Showing papers in "Ussr Computational Mathematics and Mathematical Physics in 1968"

Algorithm for discovering the set of all the solutions of a linear programming problem

Abstract: IN this paper two versions of a canonical algorithm for discovering all the optimal solutions of a linear programming problem with the condition of non-negativeness of the variables are presented: the first for the case of canonical notation, the second for the standard notation. The algorithm is obtained by applying the results of [1] (respectively [2]) to the following problem which is obviously equivalent to the linear programming problem indicated: for a finite system of linear equations (inequalities), one of the free terms of which is a parameter, to find the greatest value of this parameter for which the system is consistent, and all its non-negative solutions for this greatest value of the parameter.

Evaluation of the integrals of oscillating functions by interpolation at nodes of gaussian quadratures

Abstract: The direct computation of these integrals by the Newton-Cotes quadrature formulas is not very efficient. Vhen such formulas are applied the integrand, in this case f(x) exp (iox), is approximated by polynomials. Therefore, in order to obtain a good approximation it is necessary for the integration step to be smaller than the typical segment of variation of the integrand, which is of the order of 2R/O.

The parametric problem of shortest distances

Abstract: THE problem of finding shortest distances (and paths) is a component part of many network problems, though it is also of independent interest Solving such problems, for example, by the method of successive calculations [21, involves the repeated calculation of the extreme value of a target function by the variation of one or several parameters (for example, the length of arc). Therefore, the computational process includes as a very essential element the discovery of all the shortest distances for the new values of the parameters.

On the convergence of difference schemes approximating the second and third boundary value problems for elliptic equations

Abstract: THE present paper considers difference schemes approximating the second and third boundary value problems for a self-adjoint elliptic equation without mixed derivatives in a rectangular region, and represents a direct continuation of [1].

Solution of a parabolic equation of diffraction theory by a finite difference method

Abstract: THE method of transverse diffusion developed in papers by M. A. Leontovich, V. A. Fok, and G. D. Malyuzhinets (there is a review of these papers in [1]), has recently been extensively applied to find the short wave asymptotic behaviour of stationary diffraction problems.

A finite-difference method for the solution of one-dimensional non-stationary problems in magneto-hydrodynamics☆

Abstract: IN theoretical investigations of a number of applied problems in magneto-hydra dynamics (various types of MHD-generators, problems of astrophysics etc.) there is particular interest in the study of interaction processes between a compressible electrically conducting gas and a magnetic field for arbitrary Reynolds numbers Rem and the magnetic interaction parameters R, = W/&p, where H is the magnetic field strength and p the pressure. In this case and in physical experiments an important role is played by the investigation of mathematical models which take into account mainly the non-linear relations between the non-stationary processes of magneto-hydrodynamics. In the one-dimensional approximation numerical methods also enable us not only to study the quantitative sides of the processes, but also to establish a number of new qualitative regularities. Thus the use of numerical methods for equations of magnetohydrodynamics, taking into account complex non-linear dissipative processes, has made possible the solution of a number of actual physical problems 11-61. In [6l a new physical phenomenon is described, the so-called T-layer effect a high-temperature, electricallyconducting, self-sustaining layer of gas, arising at a definite part of the mass due to Joule heating.

Differentiability of a maximum function. I

Abstract: A topic discussed in [1] was the differentiability with respect to a direction of a function of the form ϑ(y) = max x∈Ω x ƒ (x, y) . A variety of problems can be solved by means of θ ( y ), including the problem of best approximation of functions, the problem of finding a minimax, and time-optimal problems. Such functions have been discussed e.g. in [2]. It is sometimes of interest to investigate the differentiability with respect to a direction of the function ϑ(z) = max x min y ƒ(x,y,z) . Here, finite-dimensional sets are considered. The result can also be extended to the case of infinite-dimensional spaces.

An optimal algorithm for finding the roots of an approximately computed function

Abstract: IN a wide range of practical and computational problems we have to find the roots of a function which can only be computed approximately for given values of its argument. Examples include the boundary value problem for a nonlinear system of differential equations, solved by specifying missing initial conditions (at one end), or the problem of the experimental selection of the parameters of a process or device in order to satisfy a given condition. Since computation of the function whose root is required often involves a laborious, lengthy or expensive procedure, it seems natural to consider optimization of the method of finding the root. Two problems arise here: first, assuming that the errors in computing the function are known, how to select the points at which it is to be computed; and second, how optimally to distribute the existing resources over the computational stages. By resources we mean what the computational accuracy depends on in a specific case, e.g. computer time, labour or cost of the computations, etc. The present paper states mathematically and solves the problem of an optimal search method for one class of functions. The statement comes in Section 1, while Section 2 defines the optimal search algorithm for the case when the function is computed with known errors (and in particular, accurately). The question of the optimal distribution of the resources over the computations is discussed in Section 3. Some examples are considered, and the results of calculating optimal algorithms are given. In the course of the solution we use ideas of the method of dynamic programming, which were earlier used for devising opt1imal search algorithms in which computational errors were ignored [1].

Calculation of the equations of a laminar boundary space layer by the method of integral relations

Abstract: THE method of integral relations, put forward by A. A. Dorodnitsyn [1] for the solution of non-linear problems in aerodynamics, is widely used for numerical integration of the equation of a laminar boundary layer. Integration of the equation for a plane and axially-symmetric laminar boundary layer by this method [2–6] has shown that when the number of strips into which the field of flow is divided is increased the difference between two successive approximations is reduced, and the solution obtained agrees well with the results of calculations by other methods. This indicates that as the number of strips increases the accuracy of the claculation increases, and the solution itself is the solution of the original differential equation. In [3–5] the accuracy of the calculation of the laminary boundary layer equations by the method of integral relations is evaluated over the values of the local friction stress and the local heat flux. In particular, it was shown in [3] that in the neighbourhood of the plane critical point the local friction stress and local heat flux are calculated with an error of the order of 0.2 % in the fourth approximation. The solution of the laminar boundary layer on a blunt cylindrical slate [4] showed that if x ⩽ π 2 the fourth and third approximations differ by less than 3%; if x > π 2 the convergence deteriorates and the difference is about 6%. The fourth and fifth approximations on the cylindrical blunted part differ by less than 1.5% and outside it the difference is about 3% [5]. A comparison of the results of calculation of the characteristics of a laminar axially-symmetric boundary layer by the method of integral relations with the data of numerical integration [6] shows that if the field of flow is divided into five strips the method of integral relations in a region with zero or negative pressure gradient ensures an accuracy of calculation of the order of 1.5% in the case of strong heat transfer and of the order of 1.5–20% with no heat transfer. In a region with positive pressure gradient the results of calculations by these two methods differ significantly from one another. In the present paper the method of integral relations is used for the numerical integration of the equations for a laminar space boundary layer with finite external flow. In three particular examples we evaluate the accuracy of calculation of local coefficients of frictional resistance and local heat flow, and also of the profile of frictional stress, the rate of secondary flow and the total enthalpy. The author earlier used the method of integral relations to integrate the equations of a laminar boundary layer on infinite moving elliptic cylinders [7], but there the accuracy of the calculations was not evaluated.

The least squares and Bubnov-Galerkin methods

Abstract: IN this paper the rate of convergence of the approximate solution, constructed by the least squares method, to the exact solution, is established subject to certain conditions. A similar problem is investigated for a generalized Bubnov-Gal'erkin method.

The short-wave asymptotic behaviour of Green's function for the n-dimensional wave equation in an inhomogeneous medium

Abstract: IN the present paper the asymptotic behaviour of Green's function is studied for the wave equation [Δ + k 2 n 2 (x)]G = δ(x − x 0 ) , G = O( 1 |x| (N − 1> 2 ), ∂G ∂|x| + ik √EG = o( 1 |x| (N − 1) 2 ) , (1) x = (x 1 , …, x N ), E = lim |x|→+∞ n 2 (x), n 2 = E − V(x) > 0 in an N-dimensional space as k → + ∞. An asymptotic function will be constructed subject to the following conditions on the function n2(x).

The collocation method

Abstract: WHEN solving boundary value problems of mathematical physics by variational methods difficulties are often encountered in evaluating the integrals; and the difficulties are even greater whem empirical functions enter into the equations. In the latter case the collocation method [1–4] is best, since the interpolation of the empirical functions can then be completely avoided and we can operate with the available discrete set of data. The points where the functions are specified are usually equally spaced in practice. In view of this, a modification of the collocation method can be useful, based on the trigonometric interpolation over equidistant base-points discussed in Sections 1–4 below.