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This book provides an accessible yet comprehensive description of the application methods of group analysis to integro-differential equations. It offers both fundamental theoretical and algorithmic aspects of these methods and includes instructive examples.

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The present book is an introduction to a new eld in applied group analysis. The book deals with symmetries of integro-differential, stochastic and delay equations that form the basis of a large variety of mathematical models, used to describe va- ous phenomena in uid mechanics and plasma physics and other elds of nonlinear science. Because of its baf ing complexity the mathematical study of nonlocal equations is far from completion, although the equations have been intensively studied in - merous applications over more than fty last years using both numerical and anal- ical methods. The principal aim of analytical approaches is to obtain exact solutions, admitted symmetries, conservation laws and other mathematical properties, which allow one to make sound decisions in more detailed applied investigations. Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. Consequently, group theoretical me- ods appear ef cient in analyzing different phenomena using mathematical models that employ differential equations. However Lie s methods cannot be directly - plied to integro-differential equations, in nite systems of differential equations, - lay equations, etc. Hence it is natural to extend the ideas of modern group analysis to these mathematical objects that up to recently were not in mainstream of classical group theoretical approaches.

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to Group Analysis of Differential Equations.- to Group Analysis and Invariant Solutions of Integro-Differential Equations.- The Boltzmann Kinetic Equation and Various Models.- Plasma Kinetic Theory: Vlasov-Maxwell and Related Equations.- Symmetries of Stochastic Differential Equations.- Delay Differential Equations.

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From the reviews:

"The present book deals with a large number of topics concerning symmetries of integro-differential, stochastic and delay equations that are met in many fields, from fluid mechanics and plasma physics to biology and so on. ... In conclusion, the book is rich in results and is of interest not only to physicists and applied scientists, but also to mathematicians working with (nonlinear) integrodifferential equations." (Alfredo Lorenzi, Mathematical Reviews, Issue 2011 i)

"The reviewed collective monograph is very useful to both physicists and mathematicians interested in general methods for investigation of nonlinear problems using symmetries. It can also serve as a textbook on symmetries of I-DEs, stochastic and delay equations for graduate students in applied mathematics, physics and engineering." (Boris V. Loginov, Zentralblatt MATH, Vol. 1203, 2011)

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