Stable, economical methods, based on the phase function method, are proposed for computing ellipsoidal and spheroidal radial wave functions as eigenfunctions of the continuous spectrum of ordinary differential equations of second order in a semi-infinite interval. The methods are quite universal, permitting the computation of radial functions over a broad range of parameter values, and capable of a natural generalization to other classes of equations that arise upon separation of variables in the Helmholtz and Schrodinger equations in various coordinate systems. The Lame radial wave functions are computed for the first time.
An iterative method is proposed for solving the problem of diffraction by a spherical segment (scalar case). The method ultimately yields effective mathematical models for investigations in the quasi-optical domain. Underlying it is a rigorous method for solving the scattering problem. The efficiency of the method is estimated and it is compared with approximate methods.
An exact and an approximate method for reducing a mathematical programming problem with a convex irregular feasible set to a mathematical programming problem with a regular feasible set is proposed.
The three-dimensional dynamical problem of the oblique impact of a rigid pellet on a deformable elastoplastic barrier is solved using a hybrid grid-characteristic scheme for the numerical solution of non-stationary systems of hyperbolic equations. Thegrid-characteristic hybrid scheme is adapted for the numerical solution of multidimensional non-stationary problems in the mechanics of deformable bodies.
The method of finite elements with one iteration for elliptic boundary value problems is investigated. Approximate solutions obtained by this method have high order of convergence than if no iteration is used.
An explicit solution is obtained for the initial-boundary value problem of non-stationary internal waves in a vertical channel containing a stratified liquid generated by small vibrations of the bottom of the channel. The behaviour of the energy and the flow of energy of internal waves is studied as a function of time and the frequency of the force generating the vibrations. Asymptotic formulae for long times are obtained.
A method of interpolating a function of one variable whose values and first derivatives are fixed at the end-points of an interval is described. The solution of the problem is a one-parameter family of second-order curves. An example of the application of the method to the global construction of an interpolating curve is given.
A first-order ordinary differential equation is obtained to study the motion of the trajectory of motion of a material particle in a two-dimensional (axisymmetric) conservative field. The advantages of using the proposed equation in the numerical analysis of electron-optical systems are demonstrated.
The problem of the stationary flow of a heavy fluid over an uneven, periodically varying bottom with the formation of standing waves on the free surface is treated using the methods of the theory of the branching of the solutions of non-linear operators. New classes of solutions of this problem, which are expressed in terms of non-analytical functions, are obtained in the quasiresonance case.
Stability bounds for the solutions of conditionally well-posed problems for the case when the variation and the maximum absolute value of the solutions are bounded by known constants are derived. Differentiation problems, Volterra and Abel integral equations, and convolution equations are considered.
The approach is based on the projection of surfaces on to coordinate axes. The proposed algorithm takes account of the special properties of the functions occurring in the non-linear equations. These properties are convexity, separability, and monotonicity with respect to groups of variables or individual variables.
The two- and three-layer versions of a symmetric implicit difference scheme on distributed nets in the case of the one-dimensional equation of gas dynamics in the Euler form are considered. The approximate viscosity of the scheme is investigated by the method of differential approximation. The change in the approximation properties of the scheme when a mobile net is employed is analyzed. The results of a numerical experiment are presented.
In the context of the numerical treatment of the two-dimensional Navier-Stokes equations, the boundary conditionsatasolid surface may be realized in different ways, some of which are examined. The approach considered here assumes that the equations of the system are solved separately. It is shown that then, irrespective of the specific formulation of the Navier-Stokes equations for an incompressible viscous liquid - in terms of velocity-pressure, velocity-vorticity or vorticity-stream function - the boundary conditions can be realized in an algorithmically universal way, based on a two-parameter formula previously proposed to approximate vorticity on a wall.
The problem of finding solutions of a system of equations F(x) = 0 is investigated, where F is a non-linear operator from En to Em, m < n. An approach to the solution of such systems in the non-regular case is proposed.
The method of collocation and the “integral equation” method, which is based on the use of the fast Fourier transform, are compared. The two algorithms are described and the results of numerical calculations of the surface currents and amplitudes of the non-decaying reflected plane waves obtained by the two methods are presented. Tables of the energy defects, the normalized difference of the solutions in L2[0,2π] and the times required to carry out calculations on a ES-1055 digital computer are given. The operation of the two algorithms is analysed.
Sufficient conditions are derived for the Pareto-optimality of an equilibrium. A class of positional differential games satisfying these conditions are considered. In other words, equilibria that are unimprovable in the equilibrium set are Pareto-optimal, i.e., unimprovable among all the situations of the game.
