#### Foreword

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

#### Preface

Partial difference equations arise naturally in many areas of science and are used for the description of many realistic problems, such as probability problems, problems in queuing theory, physical problems, biological problems, etc. It could be stated that partial differnce eqautions are on one side of a coin, where on the other side stand partial differential equations. However, the main application of partial difference equations is probably in numerical analysis, where they arise naturally when discretizing a partial diffrential equation.

Lately, there is an increasing intrest in partial difference equations demonstrated by the enormous amount of research papers devoted to them . The initial reason for this increasing interest was probably the development of computers and the area of numerical analysis. However, there are vey few books devoted exclusively to partial difference equations in contrast to their continuos analog of partial differential equations on which hundreds of books have been written. This e-book is an attempt to present a few recent advances in partial difference equations including oscillation results, which are in the front line of research, functional analytic methods for studying partial difference equations in systems theory and numerical analysis. Basic techniques for solving or studying partial difference equations were not included, since these can be found in several chapters of books on difference equations. Of course, several other types of results could have been included, but this was not possible due to the lack of time. Probably in the future a more complete book on the subject, will be published!

The e-book constitutes of four chapters. Each chapter could be characterized as a review paper. Moreover, in all cahapters there are several examples in order to help the interseted reader get more acquainted with the presented methods and results.

I would like to express also from this position, my sincerest thanks to the colleagued who contributed in this e-book and made possible its publictaion. Also, I would like to thank the personnel of the Bentham Science Publishers and espesially Bushra Siddiqui for their continuous collaboration and support.

*
***
Eugenia N. Petropoulou**

*Assist. Professor*

*University of Patras Greece*

#### List of Contributors

##### Editor(s):

**Eugenia N. Petropoulou **

University of Patras

Greece

##### Contributor(s):

**Patricia J. Y. Wong **

School of Electrical and Electronics Engineering, Nanyang Technological University

50 Nanyang Avenue

Singapore , 639798

Singapore

**Eugenia N. Petropoulou **

Department of Engineering Sciences

Division of Applied Mathematics & Mechanics, University of Patras

Patras, 26500

Greece

**Panayiotis D. Siafarikas **

Department of Mathematics

Division of Applied Analysis, University of Patras

Patras, 26500

Greece

**Josef Hekrdla **

Department of Mathematics

Faculty of Electrical Engineering, Czech Technical university

Technika 2

Paraha 6

166 27

Czech Republic

**Jiří Gregor **

Department of Mathematics

Faculty of Electrical Engineering, Czech Technical university

Technika 2, 166 27 Paraha 6

Czech Republic

**Efstratios E. Tzirtzilakis **

Department of Mechanical Engineering & Water Resources

Technological Educational Institute of Messolongi

Messolongi, 30200

Greece

**Nikolas G. Kafoussias **

Section of Applied Analysis, University of Patras

Patras, 26500

Greece