This volume provides a comprehensive introduction to the modern theory of differential-operator and kinetic models including Vlasov-Maxwell, Fredholm, Lyapunov-Schmidt branching equations to name a few. This book will bridge the gap in the considerable body of existing academic literature on the analytical methods used in studies of complex behavior of differential-operator equations and kinetic models. This monograph will be of interest to mathematicians, physicists and engineers interested in the theory of such non-standard systems.

Foreword vii

Introduction xv

Acknowledgments xxi

Part I: Operator and Differential-Operator

Equations 1

1. Auxiliary Information on the Theory of Linear Operators 3

1.1 Generalized Jordan Chains, Sets and Root Numbers of Linear Operators 3

1.2 Regularization of Linear Equations with Fredholm Operators 9

1.3 Principal Theorem of Regularization of Linear Equations by the Perturbation Method 16

1.4 Regularization of Linear Equations Based on the Perturbation Theory in Hilbert Spaces 22

1.5 Regularization with Vector Regularizing Parameter for the First Kind Equations 31

2. Volterra Operator Equations with Piecewise Continuous Kernels: Solvability and Regularized Approximate Methods 39

2.1 Theory of the Volterra Operator Equations with Piecewise Continuous Kernels 39

2.2 NumericalMethods 62

3. Nonlinear Differential Equations Near Branching Points 83

3.1 Problem Statement 84

3.2 Open Problems and Generalizations 93

3.3 Magnetic InsulationModel Example 95

4. Nonlinear Operator Equations with a Functional Perturbation of the Argument 97

4.1 Nonlinear Operator Equations 97

4.2 Conclusion 103

5. Nonlinear Systems’ Equilibrium Points: Stability, Branching, Blow-Up 105

5.1 Reduction of a Nonlinear System in the Neighborhood of an Equilibrium Point to a Single Differential Equation 108

5.2 The Construction of a Solution of a Nonlinear System by the Successive Approximations Method 111

5.3 Open Problems 114

6. Nonclassic Boundary Value Problems in the Theory of Irregular Systems of Equations with Partial

Derivatives 117

6.1 Skeleton Chains of Linear Operators 120

6.2 Abstract Irregular Equation Reduction to the Sequence of Regular Equations 122

6.3 Skeleton Decomposition in the Theory of Irregular Ordinary Differential Equation in Banach Space 129

7. Epilogue for Part I 131

Part II: Lyapunov Methods in Theory of Nonlinear Equations with Parameters 133

8. Lyapunov Convex Majorants in the Existence Theorems 135

8.1 Parameter-IndependentMajorants 137

8.2 Majorants Depending on a Parameter 147

8.3 Solution Existence Domain 153

9. Investigation of Bifurcation Points of Nonlinear Equations 159

9.1 Lyapunov–Schmidt Method in the Problem of a Bifurcation Point 161

9.2 Open Problems 164

10. General Existence Theorems for the Bifurcation Points 169

10.1 Open Problem 179

11. Construction of Asymptotics in a Neighborhood of a Bifurcation Point 183

11.1 Analytic Lyapunov–Schmidt Method in the Study of Branching Equations 183

11.2 Variational Methods in the Study of Branching Equations 187

12. Regularization of Computation of Solutions in a Branch Point Neighborhood 205

12.1 Construction of the Regularization Equation in the Problemat a Branch Point 206

12.2 Definition and Properties of Simple Solutions 207

12.3 Construction of Regularization Equation of Simple Solutions 210

13. Iteration Methods, Analytical Initial Approximations, Interlaced Equations 213

13.1 Iterations and Uniformization of Branching Solutions 213

13.2 Branching Equation and the Selection of Initial Approximation 214

14. Iterative Methods Using Newton Diagrams 221

14.1 One-Step Iteration Method 224

14.2 N-step Iteration Method 228

14.3 Iteration Method for Nonlinear Equation Invariant Under Transformation Groups 234

15. Small Solutions of Nonlinear Equations with Vector Parameter in Sectorial Neighborhoods 241

15.1 Construction of the Minimal Branch of Solutions of Equation with Fredholm Operator 243

15.2 Sufficient Conditions of the Minimal Branch Existence 249

16. Successive Approximations to the Solutions to Nonlinear Equations with a Vector Parameter 257

16.1 Existence Theorem and Successive Approximations 258

17. Interlaced and Potential Branching Equation 265

17.1 Property of (S, K)-interlacing of an Equation and Its Inheritance by Branching Equation 266

17.2 (T,M)-interlaced and (T2,M)-interlaced Branching Equation 271

17.3 α-Parametric Interlaced Branching Equation 274

17.4 Interlaced Branching Equation of Potential Type 277

18. Epilogue for Part-II 285

Part III: Kinetic Models 287

19. The Family of Steady-State Solutions of Vlasov–Maxwell System 289

19.1 Ansatz of the Distribution Function and Reduction of Stationary Vlasov–Maxwell Equations to Elliptic System 289

20. Boundary Value Problems for the Vlasov–Maxwell System 313

20.1 Introduction 313

20.2 Collisionless Kinetic Models (Classical and Relativistic Vlasov–Maxwell Systems) 319

20.3 Quantum Models: Wigner–Poisson and Schr¨odinger–Poisson Systems 320

20.4 Mixed Quantum-Classical Kinetic Systems 320

21. Stationary Solutions of Vlasov–Maxwell System 323

21.1 Problem Reduction to the System of Nonlinear Elliptic Equations 324

21.2 System Reductions 329

22. Existence of Solutions for the Boundary Value Problem 335

22.1 Existence of Solution for Nonlocal Boundary Value Problem 342

23. Nonstationary Solutions of the Vlasov–Maxwell System 349

23.1 Reduction of the Vlasov–Maxwell System to NonlinearWave Equation 349

23.2 Existence of Nonstationary Solutions of the Vlasov–Maxwell System in the Bounded Domain 357

24. Linear Stability of the Stationary Solutions of the Vlasov–Maxwell System 363

25. Bifurcation of Stationary Solutions of the Vlasov–Maxwell System 373

25.1 Bifurcation of Solutions of Nonlinear Equations in Banach Spaces 377

25.2 Conclusions 386

26. Statement of the Boundary Value Problem and the Bifurcation Problem 387

27. Resolving Branching Equation 401

27.1 The Existence Theorem for Bifurcation Points and the Construction of Asymptotic Solutions 404

28. Numerical Modeling of the Limit Problem for the Magnetically Noninsulated Diode 415

28.1 Introduction 415

28.2 Description of Vacuum Diode 416

28.3 Shot Noise in a Diode 417

28.4 Description of the Mathematical Model 419

28.5 Solution Trajectory, Upper and Lower Solutions . 424

28.6 Second Lower Solution Hypothesis 434

28.7 NumericalMethods 438

28.8 NumericalModeling 448

29. Open Problems 453

Bibliography 459

Index 471

Alexander Sinitsyn received the PhD and the Dr habil. degrees in 1989 and 2005 respectively. He is a Professor of Universidad Nacional de Colombia. He worked and lectured in the leading mathematical centers of Germany, France, Austria, Israel, China, Hong Kong, Canada. His research interests include kinetic equations and applications. He has authored 80 scientific papers and two monographs.

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