Deriving a discrete approximate solution for a nonlinear dynamic system of two-phase soil media

A discrete approximate generalized solution is derived for a nonlinear differential model of the dynamics of two-phase soil media and its convergence is estimated for the corresponding generalized solution in the space W ₂ ¹ (Ω). Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 69–80, July–August 2009     

Approximate solution of a nonlinear system of equations for two-phase media

A mixed initial-boundary-value problem for the nonlinear equations describing the dynamic consolidation of water-saturated soils is considered. The error of a time-continuous approximate generalized solution is estimated using the finite-element method. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 73–88, July–August 2008.     

Choosing the parameter of a modified additive-averaged splitting algorithm

The problem-solving time and the solution accuracy are expressed as functions of a parameter. An optimization problem of choosing the parameter using a “time cost–solution accuracy” criterion is considered. A Pareto-optimal set of solutions is obtained. The best value of the parameter is chosen by the ideal-point method. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 98–105, July–August 2009.     

A priori determination of bounds for the optimal value of variables in integer linear programming problems

The paper outlines a simple technique for determining a narrower interval containing the coordinates of the optimal solution to integer linear programming problems. Computational experiments show that the lengths of the given ranges of variables decreases by 30–90% on average. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 86–93, March–April 2006.     

Algorithms for solution of systems of linear diophantine equations in residue fields

Algorithms are proposed for computing the basis of the solution set of a system of linear Diophantine homogeneous or inhomogeneous equations in the residue field modulo a prime number. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 15–23, March–April 2007.     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

An approach to solution uniqueness in game problems

A new concept of strong conflict equilibrium is proposed that supplements the well-known fundamental system of conflict equilibria and considerably increases the possibility of finding a unique strongest equilibrium (solution) in any game problem. The efficiency of this new equilibrium is illustrated by static and dynamic game problems. This work was carried out under the program “Basic foundations of information technologies and systems” of the Russian Academy of Science (Project No. 1–3). Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 116–127, March–April 2009.     

Analysis of systems of singular ordinary differential equations

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones. Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.     

Methods of solution and criteria of consistency of systems of linear diophantine equations over the set of natural numbers1

Some methods and algorithms of solution of systems of linear Diophantine equations over the naturals are briefly reviewed. Criteria and an incremental algorithm of efficient solution of the problem of consistency for a system of linear Diophantine equations and inequalities over the naturals are given. This research was supported under grant INTAS-RFBR 95-0095. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 12–36, July–August, 1999.     

Stability of solutions of stochastic functional-differential equations with poisson switchings and entire prehistory

The Markov properties of the solutions of Ito-Skorokhod stochastic functional-differential equations (SFDEs) with entire prehistory are considered, the concept of a weak infinitesimal operator is introduced for a Markov process that is a solution of an SFDE, and the strong solution of the SFDE is analyzed for stability. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 123–134, January–February 2009.     

Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics

Necessary and sufficient conditions of existence and uniqueness of solutions of integral equations are established for ruin probability as a function of the initial capital of an insurance company. Several sufficient conditions and correctness conditions of the problem of finding ruin probability are also established. A general method of successive approximations for solution of this problem is substantiated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 157–164, September–October 2006.     

Application of a numerical increased-accuracy method to the solution of the poisson equation

To solve the Poisson equation in a rectangular domain, a numerical increased-accuracy method is used. This method is based on the relationship be Ween the values of the function and its derivatives at neighboring nodes of a net domain. Results of solution of model problems using this method are presented. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 176–182, November–December, 1999.     

Solving the Wiener-Hopf equation with a probabilistic kernel

Inhomogeneous Wiener-Hopf equations on a halfline with probabilistic kernels are considered. They are analyzed using the theory of isotonic and contraction operators. Necessary and sufficient conditions of existence and general sufficient conditions of uniqueness for the solutions of these equations are established. The conditions for and the rate of convergence of successive approximations of the solution are found. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–47, March–April 2006.     

Formulation and analysis of problems for dynamic systems of inhomogeneous two-phase media

A mixed initial—boundary-value problem with a discontinuous solution for a system of dynamic equations of water-saturated soil consolidation is analyzed. The classical solution is proved to be unique. Error estimates are obtained for time-continuous and fully discrete approximate generalized solutions constructed by the finite-element method. The study was sponsored by the State Fund for Basic Research (GFFI), Grant No. F7/307-2001. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 85–100, November–December 2005.     

Analysis of the properties of a linear system using the method of artificial basis matrices

Formulas relating elements of the method in adjacent basis matrices are used to solve a system of linear algebraic equations and to represent analytically the general solutions to the corresponding system of linear algebraic inequalities for a nondegenerate constraint matrix. Sponsored by the ICS NATO program of April 18 2006, in line with the Project “Optimal replacement of information technologies and stable development (in Kazakhstan, Ukraine, and the USA),” NATO Grant CLG 982209. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 119–127, July–August 2007.     

An approach to solving discrete vector optimization problems over a combinatorial set of permutations

Complex discrete multicriteria problems over a combinatorial set of permutations are analyzed. Some properties of an admissible domain for a combinatorial multicriteria problem embedded into an arithmetic Euclidian space are considered. Optimality conditions are obtained for different types of effective solutions. A new approach to solving the problems formulated is constructed and substantiated. This work was supported by the Fundamental Research Fund of Ukraine (project Φ251/094). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 158–172, May–June 2008.     

Fokker–Planck approximation of the master equation in molecular biology

The master equation of chemical reactions is solved by first approximating it by the Fokker–Planck equation. Then this equation is discretized in the state space and time by a finite volume method. The difference between the solution of the master equation and the discretized Fokker–Planck equation is analyzed. The solution of the Fokker–Planck equation is compared to the solution of the master equation obtained with Gillespie’s Stochastic Simulation Algorithm (SSA) for problems of interest in the regulation of cell processes. The time dependent and steady state solutions are computed and for equal accuracy in the solutions, the Fokker–Planck approach is more efficient than SSA for low dimensional problems and high accuracy.     

Remarks on stability of solutions of the property domination problem

A class of combinatorial problems is considered whose investigation and solution require the notions of the theory of fuzzy sets. The necessary and sufficient conditions of stability are given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 36–40, July–August, 1999.     

Computer-based techniques for solving problems of applied and computational mathematics with given quality characteristics

The paper is concerned with computer-based techniques for the choice and development of computational resources and their efficient use to find an approximate solution with a given accuracy in a limited processor time. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 33–41, September–October 2006.     

Algorithms for solving systems of linear diophantine equations in residue rings

Algorithms are proposed that construct the basis of the set of solutions to a system of homogeneous or inhomogeneous linear Diophantine equations in a residue ring modulo n when the prime factors of n are known. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 27–40, November–December 2007.