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Hyperbolic Functional Differential Inequalities and Applications / by Zdzislaw Kamont.

This monograph is a self-contained exposition of hyperbolic functional differential inequalities and their applications, on which topic the present author initiated research. It aims to give a systematic and unified presentation of recent developments in the following problems: functional differenti...

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Author: Kamont, Zdzislaw
Format: eBook
Language:English
Published: Dordrecht : Springer Netherlands : Imprint : Springer, 1999.
Series:Mathematics and its applications (Springer Science+Business Media) ; 486.
Subjects:
Differential inequalities.
Functional differential equations > Numerical solutions.
Differential equations, Hyperbolic > Numerical solutions.
Table of Contents:
  • 1 Initial Problems on the Haar Pyramid
  • 1.1 Introduction
  • 1.2 Functional differential inequalities
  • 1.3 Weak functional differential inequalities
  • 1.4 Comparison theorems for classical solutions
  • 1.5 Applications of comparison theorems
  • 1.6 Kamke functions
  • 1.7 Uniqueness of classical solutions
  • 1.8 Nonlinear systems
  • 1.9 Haar inequality for nonlinear systems
  • 1.10 Uniqueness and continuous dependence
  • 1.11 Chaplygin method for initial problems
  • 2 Existence of Solutions on the Haar Pyramid
  • 2.1 Introduction
  • 2.2 Function spaces
  • 2.3 Existence of classical solutions
  • 2.4 Examples
  • 2.5 Quasi
  • linear systems
  • 2.6 Bicharacteristics of quasilinear systems
  • 2.7 Integral operators for initial problems
  • 2.8 Existence of Carathéodory solutions
  • 2.9 Uniqueness of generalized solutions
  • 3 Numerical Methods for Initial Problems
  • 3.1 Introduction
  • 3.2 Functional difference inequalities
  • 3.3 Applications of functional difference inequalities
  • 3.4 Almost linear problems
  • 3.5 Error estimates of approximate solutions
  • 3.6 Difference methods for nonlinear equations
  • 3.7 Interpolating operators on Haar pyramid
  • 3.8 The Euler method for the Cauchy problem
  • 3.9 Error estimates for the Euler method
  • 3.10 Difference methods for almost linear equations
  • 4 Initial Problems on Unbounded Domains
  • 4.1 Introduction
  • 4.2 Bicharacteristics for quasilinear systems
  • 4.3 Operator U? and its properties
  • 4.4 Existence of weak solutions
  • 4.5 Integral operators for quasilinear systems
  • 4.6 Quasilinear systems in the second canonical form
  • 4.7 Uniqueness of solutions
  • 4.8 Function spaces
  • 4.9 Bicharacteristics of nonlinear functional differential equations
  • 4.10 Integral functional equations
  • 4.11 The existence of the sequence of successive approximations
  • 4.12 Convergence of the sequence {z(m), u(m)}
  • 4.13 The main theorem
  • 4.14 Some noteworthy particular cases
  • 5 Mixed Problems for Nonlinear Equations
  • 5.1 Introduction
  • 5.2 Functional differential inequalities
  • 5.3 Comparison theorems for mixed problems
  • 5.4 Chaplygin method for mixed problems
  • 5.5 Approximate solutions
  • 5.6 Difference methods for mixed problems
  • 5.7 Functional difference equations with mixed conditions
  • 5.8 Convergence of difference methods
  • 5.9 Interpolating operators
  • 5.10 The Euler method for mixed problems
  • 5.11 Bicharacteristics for mixed problems
  • 5.12 Functional integral equations
  • 5.13 Bicharacteristics of nonlinear mixed problems
  • 5.14 Integral functional equations
  • 5.15 The existence of solutions of nonlinear mixed problems
  • 5.16 Uniqueness of weak solutions of mixed problems
  • 6 Numerical Method of Lines
  • 6.1 Introduction
  • 6.2 Comparison theorem
  • 6.3 Existence theorem and stability
  • 6.4 Convergence of the method of lines
  • 6.5 Examples of the numerical methods of lines
  • 6.6 Differential difference inequalities for mixed problems
  • 6.7 Method of lines for mixed problem
  • 6.8 Modified method of lines
  • 7 Generalized Solutions
  • 7.1 Introduction
  • 7.2 Quasi
  • equicontinuous operators for semilinear systems
  • 7.3 Existence of solutions
  • 7.4 Functional differential inequalities
  • 7.5 Extremal solutions of semilinear systems
  • 7.6 Carathéodory solutions of functional differential inequalities
  • 7.7 Existence of Carathéodory solutions
  • 7.8 Functional differential problems with unbounded delay
  • 7.9 Viscosity solutions of functional differential inequalities
  • 8 Functional Integral Equations
  • 8.1 Introduction
  • 8.2 Properties of a comparison problem
  • 8.3 The existence and uniqueness of solutions
  • 8.4 Examples of comparison problems
  • 8.5 A certain functional equation
  • 8.6 Properties of the operator U
  • 8.7 Nonlinear functional integral equations
  • 8.8 Discretization of the Darboux problem
  • 8.9 Solvability of difference problems
  • 8.10 Nonlinear estimates
  • 8.11 Implicit difference methods.

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