No 1 (2015)

In this work a general method, allowing to find solutions of differential-operator equations of some type with variable coefficients by means of analitical vector-valued functions, is described. Examined equations include as particular case differential equations in partial derivatives, difference-differential and integral equations, and other functional-operator equations. Solutions are realized by uniformly converged functional vector-valued series, generated by set of solutions of ordinary differential equation of n-th order and some set of elements of locally convex space. Sufficient conditions of continuous dependence of solutions from generating set are found. Solution of Cauchy problem for examined equations is found as well and conditions of its uniqueness are specified. Besides that the so-called general solution of examined equations (the function of the most general view, from which any particular solution can be obtained) is found. The investigation is realized by means of characteristics (order and type) of operator and operator characteristics (operator type and operator order) of vector relative to operator. In this work in investigation a convergence of operator series relative to equicontinuous bornology is used.

Traditionally ideas of geometrical optics apply to research of the approximate solutions corresponding to high-frequency limit, but it is known that, e.g. jumps of solutions of the equations of Maxwell satisfy to Huygens’s law also. In the article we indicate the class of exact solutions of the Maxwell’s equations for which the approach of the geometrical optics can be still used. We consider solutions of the Maxwell’s equations with which it is possible to associate orthogonal system of coordinates of (x1,x2,x3) so that the directions of vectors ⃗E and ⃗e2 and also ⃗H and ⃗e3 are coincided. Conditions on Lamé coefficients of this system of coordinates are found: euh1 doesn’t depend on x2 and x3 and logarithmic derivatives eh1h3 h2 and μh1h2 h3 with respect to x1 don’t depend on x2 and x3 respectively. The first condition means that x1-lines are rays of geometrical optics and it gives a reason to call such systems of coordinates as ray systems how it is accepted in geometrical optics. Thus the solution of the Maxwell’s equations can be described as a wave extending along a ray, that is as the solution of the two-dimensional hyperbolic equation. Necessary and sufficient conditions are found for association of such coordinates systems with the solution of the equations of Maxwell: the directions of vectors ⃗E and ⃗H don’t change over time, they are orthogonal each other and consist in involution, that is ⃗E × ⃗ H,rot ⃗E × ⃗ H = 0.

The computational scheme for solving heat conduction problem with periodic source function in multilayer cylindrical domain is suggested. The domain has a non-trivial geometry and the thermal coefficients are non-linear functions of temperature and have discontinuity of the first kind at the borders of the layers. The computational scheme is based on an algorithm for solving difference problem using the explicit-implicit method. The OpenCL realization of the suggested algorithm for calculations performed on a GPU is also compared to calculations performed using a CPU. It is shown that the scheme can be successfully applied to simulations of thermal processes in pulsed cryogenic cell, which is intended for pulse feeding the working gases into the working space of the ion source within the millisecond range. The results are given for a simulation of one of the particular cell structures, which is assumed to correspond to the practical realization. The computational scheme can be used for the optimization problem of the cell model parameters.

The control problem of dynamic system, containing different physical elements, is solved. Using known dynamic analogies, processes in difficult system are described by the differential-algebraic equations of the classical mechanics. The corresponding differential-algebraic equations include the dynamic equations, the constraints equations and the formulation of purpose of control. Dynamics of system is described by Lagrange equations or by equations in the canonical variables, containing indeterminate multipliers in the right hand sides. The problem of definition of Lagrange multipliers or control functions corresponding to the constraints equations, is reduced to construction of the differential equations systems having partial integrals. Definition of solutions stability of the dynamics equations in relation to the constraints equations is given. The dynamic indicators considering deviations from the constraints equations are entered for ensuring asymptotic stability and constraints stabilization at the numerical solution of the differential equations. The expanded system of dynamics equations, consisting of the initial system dynamics equations and the constraints perturbations equations is under construction. The constraints perturbations equations, constructed on the modified dynamic indicators, allow to define stability conditions and constraints stabilization. Conditions of constraints stabilization, corresponding to the numerical solution of the dynamics equations are given by Euler method and Runge-Kutta method. The solution of a problem of stabilization of vertical position of the rod fixed by cylindrical joint on the cart, making rectilinear movement, is proposed. The control is performed by force acting on the cart and moment applied to the rod.