This volume is an outcome of the EQUADIFF 87 conference in Greece. It addresses a wide spectrum of topics in the theory and applications of differential equations, ordinary, partial, and functional. The book is intended for mathematics and scientists.

The content of this book is multidisciplinary by nature. It uses mathematical tools from the theories of probability and stochastic processes, partial differential equations, and asymptotic analysis, combined with the physics of wave propagation and modeling of time reversal experiments. It is addressed to a wide audience of graduate students and researchers interested in the intriguing phenomena related to waves propagating in random media. At the end of each chapter there is a section of notes where the authors give references and additional comments on the various results presented in the chapter.

The content of this book is multidisciplinary by nature. It uses mathematical tools from the theories of probability and stochastic processes, partial differential equations, and asymptotic analysis, combined with the physics of wave propagation and modeling of time reversal experiments. It is addressed to a wide audience of graduate students and researchers interested in the intriguing phenomena related to waves propagating in random media. At the end of each chapter there is a section of notes where the authors give references and additional comments on the various results presented in the chapter.

This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.