Toward General Theory of Differential-Operator and Kinetic Models

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.
Contents: Operator and Differential-Operator Equations:Auxiliary Information on the Theory of Linear OperatorsVolterra Operator Equations with Piecewise Continuous Kernels: Solvability and Regularized Approximate MethodsNonlinear Differential Equations Near Branching PointsNonlinear Operator Equations with a Functional Perturbation of the ArgumentNonlinear Systems' Equilibrium Points: Stability, Branching, Blow-UpNonclassic Boundary Value Problems in the Theory of Irregular Systems of Equations with Partial DerivativesEpilogue for Part ILyapunov Methods in Theory of Nonlinear Equations with Parameters:Lyapunov Convex Majorants in the Existence TheoremsInvestigation of Bifurcation Points of Nonlinear EquationsGeneral Existence Theorems for the Bifurcation PointsConstruction of Asymptotics in a Neighborhood of a Bifurcation PointRegularization of Computation of Solutions in a Branch Point NeighborhoodIteration Methods, Analytical Initial Approximations, Interlaced EquationsIterative Methods Using Newton DiagramsSmall Solutions of Nonlinear Equations with Vector Parameter in Sectorial NeighborhoodsSuccessive Approximations to the Solutions to Nonlinear Equations with a Vector ParameterInterlaced and Potential Branching EquationEpilogue for Part IIKinetic Models:The Family of Steady-State Solutions of Vlasov–Maxwell SystemBoundary Value Problems for the Vlasov–Maxwell SystemStationary Solutions of Vlasov–Maxwell SystemExistence of Solutions for the Boundary Value ProblemNonstationary Solutions of the Vlasov–Maxwell SystemLinear Stability of the Stationary Solutions of the Vlasov–Maxwell SystemBifurcation of Stationary Solutions of the Vlasov–Maxwell SystemStatement of the Boundary Value Problem and the Bifurcation ProblemResolving Branching EquationNumerical Modeling of the Limit Problem for the Magnetically Noninsulated DiodeOpen Problems
Readership: Graduate students and researchers interested in mathematical physics, differential equations and mathematical modeling.Lyapunov–Schmidt Method;Bifurcation;Branching Equation;Conley Index;Fredholm Operator;Ill-Posed Problem;Interlaced Equation;Lyapunov Majorants;Stabilizing Operator;Vlasov–Maxwell System;Elliptic System;Lower-Upper Solution;Magnetic Insulation0Key Features:The first book to consider coupling between the investigation of the various kinds of differential-operator, kinetic equation and branching theoryReviews the authors' recent developments in these fieldsNew contribution and unified view on differential-operator and kinetic models