Layer Resolving Grids and Transformations for Singular Perturbation Problems

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).