Разностная схема 4-го порядка точности для дифференциального уравнения с переменными коэффициентами
We present compact difference scheme on three-point stencil for unknown function. The scheme approximates linear second order differential equation with variable smooth coefficient. Our numerical experiments confirmed 4-th accuracy order of solutions of the difference scheme and of eigenvalues’ approximation for the boundary problem. The difference operator is almost self-conjugate, and its spectrum is real. The Richardson extrapolation method improves the accuracy order.
We develop a new compact scheme for second-order PDE (parabolic and Schrodinger type) with a variable time-independent coecient. It has a higher order and smaller error than classic implicit scheme. The Dirichlet and Neumann boundary problems are considered. The relative nite-dierence operator is almost self-adjoint.
The paper is devoted to the study of the unconditional extremal problem for a fractional linearintegral functional defined on a set of probability distributions. In contrast to results proved earlier,the integrands of the integral expressions in the numeratorand the denominator in the problem underconsideration depend on a real optimization parameter vector. Thus, the optimization problem isstudied on the Cartesian product of a set of probability distributions and a set of admissible values ofa real parameter vector. Three statements on the extremum ofa fractional linear integral functionalare proved. It is established that, in all the variants, the solution of the original problem is completelydetermined by the extremal properties of the test function of the linear-fractional integral functional;this function is the ratio of the integrands of the numeratorand the denominator. Possible applicationsof the results obtained to problems of optimal control of stochastic systems are described.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.