The Resource Basic Theory of Ordinary Differential Equations, by Po-Fang Hsieh, Yasutaka Sibuya, (electronic resource)

Basic Theory of Ordinary Differential Equations, by Po-Fang Hsieh, Yasutaka Sibuya, (electronic resource)

Label
Basic Theory of Ordinary Differential Equations
Title
Basic Theory of Ordinary Differential Equations
Statement of responsibility
by Po-Fang Hsieh, Yasutaka Sibuya
Creator
Contributor
Author
Author
Subject
Language
  • eng
  • eng
Summary
The authors' aim is to provide the reader with the very basic knowledge necessary to begin research on differential equations with professional ability. The selection of topics should provide the reader with methods and results that are applicable in a variety of different fields. The text is suitable for a one-year graduate course, as well as a reference book for research mathematicians. The book is divided into four parts. The first covers fundamental existence, uniqueness, smoothness with respect to data, and nonuniqueness. The second part describes the basic results concerning linear differential equations, the third deals with nonlinear equations. In the last part the authors write about the basic results concerning power series solutions. Each chapter begins with a brief discussion of its contents and history. The book has 114 illustrations and 206 exercises. Hints and comments for many problems are given
Member of
http://library.link/vocab/creatorName
Hsieh, Po-Fang
Dewey number
515
http://bibfra.me/vocab/relation/httpidlocgovvocabularyrelatorsaut
  • tgJgPP8sqnY
  • DbTiDFYdByM
Image bit depth
0
Language note
English
LC call number
QA299.6-433
Literary form
non fiction
Nature of contents
dictionaries
http://library.link/vocab/relatedWorkOrContributorName
Sibuya, Yasutaka.
Series statement
Universitext,
http://library.link/vocab/subjectName
  • Global analysis (Mathematics)
  • Analysis
Label
Basic Theory of Ordinary Differential Equations, by Po-Fang Hsieh, Yasutaka Sibuya, (electronic resource)
Instantiates
Publication
Note
"With 114 illustrations."
Antecedent source
mixed
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
cr
Color
not applicable
Content category
text
Content type code
txt
Contents
I. Fundamental Theorems of Ordinary Differential Equations -- I-1. Existence and uniqueness with the Lipschitz condition -- I-2. Existence without the Lipschitz condition -- I-3. Some global properties of solutions -- I-4. Analytic differential equations -- Exercises I -- II. Dependence on Data -- II-1. Continuity with respect to initial data and parameters -- II-2. Differentiability -- Exercises II -- III. Nonuniqueness -- III-l. Examples -- III-2. The Kneser theorem -- III-3. Solution curves on the boundary of R(A) -- III-4. Maximal and minimal solutions -- III-5. A comparison theorem -- III-6. Sufficient conditions for uniqueness -- Exercises III -- IV. General Theory of Linear Systems -- IV-1. Some basic results concerning matrices -- IV-2. Homogeneous systems of linear differential equations -- IV-3. Homogeneous systems with constant coefficients -- IV-4. Systems with periodic coefficients -- IV-5. Linear Hamiltonian systems with periodic coefficients -- IV-6. Nonhomogeneous equations -- IV-7. Higher-order scalar equations -- Exercises IV -- V. Singularities of the First Kind -- V-1. Formal solutions of an algebraic differential equation -- V-2. Convergence of formal solutions of a system of the first kind -- V-3. TheS-Ndecomposition of a matrix of infinite order -- V-4. TheS-Ndecomposition of a differential operator -- V-5. A normal form of a differential operator -- V-6. Calculation of the normal form of a differential operator -- V-7. Classification of singularities of homogeneous linear systems -- Exercises V -- VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order -- VI- 1. Zeros of solutions -- VI- 2. Sturm-Liouville problems -- VI- 3. Eigenvalue problems -- VI- 4. Eigenfunction expansions -- VI- 5. Jost solutions -- VI- 6. Scattering data -- VI- 7. Reflectionless potentials -- VI- 8. Construction of a potential for given data -- VI- 9. Differential equations satisfied by reflectionless potentials -- VI-10. Periodic potentials -- Exercises VI -- VII. Asymptotic Behavior of Solutions of Linear Systems -- VII-1. Liapounoff’s type numbers -- VII-2. Liapounoff’s type numbers of a homogeneous linear system -- VII-3. Calculation of Liapounoff’s type numbers of solutions -- VII-4. A diagonalization theorem -- VII-5. Systems with asymptotically constant coefficients -- VII-6. An application of the Floquet theorem -- Exercises VII -- VIII. Stability -- VIII- 1. Basic definitions -- VIII- 2. A sufficient condition for asymptotic stability -- VIII- 3. Stable manifolds -- VIII- 4. Analytic structure of stable manifolds -- VIII- 5. Two-dimensional linear systems with constant coefficients -- VIII- 6. Analytic systems in ?n -- VIII- 7. Perturbations of an improper node and a saddle point -- VIII- 8. Perturbations of a proper node -- VIII- 9. Perturbation of a spiral point -- VIII-10. Perturbation of a center -- Exercises VIII -- IX. Autonomous Systems -- IX-1. Limit-invariant sets -- IX-2. Liapounoff’s direct method -- IX-3. Orbital stability -- IX-4. The Poincaré-Bendixson theorem -- IX-5. Indices of Jordan curves -- Exercises IX -- X. The Second-Order Differential Equation $$\frac{{{d^2}x}}{{d{t^2}}} + h(x)\frac{{dx}}{{dt}} + g(x) = 0 $$ -- X-1. Two-point boundary-value problems -- X-2. Applications of the Liapounoff functions -- X-3. Existence and uniqueness of periodic orbits -- X-4. Multipliers of the periodic orbit of the van der Pol equation -- X-5. The van der Pol equation for a small ?> 0 -- X-6. The van der Pol equation for a large parameter -- X-7. A theorem due to M. Nagumo -- X-8. A singular perturbation problem -- Exercises X -- XI. Asymptotic Expansions -- XI-1. Asymptotic expansions in the sense of Poincaré -- XI-2. Gevrey asymptotics -- XI-3. Flat functions in the Gevrey asymptotics -- XI-4. Basic properties of Gevrey asymptotic expansions -- XI-5. Proof of Lemma XI-2-6 -- Exercises XI -- XII. Asymptotic Expansions in a Parameter -- XII-1. An existence theorem -- XII-2. Basic estimates -- XII-3. Proof of Theorem XII-1-2 -- XII-4. A block-diagonalization theorem -- XII-5. Gevrey asymptotic solutions in a parameter -- XII-6. Analytic simplification in a parameter -- Exercises XII -- XIII. Singularities of the Second Kind -- XIII-1. An existence theorem -- XIII-2. Basic estimates -- XIII-3. Proof of Theorem XIII-1-2 -- XIII-4. A block-diagonalization theorem -- XIII-5. Cyclic vectors (A lemma of P. Deligne) -- XIII-6. The Hukuhara-Turrittin theorem -- XIII-7. An n-th-order linear differential equation at a singular point of the second kind -- XIII-8. Gevrey property of asymptotic solutions at an irregular singular point -- Exercises XIII -- References
Dimensions
unknown
Edition
1st ed. 1999.
Extent
1 online resource (XI, 469 p.)
File format
multiple file formats
Form of item
online
Isbn
9781461215066
Level of compression
uncompressed
Media category
computer
Media type code
c
Other control number
10.1007/978-1-4612-1506-6
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
  • (CKB)3400000000089575
  • (SSID)ssj0001295984
  • (PQKBManifestationID)11679683
  • (PQKBTitleCode)TC0001295984
  • (PQKBWorkID)11346815
  • (PQKB)10720651
  • (DE-He213)978-1-4612-1506-6
  • (MiAaPQ)EBC3075122
  • (EXLCZ)993400000000089575
Label
Basic Theory of Ordinary Differential Equations, by Po-Fang Hsieh, Yasutaka Sibuya, (electronic resource)
Publication
Note
"With 114 illustrations."
Antecedent source
mixed
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
cr
Color
not applicable
Content category
text
Content type code
txt
Contents
I. Fundamental Theorems of Ordinary Differential Equations -- I-1. Existence and uniqueness with the Lipschitz condition -- I-2. Existence without the Lipschitz condition -- I-3. Some global properties of solutions -- I-4. Analytic differential equations -- Exercises I -- II. Dependence on Data -- II-1. Continuity with respect to initial data and parameters -- II-2. Differentiability -- Exercises II -- III. Nonuniqueness -- III-l. Examples -- III-2. The Kneser theorem -- III-3. Solution curves on the boundary of R(A) -- III-4. Maximal and minimal solutions -- III-5. A comparison theorem -- III-6. Sufficient conditions for uniqueness -- Exercises III -- IV. General Theory of Linear Systems -- IV-1. Some basic results concerning matrices -- IV-2. Homogeneous systems of linear differential equations -- IV-3. Homogeneous systems with constant coefficients -- IV-4. Systems with periodic coefficients -- IV-5. Linear Hamiltonian systems with periodic coefficients -- IV-6. Nonhomogeneous equations -- IV-7. Higher-order scalar equations -- Exercises IV -- V. Singularities of the First Kind -- V-1. Formal solutions of an algebraic differential equation -- V-2. Convergence of formal solutions of a system of the first kind -- V-3. TheS-Ndecomposition of a matrix of infinite order -- V-4. TheS-Ndecomposition of a differential operator -- V-5. A normal form of a differential operator -- V-6. Calculation of the normal form of a differential operator -- V-7. Classification of singularities of homogeneous linear systems -- Exercises V -- VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order -- VI- 1. Zeros of solutions -- VI- 2. Sturm-Liouville problems -- VI- 3. Eigenvalue problems -- VI- 4. Eigenfunction expansions -- VI- 5. Jost solutions -- VI- 6. Scattering data -- VI- 7. Reflectionless potentials -- VI- 8. Construction of a potential for given data -- VI- 9. Differential equations satisfied by reflectionless potentials -- VI-10. Periodic potentials -- Exercises VI -- VII. Asymptotic Behavior of Solutions of Linear Systems -- VII-1. Liapounoff’s type numbers -- VII-2. Liapounoff’s type numbers of a homogeneous linear system -- VII-3. Calculation of Liapounoff’s type numbers of solutions -- VII-4. A diagonalization theorem -- VII-5. Systems with asymptotically constant coefficients -- VII-6. An application of the Floquet theorem -- Exercises VII -- VIII. Stability -- VIII- 1. Basic definitions -- VIII- 2. A sufficient condition for asymptotic stability -- VIII- 3. Stable manifolds -- VIII- 4. Analytic structure of stable manifolds -- VIII- 5. Two-dimensional linear systems with constant coefficients -- VIII- 6. Analytic systems in ?n -- VIII- 7. Perturbations of an improper node and a saddle point -- VIII- 8. Perturbations of a proper node -- VIII- 9. Perturbation of a spiral point -- VIII-10. Perturbation of a center -- Exercises VIII -- IX. Autonomous Systems -- IX-1. Limit-invariant sets -- IX-2. Liapounoff’s direct method -- IX-3. Orbital stability -- IX-4. The Poincaré-Bendixson theorem -- IX-5. Indices of Jordan curves -- Exercises IX -- X. The Second-Order Differential Equation $$\frac{{{d^2}x}}{{d{t^2}}} + h(x)\frac{{dx}}{{dt}} + g(x) = 0 $$ -- X-1. Two-point boundary-value problems -- X-2. Applications of the Liapounoff functions -- X-3. Existence and uniqueness of periodic orbits -- X-4. Multipliers of the periodic orbit of the van der Pol equation -- X-5. The van der Pol equation for a small ?> 0 -- X-6. The van der Pol equation for a large parameter -- X-7. A theorem due to M. Nagumo -- X-8. A singular perturbation problem -- Exercises X -- XI. Asymptotic Expansions -- XI-1. Asymptotic expansions in the sense of Poincaré -- XI-2. Gevrey asymptotics -- XI-3. Flat functions in the Gevrey asymptotics -- XI-4. Basic properties of Gevrey asymptotic expansions -- XI-5. Proof of Lemma XI-2-6 -- Exercises XI -- XII. Asymptotic Expansions in a Parameter -- XII-1. An existence theorem -- XII-2. Basic estimates -- XII-3. Proof of Theorem XII-1-2 -- XII-4. A block-diagonalization theorem -- XII-5. Gevrey asymptotic solutions in a parameter -- XII-6. Analytic simplification in a parameter -- Exercises XII -- XIII. Singularities of the Second Kind -- XIII-1. An existence theorem -- XIII-2. Basic estimates -- XIII-3. Proof of Theorem XIII-1-2 -- XIII-4. A block-diagonalization theorem -- XIII-5. Cyclic vectors (A lemma of P. Deligne) -- XIII-6. The Hukuhara-Turrittin theorem -- XIII-7. An n-th-order linear differential equation at a singular point of the second kind -- XIII-8. Gevrey property of asymptotic solutions at an irregular singular point -- Exercises XIII -- References
Dimensions
unknown
Edition
1st ed. 1999.
Extent
1 online resource (XI, 469 p.)
File format
multiple file formats
Form of item
online
Isbn
9781461215066
Level of compression
uncompressed
Media category
computer
Media type code
c
Other control number
10.1007/978-1-4612-1506-6
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
  • (CKB)3400000000089575
  • (SSID)ssj0001295984
  • (PQKBManifestationID)11679683
  • (PQKBTitleCode)TC0001295984
  • (PQKBWorkID)11346815
  • (PQKB)10720651
  • (DE-He213)978-1-4612-1506-6
  • (MiAaPQ)EBC3075122
  • (EXLCZ)993400000000089575

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