While “Mathematical Reviews” currently lists 1058 books containing “Partial Differential Equations” in their title and 128 books containing “Difference Equations” in their title, it only lists 3 books containing “Partial Difference Equations” in their title. On the other hand, 238 journal publications are listed containing “Partial Difference Equations” in their title, so research in this area is rather active and ongoing. This is due to the rich possibility of theoretical investigations and the numerous applications which partial difference equations enjoy. These facts illustrate that there is an urgent need to expand the availability of textbooks in the area of “Partial Difference Equations”.
The book at hand, “Some Recent Advances in Partial Difference Equations”, as edited and presented by Professor Eugenia Petropoulou, is a welcome, timely, and excellent contribution filling the above described gap. Professor Petropoulou has done a terrific job in putting together this volume, offering four chapters on distinct topics of current interest in the area of partial difference equations.
The first chapter covers oscillation theory of partial difference equations and is written by Professor Patricia Wong (Singapore), a world-wide leading expert in the area of differential, difference, and dynamic equations and in particular oscillation theory for these equations. Criteria for the nonexistence of positive solutions of certain partial difference equations with deviating arguments are presented and several examples are offered.
The second chapter shows a connection between functional analysis and partial difference equations and is written by the late Professor Panayiotis Siafarikas (Greece), the internationally esteemed expert in this area of research, together with his student Professor Eugenia Petropoulou (Greece), who is the editor of this volume. A functional-analytic method to study partial difference equations is developed and illustrated by two fundamental examples.
The third chapter discusses the connection of partial difference equations to systems theory and is written by Professor Jiří Gregor and Professor Josef Hekrdla (Czech Republic). Existence and uniqueness results for initial value problems and boundary value problems involving linear partial difference equations are presented and extended to systems of linear partial difference equations. These results are applied to input—output relations of linear multidimensional systems.
The fourth chapter offers some numerical schemes constituting partial difference equations and is written by Professor Efstratios Tzirtzilakis and Professor Nikolaos Kafoussias (Greece). Partial differential equations are discretized in order to obtain numerical schemes resulting in partial difference equations, and the connection of the solutions of these two equations is examined analytically and numerically.
Of course these covered topics only scratch the surface of this exciting area of research. We look forward to future developments inspired by the publication of this volume.
Martin Bohner, Rolla, Missouri (USA)
October 20, 2010