The approach of layer-damping coordinate transformations to treat singularly perturbed equations is a relatively new, and fast growing area in the field of applied mathematics. This monograph aims to present a clear, concise, and easily understandable description of the qualitative properties of solutions to singularly perturbed problems as well as of the essential elements, methods and codes of the technology adjusted to numerical solutions of equations with singularities by applying layer-damping coordinate transformations and corresponding layer-resolving grids. The first part of the book deals with an analytical study of estimates of the solutions and their derivatives in layers of singularities as well as suitable techniques for obtaining results. In the second part, a technique for building the coordinate transformations eliminating boundary and interior layers, is presented. Numerical algorithms based on the technique which is developed for generating layer-damping coordinate transformations and their corresponding layer-resolving meshes are presented in the final part of this volume.
This book will be of value and interest to researchers in computational and applied mathematics.
Chapter 1 Introduction to singularly perturbed problems: introduction; examples of singularly perturbed problems; convection-diffusion problems; momentum conservation laws; Prandtl equations; problem of a thin beam; problems of the shock wave structure; Burger's equation; one dimensional steady reaction-diffusion-convection model; Orr-Sommerfeld problem; diffusion-drift motion problem; idealized problems; semilinear problem; weakly-coupled systems of ordinary differential equations; autonomous equation; equation with a power function multiplying the second derivative; general idealized problem; invariants of equations; singular functions; definition of the singular functions; examples of singular functions; layer-type functions; notion of layers; definition of layers; examples of layers; partition of layers; scale of a layer; classification of layers; basic approaches to analyze problems with a small parameter; method of multivariable asymptotic expansions; method of matched asymptotic expansions; expansion via differential inequalities; numerical methods; method of layer-damping transformations; comments. Chapter 2 Background for qualitative analysis: introduction; differential inequalities; scalar problems; systems of the second order; theorems of inverse monotonicity; first order equations; second order equations; requirements imposed on estimates of the derivatives; formulation of an optimal univariate transformation; necessary bounds for the first derivative; bounds on the higher derivatives; uniform bounds on the total variation; inequality relations; comments. Chapter 3 Estimates of the solution derivatives to semilinear problems: introduction; initial problem; smooth problem; nonsmooth terms; second order equations; strong ellipticity; problem with the condition f(x,u) = xg(x,u); problem of population dynamics theory; generalization to mixed boundary conditions and dependence on e; equation with a power function affecting the second derivative; power singularities; exponential singularity; generalization to elliptic and parabolic equations; estimates of the solution derivatives; comments. Chapter 4 Problems for ordinary quasilinear equations: introduction; autonomous boundary value problem; preliminary results; boundary layers; interior layers; nonautonomous equation; estimates of the first derivative; graphical chart for localizing the layers; example of the problem; analysis of the limit solution; properties of the limit solution; comments. (Part contents).