Synopses & Reviews
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 differential inequalities generated by initial and mixed problems; existence theory of local and global solutions; functional integral equations generated by hyperbolic equations; numerical methods of lines for hyperbolic problems; and difference methods for initial and initial-boundary value problems. Besides classical solutions, some classes of weak solutions are also treated, such as Carathéodory solutions for quasilinear equations, entropy solutions and viscosity solutions for nonlinear problems, and solutions in the Friedrichs sense for almost linear equations. The theory of difference and differential difference equations generated by original problems and its applications to the construction of numerical methods for functional differential problems is also discussed. Audience: This volume will be valuable to pure mathematicians and graduate students whose work involves the theory of functional differential problems at an advanced level. Applied mathematicians and research engineers will find the numerical algorithms for many hyperbolic problems of interest.
Synopsis
This book is intended as a self-contained exposition of hyperbolic functional dif- ferential inequalities and their applications. Its aim is to give a systematic and unified presentation of recent developments of the following problems: (i) functional differential inequalities generated by initial and mixed problems, (ii) existence theory of local and global solutions, (iii) functional integral equations generated by hyperbolic equations, (iv) numerical method of lines for hyperbolic problems, (v) difference methods for initial and initial-boundary value problems. Beside classical solutions, the following classes of weak solutions are treated: Ca- ratheodory solutions for quasilinear equations, entropy solutions and viscosity so- lutions for nonlinear problems and solutions in the Friedrichs sense for almost linear equations. The theory of difference and differential difference equations ge- nerated by original problems is discussed and its applications to the constructions of numerical methods for functional differential problems are presented. The monograph is intended for different groups of scientists. Pure mathemati- cians and graduate students will find an advanced theory of functional differential problems. Applied mathematicians and research engineers will find numerical al- gorithms for many hyperbolic problems. The classical theory of partial differential inequalities has been described exten- sively in the monographs 138, 140, 195, 225). As is well known, they found applica- tions in differential problems. The basic examples of such questions are: estimates of solutions of partial equations, estimates of the domain of the existence of solu- tions, criteria of uniqueness and estimates of the error of approximate solutions.
Description
Includes bibliographical references (p. 289-301) and index.
Table of Contents
Preface. 1. Initial Problems on the Haar Pyramid. 2. Existence of Solutions on the Haar Pyramid. 3. Numerical Methods for Initial Problems. 4. Initial Problems on Unbounded Domains. 5. Mixed Problems for Nonlinear Equations. 6. Numerical Method of Lines. 7. Generalized Solutions. 8. Functional Integral Equations. Bibliography. Index.