Preface ........................................................ ix
Chapter 1. Introduction to singularly perturbed problems
1.1 Introduction ............................................... 1
1.2 Examples of singularly perturbed problems .................. 3
1.2.1 Convection-diffusion problem ........................ 3
1.2.2 Momentum conservation laws .......................... 4
1.2.3 Prandtl equations ................................... 4
1.2.4 Problem of a thin beam .............................. 5
1.2.5 Problems of the shock wave structure ................ 5
1.2.6 Burger's equation ................................... 5
1.2.7 One-dimensional steady reaction-diffusion-
convection model .................................... 6
1.2.8 Orr-Sommerfeld problem .............................. 6
1.2.9 Diffusion-drift motion problem ...................... 7
1.3 Idealized problems ......................................... 7
1.3.1 Semilinear problem .................................. 8
1.3.2 Weakly-coupled systems of ordinary differential
equations ........................................... 9
1.3.3 Autonomous equation ................................. 9
1.3.4 Equation with a power function multiplying the
second derivative .................................. 10
1.3.5 General idealized problem .......................... 10
1.3.6 Invariants of equations ............................ 11
1.4 Singular functions ........................................ 12
1.4.1 Definition of the singular functions ............... 12
1.4.2 Examples of singular functions ..................... 13
1.4.3 Layer-type functions ............................... 16
1.5 Notion of layers .......................................... 17
1.5.1 Definition of layers ............................... 17
1.5.2 Examples of layers ................................. 18
1.5.3 Partition of layers ................................ 20
1.5.4 Scale of a layer ................................... 21
1.5.5 Classification of layers ........................... 21
1.6 Basic approaches to analyze problems with a small
parameter ................................................. 24
1.6.1 Method of multivariable asymptotic expansions ...... 25
1.6.2 Method of matched asymptotic expansions ............ 26
1.6.3 Expansion via differential inequalities ............ 27
1.6.4 Numerical methods .................................. 27
1.6.5 Method of layer-damping transformations ............ 28
1.7 Comments .................................................. 34
Chapter 2. Background for qualitative analysis
2.1 Introduction .............................................. 37
2.2 Differential inequalities ................................. 37
2.2.1 Scalar problems .................................... 37
2.2.2 Systems of the second order ........................ 40
2.3 Theorems of inverse monotonicity .......................... 43
2.3.1 First order equations .............................. 43
2.3.2 Second order equations ............................. 47
2.4 Requirements imposed on estimates of the derivatives ...... 55
2.4.1 Formulation of an optimal univariate
transformation ..................................... 56
2.4.2 Necessary bounds for the first derivative .......... 56
2.4.3 Bounds on the higher derivatives ................... 58
2.4.4 Uniform bounds on the total variation .............. 59
2.5 Inequality relations ...................................... 62
2.6 Comments .................................................. 63
Chapter 3. Estimates of the solution derivatives to
semilinear problems
3.1 Introduction .............................................. 65
3.2 Initial problem ........................................... 65
3.2.1 Smooth terms ....................................... 65
3.2.2 Nonsmooth terms ................................... 70
3.3 Second order equations .................................... 72
3.3.1 Strong ellipticity ................................. 72
3.3.2 Problem with the condition ƒ(x,u) = xg(x,u) ........ 97
3.3.3 Problem of population dynamics theory ............. 103
3.3.4 Generalization to mixed boundary conditions and
dependence on ε ................................... 105
3.4 Equation with a power function affecting the second
derivative ............................................... 106
3.4.1 Power singularities ............................... 106
3.4.2 Exponential singularity ........................... 110
3.5 Generalization to elliptic and parabolic equations ....... 111
3.5.1. Estimates of the solution derivatives ............. 112
3.6 Comments ................................................. 113
Chapter 4. Problems for ordinary quasilinear equations
4.1 Introduction ............................................. 115
4.2 Autonomous boundary value problem ........................ 115
4.2.1 Preliminary results ............................... 115
4.2.2 Boundary layers ................................... 117
4.2.3 Interior layers ................................... 119
4.3 Nonautonomous Equation ................................... 127
4.3.1 Estimates of the first derivative ................. 127
4.3.2 Graphical chart for localizing the layers ......... 142
4.3.3 Example of the problem ............................ 143
4.4 Analysis of the limit solution ........................... 144
4.4.1 Properties of the limit solution .................. 145
4.5 Comments ................................................. 148
Chapter 5. Systems of ordinary differential equations
5.1 Introduction ............................................. 149
5.2 Equations of the first order ............................. 149
5.2.1 Initial-value problem without turning points ...... 149
5.2.2 Equation with a turning point ..................... 152
5.3 Semilinear equations of the second order ................. 154
5.3.1 Estimates of the derivatives ...................... 156
5.3.2 Estimates of the first derivative ................. 158
5.3.3 Estimates of the higher derivatives ............... 169
5.4 Derivative estimates and location of the shock layer ..... 180
5.5 Comments ............................................ 182
Chapter 6. Generation of transformations and layer-resolving
grids
6.1 Introduction ............................................. 183
6.2 Implicit generation of layer-damping transformations ..... 185
6.2.1 One-dimensional case .............................. 185
6.2.2 Extension to multidimensions ...................... 191
6.3 Explicit generation of layer-damping transformations ..... 193
6.3.1 Basic majorants ................................... 193
6.3.2 Generation of univariate local stretching
transformations ................................... 197
6.3.3 Basic layer-damping transformations ............... 202
6.3.4 Construction of basic intermediate
transformations ................................... 211
6.4 Algebraic method for generating coordinate
transformations .......................................... 214
6.4.1 Lagrange interpolation ............................ 218
6.4.2 Hermite interpolation ............................. 220
6.4.3 Control of grid resolution ........................ 222
6.5 Comments ................................................. 224
Chapter 7. Analysis of numerical algorithms
7.1 Introduction ............................................. 227
7.2 Relations between the solution and truncation errors ..... 227
7.2.1 Initial problem ................................... 228
7.2.2 Two-point boundary value problem .................. 230
7.3 Uniform convergence for initial-value problems ........... 231
7.3.1 Equation with a turning point ..................... 231
7.3.2 Equation without a turning point .................. 233
7.4 Uniform convergence for two-point boundary value
problems ................................................. 233
7.4.1 Equation without a boundary turning point ......... 234
7.4.2 Boundary turning points ........................... 239
7.4.3 Interior turning points ........................... 243
7.4.4 General case ...................................... 244
7.4.5 Problem with the condition ƒ(x, u) = xg(x, u) ..... 246
7.4.6 Uniform convergence of the numerical friction ..... 254
7.5 Problem with a variable coefficient before the second
derivative ............................................... 256
7.5.1 Case 0 < α(0) < 1 .................................. 258
7.5.2 Case α(0) > 1 ..................................... 259
7.5.3 Case α(0) = 1 ..................................... 262
7.5.4 Algorithm for generating layer-resolving grids
263
7.5.5 Generalization to systems ......................... 263
7.5.6 Generalization to elliptic and parabolic
equations ......................................... 264
7.5.7 Uniform convergence at the points of a uniform
grid .............................................. 266
7.6 Comments ................................................. 268
References .................................................... 270
Index ......................................................... 283
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