## Layer Resolving Grids and Transformations for Singular Perturbation ProblemsThe 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. |

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### Contents

Introduction to singularly perturbed problems | 1 |

Background for qualitative analysis | 37 |

Estimates of the solution derivatives to semilinear problems | 65 |

Problems for ordinary quasilinear equations | 115 |

Systems of ordinary differential equations | 149 |

### Other editions - View all

Layer Resolving Grids and Transformations for Singular Perturbation Problems V. D. Liseĭkin No preview available - 2001 |

### Common terms and phrases

algorithm Analogously applied approach approximation arbitrary assume asymptotic expansions barrier function blending functions boundary and interior boundary layer boundary point boundary turning point boundary value problem Chap coefficients considered contraction functions defined differential inequalities domain e-uniform convergence e-uniformly bounded example exponential function following estimate formula function u(x,e grid clustering Hermite interpolations inequality interior layers interior turning points interval 0,1 inverse monotonicity layer-damping coordinate transformations layer-damping transformations layer-resolving grids layer-type layers of singularities Lemma Let u(x,e limit solution linear Liseikin mapping method nodes numerical solution operator ordinary differential equations point x problem L[u proved qualitative behavior respect satisfies the condition scalar singular functions singularly perturbed equations singularly perturbed problems small parameter smooth solution derivatives solution u(x,e stretching functions taking into account techniques Theorem tions transformation x(£,e transformed problem two-point boundary value ui(x uniform grid univariate valid variable vector-valued functions vicinity weight function

### References to this book

Hp-Finite Element Methods for Singular Perturbations, Issue 1796 Jens M. Melenk Limited preview - 2002 |