Applied Partial Differential Equations

The purpose of this book is to provide the reader with a comprehensive introduction to the applications of symmetry analysis to ordinary and partial differential equations. The theoretical background of physics is illustrated by modem methods of computer algebra. The presentation of the material in the book is based on Mathematica 3.0 note­ books. The entire printed version of this book is available on the accompanying CD.

The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Bäcklund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg-de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability.

Many real-world problems in science and engineering, including physical, biological, and social phenomena, can be mathematically formulated and rigorously solved by modeling them in linear and nonlinear differential and partial differential equations. This Special Issue titled “Differential Equations and Asymptotic Analysis: Recent Advances and Applications” consists of a collection of papers written by eminent mathematicians and experts in their fields, covering many different areas of nonlinear analysis, both theoretical and applied, related to differential equations, including fixed point theory, monotone operator theory, equilibrium problems and optimization, asymptotic analysis, mathematical biology, and numerical computations. We hope that this Special Issue will be of interest to many researchers, as well as graduate students working in these fields.

This book started its life as a series of lectures given by the second author from the 1970’s onwards to students in their third and fourth years in the Department of Mechanics and Mathematics at Rostov State University. For these lectures there was also an audience of engineers and applied mechanicists who wished to understand the functional analysis used in contemporary research in their fields. These people were not so much interested in functional analysis itself as in its applications; they did not want to be told about functional analysis in its most abstract form, but wanted a guided tour through those parts of the analysis needed for their applications. The lecture notes evolved over the years as the first author started to make more formal typewritten versions incorporating new material. About 1990 the first author prepared an English version and submitted it to Kluwer Academic Publishers for inclusion in the series Solid Mechanics and its Applications. At that state the notes were divided into three long chapters covering linear and nonlinear analysis. As Series Editor, the third author started to edit them. The requirements of lecture notes and books are vastly different. A book has to be complete (in some sense), self contained, and able to be read without the help of an instructor.