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Vol 27, No 2 (2016)

I. Numerical Methods

Finite-Difference–Integral Method for Computing Low-Frequency Electromagnetic Fields in a Nonhomogeneous Medium

Dmitriev V.I., Barashkov I.S.

Abstract

We derive integral relationships for electromagnetic fields, integral recalculation formulas for anomalous electric and magnetic fields, and integral boundary conditions. The efficiency of the finite-difference–integral method is checked by computing the apparent resistance curves for the graben model. Apparent resistance curves computed from finite-difference formulas and from formulas with integral smoothing are compared. Finite-difference graphs are polygonal curves that for small wavelengths lie below the smoothed curves, because the finite-difference computation of the derivative as if reduces the impact of the surface skin effect on the apparent resistance. As the wavelength increases, the surface skin effect diminishes and the finite-difference curves approach the integral curves.

Computational Mathematics and Modeling. 2016;27(2):145-161
pages 145-161 views

Article

Numerical Solution of the Inverse Problem for the Mathematical Model of Cardiac Excitation

Solov’eva S.I., Tuikina S.R.

Abstract

We consider the problem of localizing the region of the heart damaged by myocardial infarct. For the two-dimensional modified FitzHugh–Nagumo mathematical model, this inverse problem involves determining the coefficient dependent on spatial variables for a system of partial differential equations in a region with a localized source of cardiac excitation. Additional dynamical measurements of the potential are carried out on the inner boundary of the region representing the section of the heart and its ventricles by a horizontal plane. Potential measurements on the inner boundary correspond to data obtained from ventricular catheters. A numerical method is proposed for the solution of this inverse problem and results of computer experiments are reported.

Computational Mathematics and Modeling. 2016;27(2):162-171
pages 162-171 views

Nonlinear Dynamical Model of Microorganism Growth in Soil

Vasilyeva N.A., Ingtem J.G., Silaev D.A.

Abstract

We propose a mathematical description of the soil as a complex dynamical system with components of living, organic and physical nature. The model is a system of partial differential equations of parabolic type with nonlinear sources and nonlinear diffusion. Known oxygen and water dependent feedbacks are described as the sum of variables products: microorganism growth; switching from easily degradable to complex feed substrate; biotic/abiotic/autocatalytic transformations between substances. A selfoscillatory regime with self-organization of stable irregular dynamical structures, in particular the porous space (the the habitat), is detected in the model. The model with linear and nonlinear diffusion reflects processes that occur on different scales. The characteristic size of the problem is determined by the magnitude of the diffusion coefficient, which in capillaries is an order of magnitude lower than in free water. The model makes it possible to describe the macroscopic behavior of the soil system as a whole based on the dynamics of microscale spatially nonhomogeneous structures and allows simulation of various natural soil regimes for which experimental data are available.

Computational Mathematics and Modeling. 2016;27(2):172-180
pages 172-180 views

Numerical Investigation of Spatial Solitons

Laponin V.S.

Abstract

The numerical method we have previously developed to search for soliton solutions is applied in this article to find spatial solitons described by a nonlinear Schroedinger equation. We show in detail how to derive the Schroedinger equation from Maxwell’s equations and now to construct a one-soliton analytical solution for the one-dimensional Schroedinger equation. The proposed numerical method is appropriate to search for spatial solitons, which is corroborate by comparing the numerical results with the analytical solution.

Computational Mathematics and Modeling. 2016;27(2):181-189
pages 181-189 views

Spectral-Domain Integral Equation Method for Modeling the Scattering Properties of a Group of Planar Particles in the Presence of a Substrate

Lopushenko V.V.

Abstract

Volume integral equation in the spectral domain is used to develop an algorithm for the analysis of the scattering properties of a group of planar objects in the shape of flattened cylinders located on or near a substrate. The algorithm features are demonstrated in application to some particle clusters.

Computational Mathematics and Modeling. 2016;27(2):190-205
pages 190-205 views

Combining Compact Finite Difference Schemes with Filters for Image Restoration

Aghazadeh N., Akbarifard F.

Abstract

In this study, the Rudin, Osher, and Fatemi (ROF) model is considered to restore images. To this end, the compact finite difference (CFD) method is introduced to approximate the spatial derivatives in the ROF model. Moreover, two filters are presented to improve the performance of the first- and second-order derivative approximation. The third-order total variation diminishing Runge-Kutta (TVD-RK3) method is applied to solve the obtained system along the time axis. Two examples are given to show the efficiency and accuracy of the method.

Computational Mathematics and Modeling. 2016;27(2):206-216
pages 206-216 views

Overlapping Domain Decomposition Method for a Noncoercive System of Quasi-Variational Inequalities Related to the Hamilton–Jacobi–Bellman Equation

Djemaoune L., Haiour M.

Abstract

This paper deals with numerical analysis of an overlapping Schwarz method on nonmatching grids for a noncoercive system of quasi-variational inequalities related to the Hamilton–Jacobi–Bellman equation. We prove that the discrete alternating Schwarz sequences on every subdomain converge monotonically into the unique solution of the discrete problem and geometrically in uniform norm. Optimally L-error estimates are also established.

Computational Mathematics and Modeling. 2016;27(2):217-227
pages 217-227 views

Some Existence and Convergence Theorems for Solving a System of Hierarchical Optimization Problems

Wairojjana N., Kumam P.

Abstract

In this article, we prove the existence and convergence theorems of the solution for a system of hierarchical variational inequality problem in Hilbert spaces. In this paper, we use Maingé’s approach for finding a solution of the system of hierarchical variational inequality problems. Our result in this article improves, extends, and generalizes some well-known corresponding results in the literature.

Computational Mathematics and Modeling. 2016;27(2):228-246
pages 228-246 views

A Model of Overpowering a Multilevel Defense System by Attack

Reshetov V.Y., Perevozchikov A.G., Lesik I.A.

Abstract

We consider a multilevel defense system on a given direction. This is a particular model of terminal discrete optimal control that may be solved by the gradient descent method. The main difficulty is the nondifferentiability of the Lipschitzian functions in the right-hand sides of the equation of motion and their derivatives with respect to all variables, which leads to an ill-posed problem when applying the classical results on differentiability of the terminal function and constructing its gradient from the conjugate system. A method is proposed for the solution of the problem by averaging the right-hand sides in combination with the stochastic gradient projection method. The study develops Germeier’s defense-attack model by allowing for a multilevel defense structure, which in general leads to the synthesis problem for a discrete optimal control system.

Computational Mathematics and Modeling. 2016;27(2):254-269
pages 254-269 views

Stability of Finite-Difference Problems with Nonreflecting Boundary Conditions

Dorodnitsyn L.V.

Abstract

The article considers a three-point central-difference scheme for the transfer equation, with different kinds of special boundary conditions that reduce wave reflection from artificial boundaries. For the corresponding problems, we prove stability in grid energy norms analogous to the ordinary L2-norm or the differential (Sobolev) norm.

Computational Mathematics and Modeling. 2016;27(2):270-274
pages 270-274 views

II. Informatics

Parallel Algorithm to Detect Structural Changes in Time Series

Nikol’skii I.M., Furmanov K.K.

Abstract

Analysis of long time series is relevant in many current applications. These investigations are usually carried out on multiprocessor computers (supercomputers). However, supercomputer calculations are efficient only if the time-series processing algorithms are sufficiently parallelized. In this article, we propose a parallel algorithm that detects shift points of the time-series mean — a highly important task in many applications. The algorithm breaks the time series into segments and looks for shift points on each segment using a statistical test. The critical values have been calculated for this test. An additional test reduces the number of false detections.

Computational Mathematics and Modeling. 2016;27(2):247-253
pages 247-253 views