Preface xiDmitri KOROLIOUK Chapter 1 The Hydrodynamic-type Equations and the Solitary Solutions 1Sergiy LYASHKO, Valerii SAMOILENKO, Yuliia SAMOILENKO and Ihor GAPYAK 1.1 Introduction 1 1.2 The Korteweg-de Vries equation and the soliton solutions 3 1.3 The Korteweg-de Vries equation with a small perturbation 4 1.4 The linear WKB technique and its generalization 7 1.5 Acknowledgments 11 1.6 References 11 Chapter 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 VAKAL 2.1 Introduction 16 2.2 Main notations and definitions 18 2.3 The structure of the asymptotic one-phase soliton-like solution 19 2.4 The KdV equation with quadratic singularity 20 2.5 Equations for the regular part of the asymptotics and their analysis 22 2.6 Equations for the singular part of the asymptotics and their analysis 24 2.6.1 The main term of the singular part 25 2.6.2 The higher terms of the singular part and the orthogonality condition 26 2.6.3 The orthogonality condition and the discontinuity curve 29 2.6.4 Prolongation of the singular terms from the discontinuity curve 34 2.7 Justification of the algorithm 38 2.8 Discussion and conclusion 44 2.9 Acknowledgments 45 2.10 References 45 Chapter 3 Asymptotic Analysis of the vcKdV Equation with Weak Singularity 49Sergiy LYASHKO, Valerii SAMOILENKO, Yuliia SAMOILENKO and Nataliia LYASHKO 3.1 Introduction 50 3.2 The asymptotic soliton-like solutions 51 3.3 The examples of the asymptotic soliton-like solutions 56 3.3.1 The asymptotic step-wise solutions 57 3.3.2 The asymptotic solutions of soliton type 61 3.4 Discussion and conclusion 66 3.5 Acknowledgments 66 3.6 References 66 Chapter 4 Modeling of Heterogeneous Fluid Dynamics with Phase Transitions and Porous Media 69Gennadiy V SANDRAKOV 4.1 Introduction 69 4.2 The large particle method 72 4.3 The particle-in-cell method 79 4.4 Modeling of heterogeneous fluid dynamics 83 4.5 Modeling of heterogeneous fluid dynamics with phase transitions 88 4.6 Modeling of viscous fluid dynamics and porous media 94 4.7 References 98 Chapter 5 Mathematical Models and Control of Functionally Stable Technological Process 101Volodymyr PICHKUR, Valentyn SOBCHUK and Dmytro CHERNIY 5.1 Introduction 101 5.2 Analysis of production process planning procedure 104 5.3 Mathematical model of the production process management system of an industrial enterprise 108 5.4 Control design 111 5.5 Algorithm of control of production process 115 5.6 Conclusion 116 5.7 Acknowledgments 117 5.8 References 118 Chapter 6 Alternative Direction Multiblock Method with Nesterov Acceleration and Its Applications 121Vladislav HRYHORENKO, Nataliia LYASHKO, Sergiy LYASHKO and Dmytro KLYUSHIN 6.1 Introduction 121 6.2 Proximal operators 122 6.3 ADMM (alternating direction method of multipliers) 128 6.4 Bregman iteration 131 6.5 Forward-backward envelope (FBE) 132 6.6 Douglas-Rachford envelope (DRE) 133 6.7 Proximal algorithms for complex functions 134 6.8 Fast alternative directions methods 137 6.9 Numerical experiments 142 6.9.1 Exchange problem 142 6.9.2 Basis pursuit problem 143 6.9.3 Constrained LASSO problem 144 6.10 Conclusion 145 6.11 References 145 Chapter 7 Modified Extragradient Algorithms for Variational Inequalities 149Vladimir V SEMENOV and Sergey V DENISOV 7.1 Introduction 149 7.2 Preliminaries 149 7.3 Overview of the main algorithms for solving variational inequalities and approximations of fixed points 156 7.4 Modified extragradient algorithm for variational inequalities 164 7.5 Modified extragradient algorithm for variational inequalities and operator equations with a priori information 173 7.6 Strongly convergent modified extragradient algorithm 177 7.6.1 Algorithm variant for variational inequalities 178 7.6.2 Variant for problems with a priori information 193 7.7 References 199 Chapter 8 On Multivariate Algorithms of Digital Signatures on Secure El Gamal-Type Mode 205Vasyl USTIMENKO 8.1 On post-quantum, multivariate and non-commutative cryptography 206 8.2 On stable subgroups of formal Cremona group and privatization of multivariate public keys based on maps of bounded degree 208 8.3 Multivariate Tahoma protocol for stable Cremona generators and its usage for multivariate encryption algorithms 211 8.4 On multivariate digital signature algorithms and their privatization scheme 214 8.5 Examples of stable cubical groups 216 8.5.1 Simplest graph-based example 216 8.5.2 Other stable subgroups defined via linguistic graphs 219 8.5.3 Special homomorphisms of linguistic graphs and corresponding semigroups 222 8.5.4 Example of stable subsemigroups of arbitrary degree 223 8.6 Conclusion 225 8.7 References 227 Chapter 9 Metasurface Model of Geographic Baric Field Formation 231Dmitri KOROLIOUK, Maksym ZOZIUK, Pavlo KRYSENKO and Yuriy YAKYMENKO 9.1 Introduction 231 9.2 The parametric scalar field model principle 233 9.3 Local isobaric scalar field model 234 9.4 Modeling Chladni figures based on the proposed model 235 9.5 The frequency of forcing influences and the problem of its detection 237 9.6 Conclusion 239 9.7 References 241 Chapter 10 Simulation of the Electron-Hole Plasma State by Perturbation Theory Methods 245Andrii BOMBA, Sergiy LYASHKO and Ihor MOROZ 10.1 Introduction Nonlinear boundary value problems of the p-i-n diodes theory 245 10.2 Construction of an asymptotic solution of a boundary value problem for the system of the charge carrier current continuity equations and the Poisson equation 249 10.3 Simulation of the charge carriers’ stationary distribution in the electron-hole plasma of the p-i-n diode assembly elements 262 10.4 Modeling the charge carriers stationary distribution in the active region of the integrated surface-oriented p-i-n structures 264 10.5 Final considerations 270 10.6 References 271 Chapter 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 PRYSHCHEPA 11.1 Introduction 273 11.2 Model problem of infectious disease dynamics taking into account diffusion perturbation and asymptotics of the solution 277 11.3 Modeling of diffusion perturbations of infectious disease process taking into account the concentrated effects and immunotherapy 282 11.4 Modeling the influence of diffusion perturbations on development of infectious diseases under convection 288 11.5 Numerical experiment results 292 11.6 Conclusion 300 11.7 References 301 Chapter 12 Solitary Waves in the "Shallow Water" Environments 305Yurii TURBAL, Mariana TURBAL and Andrii BOMBA 12.1 Introduction 305 12.2 T-forms for the solitary wave approximation 307 12.3 Existence of the solution of the gas dynamics equations in the form of solitary waves 313 12.4 Analysis of the localized wave trajectories 332 12.5 Numerical results 338 12.6 Conclusion 341 12.7 References 342 Chapter 13 Instrument Element and Grid Middleware in Metrology Problems 345Pavlo NEYEZHMAKOV, Stanislav ZUB, Sergiy LYASHKO, Irina YALOVEGA and Nataliia LYASHKO 13.1 Introduction 345 13.2 Security in the grid 347 13.3 Grid element for measuring instruments 347 13.4 Grid and some problems of metrology 350 13.5 Discussion and conclusion 352 13.6 References 352 Chapter 14 Differential Evolution for Best Uniform Spline Approximation 355Larysa VAKAL and Evgen VAKAL 14.1 Introduction 356 14.2 Problem statement 356 14.3 Review of methods for spline approximation 357 14.4 Algorithm 359 14.5 Experimental results and discussion 362 14.6 Conclusion 364 14.7 References 365 Chapter 15 Finding a Nearest Pair of Points Between Two Smooth Curves in Euclidean Space 367Vladimir V SEMENOV, Nataliia LYASHKO, Stanislav ZUB and Yevhen HAVRYLKO 15.1 Introduction 367 15.2 Define the problem and notations 368 15.3 Lagrange function with energy dissipation 369 15.4 Lagrange equation 370 15.5 Hamiltonian equations 372 15.6 Numerical experiments 376 15.7 Concluding remarks 378 15.8 References 379 Chapter 16 Constrained Markov Decision Process for the Industry 381Michel BOUSSEMART and Nikolaos LIMNIOS 16.1 Introduction 381 16.2 Introduction to constrained Markov decision processes 382 16.2.1 Introduction 382 16.2.2 Model 383 16.2.3 Economic criteria 384 16.2.4 Infinite horizon expected discounted reward 386 16.2.5 Infinite horizon expected average reward 392 16.3 Markov decision process with a constraint on the asymptotic availability 396 16.3.1 Introduction 396 16.3.2 Model 397 16.3.3 Algorithm 399 16.3.4 Application 399 16.4 Markov decision process with a constraint on the asymptotic failure rate 408 16.4.1 Introduction 408 16.4.2 Model 409 16.4.3 Algorithm 413 16.4.4 Application 413 16.5 Conclusion 418 16.6 References 419 List of Authors 423 Index 427