##
**Partial difference equations.**
*(English)*
Zbl 1016.39001

Advances in Discrete Mathematics and Applications. 3. London: Taylor and Francis. xii, 267 p. (2003).

This book on two-dimensional partial difference equations over different subdomains of the lattice plane begins with introductory examples stemming mostly from discretizations of partial differential equations, but also from game theory or population dynamics. These are followed by the basic tools concerning differences, sequences, frequency measures and fixed point theorems.

In order to calculate explicit solutions of difference equations the foundations of Mikusiński’s operational calculus are laid down and transferred to the two-dimensional discrete version of W. Jentsch, but also other methods are developed as the separation of variables and the solution by means of Green’s function. Usually, the difference equations are linear two-level equations, but also three-level equations, equations with delay and nonlinear perturbations are considered.

The chapters on maximum principles, stability theory, existence or non-existence of solutions with additional properties as positive, bounded, oscillatory or traveling wave solutions, mainly contain equations stemming from the author, partly together with coauthors.

Altogether 66 examples illustrate the different topics of the book, which is addressed both to graduate students and to research workers.

In order to calculate explicit solutions of difference equations the foundations of Mikusiński’s operational calculus are laid down and transferred to the two-dimensional discrete version of W. Jentsch, but also other methods are developed as the separation of variables and the solution by means of Green’s function. Usually, the difference equations are linear two-level equations, but also three-level equations, equations with delay and nonlinear perturbations are considered.

The chapters on maximum principles, stability theory, existence or non-existence of solutions with additional properties as positive, bounded, oscillatory or traveling wave solutions, mainly contain equations stemming from the author, partly together with coauthors.

Altogether 66 examples illustrate the different topics of the book, which is addressed both to graduate students and to research workers.

Reviewer: Lothar Berg (Rostock)

### MSC:

39A05 | General theory of difference equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39A11 | Stability of difference equations (MSC2000) |

44A40 | Calculus of Mikusiński and other operational calculi |

39A10 | Additive difference equations |