Showing papers in "Ussr Computational Mathematics and Mathematical Physics in 1966"

On the convergence of a relaxation method with natural constraints on the elliptic operator

Abstract: ONE OF the most perfect methods for the solution of a network approximation of Poisson's equation in a rectangle is by a change of variables [1]. This method requires arithmetic operations of the order h−2 ln h−1 ln ϵ−1 for finding the solution of the network equation to within ϵ; here h is the network step. The combination of this method with others [2, 3] enables us with a number of operations of the same order to find the solution of network approximations of elliptic equations with operators of fixed sign. In [4, 5] a relaxation method, based on completely new ideas, is put forward for the solution of a network approximation of Poisson's equation in a square, which requires arithmetic operations of the order of h−2 ln ϵ−1 to decrease the norm of the discrepancy by a factor of ϵ. In the present paper the application of this method is described with the same order of evaluation of the number of operations in the case of an arbitrary elliptic operator with continuous coefficients on the natural assumption that the point 0 is not a point of the spectrum. In Section 5 we show that with this definite approach to the problem we can eliminate the factor ln ϵ−1 in the evaluation of the number of operations after which the method becomes optimal as regards the order of the number of operations. Because of the generality and coarseness of the investigations carried out, the evaluations of the number of operations with practically admissible values of h and ϵ. which we have obtained, may be inferior to the evaluations obtained in [1–3]. This, however, is atoned for at least by the fact that the method is used for an essentially wider circle of equations. For instance it is used in the case of the equation Δu + λu = f with large positive λ(x1, x2). Previously no methods of solving this equation with good asymptotics for the number of operations were known. The basic idea of the method and the proof of the convergence are based on general considerations which suggest the possibility of using the method in the case of arbitrary, and in particular non-linear, equations and other classes of boundary conditions.

The approximate solution of operator equations of the first kind

Abstract: A METHOD of solving operator equations of the 1st kind, dual to A.N. Tikhonov's method, is considered. It enables us to select the parameter a on which the approximate solution depends, in such a way that the substitution of the approximate solution in the equation yields a discrepancy not exceeding the error with which the initial data are assigned.

The solution of variational and boundary value problems by the method of local variations

Abstract: IN this paper we put forward an algorithm for the numerical solution of variational problems for functions of two independent variables (problems with partial derivatives) by the method of local variations [1]. The method is applicable in the solution of various variational and boundary value problems with constraints. We quote results for the solution of some problems on an electronic computer using the algorithm described, and also data on the convergence of the method.

Solution of problems of optimal control by the method of local variations

Abstract: This paper gives a description of an algorithm for the numerical solution of problems of optimal control by the method of local variations put forward in [1]. The results of a numerical solution of some variational problems which demonstrate the effectiveness of the method are given. A description of the numerical algorithm in the language ALGOL-60 is presented.

Evaluation of a generalized inverse matrix and projector

Abstract: A GENERALIZED inverse matrix or projector may be involved when solving linear systems in which the matrix of coefficients is rectangular or singular.

Calculation of the boundary layer arising in flow about a cone under an angle of attack

Abstract: We consider the problem of the flow about an infinite circular cone in an ideal gas. We describe a finite-difference method for calculating the self-similar solutions of the Prandtl boundary layer equations, and quote results. The nature of the singularity which arises in the solutions on the leeward side of the cone is discussed.

Conditions for the convergence of iterative processes on the real axis

Abstract: ONE OF the most important methods for the numerical solution of the non-linear equations F ( x ) = x is the method of iteration [1]. In the literature, conditions are formulated which are sufficient for the convergence of iterative processes [1 and 2], but these conditions are not all necessary.