Two-scale approach to oscillatory singularly perturbed transport equations (Q2399899)

Multi-scale problems in science and engineering are often described by partial differential equations (PDEs) with highly oscillating coefficients. Typical examples include flows in porous media, and turbulent transport problems. A complete analysis of these problems is extremely difficult. Two-scale convergence is a useful homogenization tool to solve these types of problems. In this lecture notes, the author applys the two-scale homogenization technique to solve ordinary differential equations (ODEs) with oscillating coefficients and oscillatory singularly perturbed ODEs. The monograph has two parts: The first part presents the two-scale convergence and the later one is the two-scale numerical methods. Chapter 1 presents a nice introduction to homogenization problems, two-scale convergence with some examples. Chapter 2 provides some basic definitions and theoretical estimates related to two-scale convergence. Chapter 3 studies the homogenization of ODEs and oscillatory singularly perturbed ODEs. Further, it deals with the homogenization of hyperbolic PDEs, and singularly perturbed hyperbolic PDEs. Part II of this monograph presents numerical methods based on two-scale convergence criteria. Chapter 5 deals with the two-scale numerical method for the long-term forecast of the drift of objects in an ocean with tide and wind. Chapter 6 focuses on the two-scale numerical method for the simulation of particle beams in a focusing channel. This is a good research monograph for people working on theoretical and numerical aspects of oscillatory singularly perturbed differential equations. The book is well-written with several examples from various applications. This book provides the complete picture of two-scale convergence approach for homogenization problems and the numerical approach. This monograph is excellent and well-written. This book will be very useful for mathematicians and engineers working on multiscale problems.