Peter Hydon has delivered his new book, Difference Equations by Differential Equation Methods, to Cambridge University Press for publication.

Around twenty years ago, an eminent numerical analyst said, “Any problem involving difference equations is an order of magnitude harder than the corresponding problem for differential equations.” Research since that time has transformed our understanding of difference equations and their solutions. The basic geometric structures that underpin difference equations are now known. From these, it has been possible to develop a plethora of systematic techniques for finding solutions, first integrals or conservation laws of a given difference equation. These look a little different to their counterparts for differential equations, mainly because the solutions of difference equations are not continuous. However, they are widely applicable and most of them do not require the equation to have special properties such as linearizability or integrability.

As our understanding of these methods has grown, it became apparent that many of them would be of use to non-specialists in a wide variety of fields. Difference equations occur in many branches of pure and applied mathematics, some of which (such as mathematical modelling and numerical simulation) are used by engineers and physicists. The purpose of this book is to cover a broad range of exact techniques for dealing with difference equations, together with a clear explanation of why they work. No prior knowledge of difference equations is assumed, as I think that some potential readers who (like me) enjoy working with differential equations will be attracted by the prospect of being able to use similar ideas in a new field.

The book will appear in the series Cambridge Monographs on Applied and Computational Mathematics.