Computational Methods and Mathematical Modeling in Cyberphysics and Engineering Applications 1

Dmitri Koroliouk is a Doctor of Sciences, Professor at the National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", and leading researcher at the Institute of Mathematics, and at the Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine. He is also Head of the Digital Innovation Laboratory at UNESCO Interdisciplinary Chair in Biotechnology and Bioethics at the University of Rome Tor Vergata, Italy.Sergiy Lyashko is Head of the Department of Computational Mathematics at the Faculty of Computer Sciences and Cybernetics, Taras Shevchenko National University of Kyiv, and a professor at the National Academy of Sciences of Ukraine. His research interests concern problems of singular optimal control of linear systems.Nikolaos Limnios is a professor at the Laboratoire de Mathématiques Appliquées, Université de Technologie de Compiègne, France. His research interests concern statistical inference for stochastic processes and semi-Markov processes.

• Preface xiDmitri KOROLIOUKChapter 1 The Hydrodynamic-type Equations and the Solitary Solutions 1Sergiy LYASHKO, Valerii SAMOILENKO, Yuliia SAMOILENKO and Ihor GAPYAK1.1 Introduction 11.2 The Korteweg-de Vries equation and the soliton solutions 31.3 The Korteweg-de Vries equation with a small perturbation 41.4 The linear WKB technique and its generalization 71.5 Acknowledgments 111.6 References 11Chapter 2 The Nonlinear WKB Technique and Asymptotic Soliton-like Solutions to the Korteweg-de Vries Equation with Variable Coefficients and Singular Perturbation 15Sergiy LYASHKO, Valerii SAMOILENKO, Yuliia SAMOILENKO and Evgen VAKAL2.1 Introduction 162.2 Main notations and definitions 182.3 The structure of the asymptotic one-phase soliton-like solution 192.4 The KdV equation with quadratic singularity 202.5 Equations for the regular part of the asymptotics and their analysis 222.6 Equations for the singular part of the asymptotics and their analysis 242.6.1 The main term of the singular part 252.6.2 The higher terms of the singular part and the orthogonality condition 262.6.3 The orthogonality condition and the discontinuity curve 292.6.4 Prolongation of the singular terms from the discontinuity curve 342.7 Justification of the algorithm 382.8 Discussion and conclusion 442.9 Acknowledgments 452.10 References 45Chapter 3 Asymptotic Analysis of the vcKdV Equation with Weak Singularity 49Sergiy LYASHKO, Valerii SAMOILENKO, Yuliia SAMOILENKO and Nataliia LYASHKO3.1 Introduction 503.2 The asymptotic soliton-like solutions 513.3 The examples of the asymptotic soliton-like solutions 563.3.1 The asymptotic step-wise solutions 573.3.2 The asymptotic solutions of soliton type 613.4 Discussion and conclusion 663.5 Acknowledgments 663.6 References 66Chapter 4 Modeling of Heterogeneous Fluid Dynamics with Phase Transitions and Porous Media 69Gennadiy V SANDRAKOV4.1 Introduction 694.2 The large particle method 724.3 The particle-in-cell method 794.4 Modeling of heterogeneous fluid dynamics 834.5 Modeling of heterogeneous fluid dynamics with phase transitions 884.6 Modeling of viscous fluid dynamics and porous media 944.7 References 98Chapter 5 Mathematical Models and Control of Functionally Stable Technological Process 101Volodymyr PICHKUR, Valentyn SOBCHUK and Dmytro CHERNIY5.1 Introduction 1015.2 Analysis of production process planning procedure 1045.3 Mathematical model of the production process management system of an industrial enterprise 1085.4 Control design 1115.5 Algorithm of control of production process 1155.6 Conclusion 1165.7 Acknowledgments 1175.8 References 118Chapter 6 Alternative Direction Multiblock Method with Nesterov Acceleration and Its Applications 121Vladislav HRYHORENKO, Nataliia LYASHKO, Sergiy LYASHKO and Dmytro KLYUSHIN6.1 Introduction 1216.2 Proximal operators 1226.3 ADMM (alternating direction method of multipliers) 1286.4 Bregman iteration 1316.5 Forward-backward envelope (FBE) 1326.6 Douglas-Rachford envelope (DRE) 1336.7 Proximal algorithms for complex functions 1346.8 Fast alternative directions methods 1376.9 Numerical experiments 1426.9.1 Exchange problem 1426.9.2 Basis pursuit problem 1436.9.3 Constrained LASSO problem 1446.10 Conclusion 1456.11 References 145Chapter 7 Modified Extragradient Algorithms for Variational Inequalities 149Vladimir V SEMENOV and Sergey V DENISOV7.1 Introduction 1497.2 Preliminaries 1497.3 Overview of the main algorithms for solving variational inequalities and approximations of fixed points 1567.4 Modified extragradient algorithm for variational inequalities 1647.5 Modified extragradient algorithm for variational inequalities and operator equations with a priori information 1737.6 Strongly convergent modified extragradient algorithm 1777.6.1 Algorithm variant for variational inequalities 1787.6.2 Variant for problems with a priori information 1937.7 References 199Chapter 8 On Multivariate Algorithms of Digital Signatures on Secure El Gamal-Type Mode 205Vasyl USTIMENKO8.1 On post-quantum, multivariate and non-commutative cryptography 2068.2 On stable subgroups of formal Cremona group and privatization of multivariate public keys based on maps of bounded degree 2088.3 Multivariate Tahoma protocol for stable Cremona generators and its usage for multivariate encryption algorithms 2118.4 On multivariate digital signature algorithms and their privatization scheme 2148.5 Examples of stable cubical groups 2168.5.1 Simplest graph-based example 2168.5.2 Other stable subgroups defined via linguistic graphs 2198.5.3 Special homomorphisms of linguistic graphs and corresponding semigroups 2228.5.4 Example of stable subsemigroups of arbitrary degree 2238.6 Conclusion 2258.7 References 227Chapter 9 Metasurface Model of Geographic Baric Field Formation 231Dmitri KOROLIOUK, Maksym ZOZIUK, Pavlo KRYSENKO and Yuriy YAKYMENKO9.1 Introduction 2319.2 The parametric scalar field model principle 2339.3 Local isobaric scalar field model 2349.4 Modeling Chladni figures based on the proposed model 2359.5 The frequency of forcing influences and the problem of its detection 2379.6 Conclusion 2399.7 References 241Chapter 10 Simulation of the Electron-Hole Plasma State by Perturbation Theory Methods 245Andrii BOMBA, Sergiy LYASHKO and Ihor MOROZ10.1 Introduction Nonlinear boundary value problems of the p-i-n diodes theory 24510.2 Construction of an asymptotic solution of a boundary value problem for the system of the charge carrier current continuity equations and the Poisson equation 24910.3 Simulation of the charge carriers’ stationary distribution in the electron-hole plasma of the p-i-n diode assembly elements 26210.4 Modeling the charge carriers stationary distribution in the active region of the integrated surface-oriented p-i-n structures 26410.5 Final considerations 27010.6 References 271Chapter 11 Diffusion Perturbations in Models of the Dynamics of Infectious Diseases Taking into Account the Concentrated Effects 273Serhii BARANOVSKY, Andrii BOMBA, Sergiy LYASHKO and Oksana PRYSHCHEPA11.1 Introduction 27311.2 Model problem of infectious disease dynamics taking into account diffusion perturbation and asymptotics of the solution 27711.3 Modeling of diffusion perturbations of infectious disease process taking into account the concentrated effects and immunotherapy 28211.4 Modeling the influence of diffusion perturbations on development of infectious diseases under convection 28811.5 Numerical experiment results 29211.6 Conclusion 30011.7 References 301Chapter 12 Solitary Waves in the "Shallow Water" Environments 305Yurii TURBAL, Mariana TURBAL and Andrii BOMBA12.1 Introduction 30512.2 T-forms for the solitary wave approximation 30712.3 Existence of the solution of the gas dynamics equations in the form of solitary waves 31312.4 Analysis of the localized wave trajectories 33212.5 Numerical results 33812.6 Conclusion 34112.7 References 342Chapter 13 Instrument Element and Grid Middleware in Metrology Problems 345Pavlo NEYEZHMAKOV, Stanislav ZUB, Sergiy LYASHKO, Irina YALOVEGA and Nataliia LYASHKO13.1 Introduction 34513.2 Security in the grid 34713.3 Grid element for measuring instruments 34713.4 Grid and some problems of metrology 35013.5 Discussion and conclusion 35213.6 References 352Chapter 14 Differential Evolution for Best Uniform Spline Approximation 355Larysa VAKAL and Evgen VAKAL14.1 Introduction 35614.2 Problem statement 35614.3 Review of methods for spline approximation 35714.4 Algorithm 35914.5 Experimental results and discussion 36214.6 Conclusion 36414.7 References 365Chapter 15 Finding a Nearest Pair of Points Between Two Smooth Curves in Euclidean Space 367Vladimir V SEMENOV, Nataliia LYASHKO, Stanislav ZUB and Yevhen HAVRYLKO15.1 Introduction 36715.2 Define the problem and notations 36815.3 Lagrange function with energy dissipation 36915.4 Lagrange equation 37015.5 Hamiltonian equations 37215.6 Numerical experiments 37615.7 Concluding remarks 37815.8 References 379Chapter 16 Constrained Markov Decision Process for the Industry 381Michel BOUSSEMART and Nikolaos LIMNIOS16.1 Introduction 38116.2 Introduction to constrained Markov decision processes 38216.2.1 Introduction 38216.2.2 Model 38316.2.3 Economic criteria 38416.2.4 Infinite horizon expected discounted reward 38616.2.5 Infinite horizon expected average reward 39216.3 Markov decision process with a constraint on the asymptotic availability 39616.3.1 Introduction 39616.3.2 Model 39716.3.3 Algorithm 39916.3.4 Application 39916.4 Markov decision process with a constraint on the asymptotic failure rate 40816.4.1 Introduction 40816.4.2 Model 40916.4.3 Algorithm 41316.4.4 Application 41316.5 Conclusion 41816.6 References 419List of Authors 423Index 427