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Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems [Kõva köide]

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  • Formaat: Hardback, 373 pages, kaal: 833 g
  • Sari: Graduate Studies in Mathematics
  • Ilmumisaeg: 30-Jan-2012
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821846949
  • ISBN-13: 9780821846940
Teised raamatud teemal:
Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value ProblemsSuurem pilt
  • Formaat: Hardback, 373 pages, kaal: 833 g
  • Sari: Graduate Studies in Mathematics
  • Ilmumisaeg: 30-Jan-2012
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821846949
  • ISBN-13: 9780821846940
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780821846940.html
Märksõnad:
This text emphasises rigourous mathematical techniques for the analysis of boundary value problems for ODEs arising in applications. The emphasis is on proving existence of solutions, but there is also a substantial chapter on uniqueness and multiplicity questions and several chapters which deal with the asymptotic behaviour of solutions with respect to either the independent variable or some parameter. These equations may give special solutions of important PDEs, such as steady state or travelling wave solutions. Often two, or even three, approaches to the same problem are described. The advantages and disadvantages of different methods are discussed.

The book gives complete classical proofs, while also emphasising the importance of modern methods, especially when extensions to infinite dimensional settings are needed. There are some new results as well as new and improved proofs of known theorems. The final chapter presents three unsolved problems which have received much attention over the years.

Both graduate students and more experienced researchers will be interested in the power of classical methods for problems which have also been studied with more abstract techniques. The presentation should be more accessible to mathematically inclined researchers from other areas of science and engineering than most graduate texts in mathematics.
Preface xiii
Chapter 1 Introduction
1(6)
§1.1 What are classical methods?
1(4)
§1.2 Exercises
5(2)
Chapter 2 An introduction to shooting methods
7(30)
§2.1 Introduction
7(1)
§2.2 A first order example
8(5)
§2.3 Some second order examples
13(4)
§2.4 Heteroclinic orbits and the FitzHugh-Nagumo equations
17(10)
§2.5 Shooting when there are oscillations: A third order problem
27(3)
§2.6 Boundedness on (--∞, ∞) and two-parameter shooting
30(3)
§2.7 Wazewski's principle, Conley index, and an n-dimensional lemma
33(1)
§2.8 Exercises
34(3)
Chapter 3 Some boundary value problems for the Painleve transcendents
37(18)
§3.1 Introduction
37(1)
§3.2 A boundary value problem for Painleve I
38(6)
§3.3 Painleve II---shooting from infinity
44(8)
§3.4 Some interesting consequences
52(1)
§3.5 Exercises
53(2)
Chapter 4 Periodic solutions of a higher order system
55(8)
§4.1 Introduction, Hopf bifurcation approach
55(2)
§4.2 A global approach via the Brouwer fixed point theorem
57(4)
§4.3 Subsequent developments
61(1)
§4.4 Exercises
62(1)
Chapter 5 A linear example
63(14)
§5.1 Statement of the problem and a basic lemma
63(2)
§5.2 Uniqueness
65(1)
§5.3 Existence using Schauder's fixed point theorem
66(3)
§5.4 Existence using a continuation method
69(4)
§5.5 Existence using linear algebra and finite dimensional continuation
73(3)
§5.6 A fourth proof
76(1)
§5.7 Exercises
76(1)
Chapter 6 Homoclinic orbits of the FitzHugh-Nagumo equations
77(26)
§6.1 Introduction
77(4)
§6.2 Existence of two bounded solutions
81(2)
§6.3 Existence of homoclinic orbits using geometric perturbation theory
83(9)
§6.4 Existence of homoclinic orbits by shooting
92(7)
§6.5 Advantages of the two methods
99(2)
§6.6 Exercises
101(2)
Chapter 7 Singular perturbation problems---rigorous matching
103(38)
§7.1 Introduction to the method of matched asymptotic expansions
103(6)
§7.2 A problem of Kaplun and Lagerstrom
109(7)
§7.3 A geometric approach
116(4)
§7.4 A classical approach
120(6)
§7.5 The case n = 3
126(2)
§7.6 The case n = 2
128(3)
§7.7 A second application of the method
131(6)
§7.8 A brief discussion of blow-up in two dimensions
137(2)
§7.9 Exercises
139(2)
Chapter 8 Asymptotics beyond all orders
141(10)
§8.1 Introduction
141(3)
§8.2 Proof of nonexistence
144(6)
§8.3 Exercises
150(1)
Chapter 9 Some solutions of the Falkner-Skan equation
151(12)
§9.1 Introduction
151(2)
§9.2 Periodic solutions
153(5)
§9.3 Further periodic and other oscillatory solutions
158(2)
§9.4 Exercises
160(3)
Chapter 10 Poiseuille flow: Perturbation and decay
163(14)
§10.1 Introduction
163(1)
§10.2 Solutions for small data
164(2)
§10.3 Some details
166(3)
§10.4 A classical eigenvalue approach
169(2)
§10.5 On the spectrum of Dε, Rε for large R
171(5)
§10.6 Exercises
176(1)
Chapter 11 Bending of a tapered rod; variational methods and shooting
177(22)
§11.1 Introduction
177(3)
§11.2 A calculus of variations approach in Hilbert space
180(7)
§11.3 Existence by shooting for p > 2
187(8)
§11.4 Proof using Nehari's method
195(2)
§11.5 More about the case p = 2
197(1)
§11.6 Exercises
198(1)
Chapter 12 Uniqueness and multiplicity
199(26)
§12.1 Introduction
199(4)
§12.2 Uniqueness for a third order problem
203(2)
§12.3 A problem with exactly two solutions
205(5)
§12.4 A problem with exactly three solutions
210(7)
§12.5 The Gelfand and perturbed Gelfand equations in three dimensions
217(2)
§12.6 Uniqueness of the ground state for Δu - u + u3 = 0
219(4)
§12.7 Exercises
223(2)
Chapter 13 Shooting with more parameters
225(12)
§13.1 A problem from the theory of compressible flow
225(6)
§13.2 A result of Y.-H. Wan
231(1)
§13.3 Exercise
232(1)
§13.4 Appendix: Proof of Wan's theorem
232(5)
Chapter 14 Some problems of A. C. Lazer
237(20)
§14.1 Introduction
237(2)
§14.2 First Lazer-Leach problem
239(9)
§14.3 The pde result of Landesman and Lazer
248(2)
§14.4 Second Lazer-Leach problem
250(2)
§14.5 Second Landesman-Lazer problem
252(3)
§14.6 A problem of Littlewood, and the Moser twist technique
255(1)
§14.7 Exercises
256(1)
Chapter 15 Chaotic motion of a pendulum
257(32)
§15.1 Introduction
257(1)
§15.2 Dynamical systems
258(7)
§15.3 Melnikov's method
265(6)
§15.4 Application to a forced pendulum
271(3)
§15.5 Proof of Theorem 15.3 when δ = 0
274(3)
§15.6 Damped pendulum with nonperiodic forcing
277(7)
§15.7 Final remarks
284(2)
§15.8 Exercises
286(3)
Chapter 16 Layers and spikes in reaction-diffusion equations, I
289(12)
§16.1 Introduction
289(2)
§16.2 A model of shallow water sloshing
291(2)
§16.3 Proofs
293(4)
§16.4 Complicated solutions ("chaos")
297(2)
§16.5 Other approaches
299(1)
§16.6 Exercises
300(1)
Chapter 17 Uniform expansions for a class of second order problems
301(14)
§17.1 Introduction
301(1)
§17.2 Motivation
302(2)
§17.3 Asymptotic expansion
304(9)
§17.4 Exercise
313(2)
Chapter 18 Layers and spikes in reaction-diffusion equations, II
315(30)
§18.1 A basic existence result
316(1)
§18.2 Variational approach to layers
317(1)
§18.3 Three different existence proofs for a single layer in a simple case
318(9)
§18.4 Uniqueness and stability of a single layer
327(5)
§18.5 Further stable and unstable solutions, including multiple layers
332(8)
§18.6 Single and multiple spikes
340(2)
§18.7 A different type of result for the layer model
342(1)
§18.8 Exercises
343(2)
Chapter 19 Three unsolved problems
345(12)
§19.1 Homoclinic orbit for the equation of a suspension bridge
345(1)
§19.2 The nonlinear Schrodinger equation
346(1)
§19.3 Uniqueness of radial solutions for an elliptic problem
346(1)
§19.4 Comments on the suspension bridge problem
346(1)
§19.5 Comments on the nonlinear Schrodinger equation
347(2)
§19.6 Comments on the elliptic problem and a new existence proof
349(6)
§19.7 Exercises
355(2)
Bibliography 357(14)
Index 371
Stuart P. Hastings, University of Pittsburgh, PA, USA.

||J. Bryce McLeod, Oxford University, England, and University of Pittsburgh, PA, USA.