Introduction to the general theory of singular perturbations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Providence, RI
American Math. Soc.
1992
|
| Series: | Translations of mathematical monographs
112 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XVIII, 375 S. |
| ISBN: | 0821845691 |
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| 240 | 1 | 0 | |a Vvedenie v obščuju teoriju singuljarnych vozmušzcenij |
| 245 | 1 | 0 | |a Introduction to the general theory of singular perturbations |c S. A. Lomov |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1992 | |
| 300 | |a XVIII, 375 S. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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| adam_text | Contents
Preface to the English Edition xi
Preface xv
Author s Preface xvii
CHAPTER 1. Introduction. General Survey 1
§1. On perturbations 7
§2. The basic idea of classical perturbation theory 9
§3. Singular perturbations 12
§4. Basic concepts. Terminology 14
1. O symbols (14).
2. Asymptotic series (14).
§5. The Schlesinger Birkhoff theorem 17
§6. The Schlesinger Birkhoff theorem and asymptotic integration 19
§7. Further development of the theory of singular perturbations 20
§8. Comparison of two types of asymptotic expansions 22
§9. Some notation and auxiliary concepts 23
Part I. Asymptotic Integration of Various Problems for Ordinary
Differential Equations
CHAPTER 2. The Method of Regularization of Singular Perturbations 27
§1. The formalism of the regularization method 27
1. Formulation of the problem (27).
2. Regularization of singularities (28).
3. Formal construction of a series for the solution (30).
§2. The space of resonance free solutions 30
1. The structure of the space (30).
2. Properties of the basic operator in the space of resonance free
solutions (32).
§3. The theory of resonance free solutions 32
1. The adjoint operator (32).
2. Normal solvability of the basic operator (33).
3. Uniqueness of the solution (34).
v
vi CONTENTS
§4. Formal regularized series 36
1. Determination of the coefficients of the series of perturbation
theory (36).
2. Uniqueness and other properties of the regularized series (38).
§5. Estimation of the remainder term of the asymptotic series for
the fundamental matrix 40
1. Formal construction of the fundamental matrix (40).
2. The asymptotic character of the series (42).
§6. Estimation of the remainder term of the asymptotic series for
the solution of the Cauchy problem 48
1. Auxiliary notation and a lemma (48).
2. Estimation of the remainder term (50).
§7. Convergence of regularized series in the usual sense 58
1. Systems with a diagonal matrix of coefficients (58).
2. Examples (61).
3. Ordinary convergence of the asymptotic series (62).
4. Convergence in a finite dimensional Hilbert space (62).
5. An example (65).
§8. The method of regularization in the case of null points of the
spectrum 66
1. Formulation of the problem (66).
2. The formalism of the regularization method (67).
3. Construction of the adjoint operator in the space of resonance
free solutions (67).
4. Questions of solvability in the space of resonance free
solutions (68).
5. A limit theorem (72).
CHAPTER 3. Asymptotic Integration of a Boundary Value Problem 75
§1. Special features of boundary value problems 75
1. Characteristic features of boundary value problems (75).
2. Formulation of the problem (78).
3. Stability of the boundary value problem (79).
§2. Construction of an algorithm for asymptotic integration of a
boundary value problem for general systems 82
1. The formalism of the method of regularization for a boundary
value problem (82).
2. Solvability theorems in the space of resonance free solutions (84).
3. Solvability of the iteration problems (89).
4. Formal asymptotic solution of the original problem (92).
§3. Construction of the Green function 93
1. Reduction of the system to quasi diagonal form (93).
2. Construction of two fundamental matrices of special form (95).
3. Construction of a fundamental matrix of a singularly perturbed
system with special boundary conditions (97).
4. Construction of the matrix [PiX(0, e) + Qu(l, e)]~l (101).
CONTENTS vii
5. Construction of the matrix Green function (103).
6. A remark on the construction of the Green function for
a more general system (105).
§4. Estimation of the remainder term 107
1. The problem for the remainder term (107).
2. An estimate theorem (107).
CHAPTER 4. Asymptotic Integration of Linear Integro Differential
Equations 109
§1. Special features of the regularization of singularities in the
presence of integrals of the desired solutions in the oscillatory
case 109
1. Formulation of the problem in the simplest case (109).
2. Partial regularization of the problems (110).
§2. Complete regularization and asymptotic integration 111
1. Regularization and the formalism of the method (111).
2. Solvability of iteration problems (115).
3. Estimation of the remainder term (116).
§3. The Cauchy problem for integro differential systems 119
1. Formulation of the problem and regularization of
singularities (119).
2. Determination of the coefficients of the formal asymptotic
series (122).
3. Estimation of the remainder term (125).
4. An example (126).
§4. Integro differential systems of Fredholm type 128
1. Auxiliary propositions (128).
2. Formulation of the problem and regularization of the
operation of differentiation (130).
3. Regularization of the integral term and of the problem for
determining the elements of the asymptotic solution (131).
4. Solvability of the iteration problem (137).
5. Estimation of the remainder term (141).
CHAPTER 5. Some Problems with Rapidly Oscillating Coefficients 143
§ 1. Construction of the asymptotic series and conditions for the
solvability of the iteration problems 144
1. Formalism of the method (144).
2. The space of solutions (146).
3. The adjoint operator (146).
4. Construction of new recurrent problems (148).
5. Solvability theorems (152).
§2. Justification of asymptotic convergence 155
1. Estimation of the remainder term (155).
2. Remark (157).
viii CONTENTS
§3. Solution of the problem of parametric amplification 157
1. An example (157).
2. Solution of the auxiliary system (164).
CHAPTER 6. Problems with an Unstable Spectrum 167
§ 1. The only point of the spectrum has a zero of arbitrary order 167
1. On the problem in the simplest formulation (167).
2. Regularization of the problem (168).
3. Asymptotic integration (168).
4. Passage to the limit (170).
§2. One of the two points of the spectrum has a zero of first order 170
1. Special features of the problem (170).
2. Choice of regularizing functions and regularization (171).
3. Special features of solving the iteration problems (173).
4. The main theorem (176).
§3. The inhomogeneous problem with a turning point 176
1. Preliminary facts regarding the problem (176).
2. Formulation of the problem (178).
3. Regularization of the problem (180).
4. Special features of the asymptotic integration of problems
with turning points (182).
5. Solvability of the iteration problems (186).
6. Estimation of the remainder term (189).
7. Proof of Lemma 18 (191).
§4. The structure of the fundamental matrix of solutions of
singularly perturbed equations with a regular singular point 194
1. The fundamental system of solutions (196).
2. Obtaining formal solutions (196).
3. Asymptotic convergence of the series (200).
4. The fundamental system in the case of two algebraic
singularities (202).
CHAPTER 7. Singularly Perturbed Problems for Nonlinear Equations 209
§1. Weakly nonlinear singularly perturbed problems in the
resonance case 211
1. Formal solutions of weakly nonlinear problems (211).
2. Questions of solvability in the space of resonance free
solutions (216).
3. The asymptotic character of solutions (223).
4. Examples (225).
§2. Regularized asymptotic solutions of strongly nonlinear
singularly perturbed problems 228
1. Regularization of strongly nonlinear problems (229).
2. Some function classes and their properties (231).
3. Theorems on the solvability of the iteration problems (235).
4. The asymptotic character of formal solutions (249).
5. An example (254).
CONTENTS a
§3. Connection of the regularization method with the averaging
method 256
1. Regularized asymptotic solutions (257).
2. Asymptotic solutions obtained by the averaging method (259).
3. Global solvability of the truncated equations (264).
Part II. Singularly Perturbed Partial Differential Equations
CHAPTER 8. Asymptotic Integration of Linear Parabolic Equations 271
§1. A parabolic singularly perturbed problem with one viscous
boundary 274
1. Few words about the Fourier method (274).
2. Formulation of the problem and basic assumptions (275).
§2. The scheme of the regularization method in the selfadjoint case 276
1. Regularization and the iteration problems (276).
2. The space of resonance free solutions (278).
3. Solvability of the iteration problems (280).
4. Asymptotic convergence of the series (284).
§3. Connection with the Fourier method and boundary layer theory 285
1. Remarks (285).
2. Example (287).
3. Remarks on the adiabatic approximation in quantum
mechanics (289).
§4. Asymptotic integration of a parabolic equation with two
viscous boundaries 290
1. Formulation of the problem for the linearized one dimensional
Navier Stokes equation (290).
2. Regularization of singularities by viscosity (292).
3. The iteration problems. The space of resonance free solutions (294).
4. Theorems on normal and unique solvability (297).
5. Construction of the series of perturbation theory (299).
6. Estimation of the remainder term (304).
§5. Unsolved problems 306
1. Problems without spectrum (306).
2. Problems with two intersecting viscous boundaries (306).
3. Multidimensional problems (307).
CHAPTER 9. Application of the Regularization Method to Some
Elliptic Problems in a Cylindrical Domain 309
§1. Formalism of the method for an elliptic problem 309
1. Formulation of the problem (309).
2. Regularization and obtaining iteration problems (310).
§2. Asymptotic well posedness and convergence of the method 312
1. Unique solvability of the iteration problems (312).
2. A theorem on asymptotic convergence of the series (314).
3. The leading term of the asymptotics (315).
x CONTENTS
CHAPTER 10. Asymptotic Integration of Some Singularly Perturbed
Evolution Equations 317
§1. Asymptotic integration of singularly perturbed problems in
Hilbert space in the case of discrete spectrum of the operator 317
1. Formulation of the problem and regularization of singularities
by a parameter (317).
2. Construction of a formal asymptotic solution of the regularized
problem (321).
3. A theorem on estimation of the remainder term (325).
4. An example (327).
§2. Generalization of the regularization method to the case of
continuous spectrum of the limit operator 337
1. Formulation of the problem and basic conditions (338).
2. Regularization and the space of resonance free solutions Hz (339).
3. Uniqueness of the asymptotic series (340).
4. Example (344).
§3. An example of a problem with continuous spectrum and a
spectral measure depending on a parameter 347
1. Regularization of the problem and the space of solutions (347).
2. Construction of the regularized series (350).
3. Conclusion (361).
References 363
Supplementary References 371
Subject Index 373
|
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| indexdate | 2024-07-09T16:57:23Z |
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| language | English Russian |
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| physical | XVIII, 375 S. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
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| publisher | American Math. Soc. |
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| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Lomov, Sergej A. Verfasser aut Vvedenie v obščuju teoriju singuljarnych vozmušzcenij Introduction to the general theory of singular perturbations S. A. Lomov Providence, RI American Math. Soc. 1992 XVIII, 375 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 112 Aus d. Russ. übers. Perturbation (Mathématiques) Perturbation (Mathématiques) ram Singulariteiten gtt Storingsrekening gtt Perturbation (Mathematics) Singuläre Störung (DE-588)4055100-3 gnd rswk-swf Singuläre Störung (DE-588)4055100-3 s DE-604 Translations of mathematical monographs 112 (DE-604)BV000002394 112 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004601409&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lomov, Sergej A. Introduction to the general theory of singular perturbations Translations of mathematical monographs Perturbation (Mathématiques) Perturbation (Mathématiques) ram Singulariteiten gtt Storingsrekening gtt Perturbation (Mathematics) Singuläre Störung (DE-588)4055100-3 gnd |
| subject_GND | (DE-588)4055100-3 |
| title | Introduction to the general theory of singular perturbations |
| title_alt | Vvedenie v obščuju teoriju singuljarnych vozmušzcenij |
| title_auth | Introduction to the general theory of singular perturbations |
| title_exact_search | Introduction to the general theory of singular perturbations |
| title_full | Introduction to the general theory of singular perturbations S. A. Lomov |
| title_fullStr | Introduction to the general theory of singular perturbations S. A. Lomov |
| title_full_unstemmed | Introduction to the general theory of singular perturbations S. A. Lomov |
| title_short | Introduction to the general theory of singular perturbations |
| title_sort | introduction to the general theory of singular perturbations |
| topic | Perturbation (Mathématiques) Perturbation (Mathématiques) ram Singulariteiten gtt Storingsrekening gtt Perturbation (Mathematics) Singuläre Störung (DE-588)4055100-3 gnd |
| topic_facet | Perturbation (Mathématiques) Singulariteiten Storingsrekening Perturbation (Mathematics) Singuläre Störung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004601409&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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