implementing the method, a lumping process allows to remove two fields from the formulation. The numerical solution has thus the same cost as the one of a problem relative to a usual Neumann boundary condition. Numerical tests confirm the ability of the method for solving this type of non‐standard boundary value problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Toward a generalized computability theory

Translated fromAlgebra i Logika, Vol. 32, No. 4, pp. 349–386, July–August, 1993.