Hyperbolic functional differential inequalities and applications. (English) Zbl 0973.35188

Kamont, Zdzisław

Mathematics and its Applications (Dordrecht). 486. Dordrecht: Kluwer Academic Publishers. xiii, 304 p. (1999).

Publisher’s description: 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.

MSC:

35R10	Partial functional-differential equations
35R45	Partial differential inequalities and systems of partial differential inequalities
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35L40	First-order hyperbolic systems
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs

Keywords:

functional differential inequalities generated by initial and mixed problems; existence theory of local and global solutions; functional integral equations; methods of lines for hyperbolic problems; difference methods; classical solutions; weak solutions; Carathéodory solutions; entropy solutions; viscosity solutions; differential difference equations