This text emphasizes rigorous 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 behavior 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 traveling 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 emphasizing 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.
Readership: Graduate students and research mathematicians interested in ODEs and PDEs.
Author(s): Stuart P. Hastings, J. Bryce Mcleod
Series: Graduate Studies in Mathematics 129
Publisher: American Mathematical Society
Year: 2012
Language: English
Pages: xviii+373
Preface
Chapter 1 Introduction
1.1. What are classical methods?
1.2. Exercises
Chapter 2 An introduction to shooting methods
2.1. Introduction
2.2. A first order example
2.2.1. An alternative formulation of shooting.
2.2.2. A problem on [0, oo).
2.3. Some second order examples
2.3.1. A linear problem.
2.3.2. A nonlinear problem
2.3.3. Airy's equation on [0, oo).
2.4. Heteroclinic orbits and the FitzHugh-Nagumo equations
2.4.1. Heteroclinic orbits.
2.4.2. 1ave1ing waves and the FitzHugh-Nagumo equations
2.4.3. Summary of the results.
2.4.4. Results of Fife and McLeod and of Xinfu Chen
2.5. Shooting when there are oscillations: A third order problem
2.5.1. Existence.
2.6. Boundedness on (-oo, oo) and two-parameter shooting
2.7. Wazewski's principle, Conley index, and an n-dimensional lemma
2.8. Exercises
Chapter 3 Some boundary value problems for the Painleve transcendents
3.1. Introduction
3.2. A boundary value problem for Painleve I
3.2.1. Shooting appears not to work.
3.2.2. An alternative approach.
3.2.3. Proof using asymptotic analysis
3.3. Painleve II shooting from infinity
3.3.1. Introduction and existence proof.
3.4. Some interesting consequences
3.5. Exercises
Chapter 4 Periodic solutions of a higher order system
4.1. Introduction, Hopf bifurcation approach
4.2. A global approach via the Brouwer fixed point theorem
4.3. Subsequent developments
4.4. Exercises
Chapter 5 A linear example
5.1. Statement of the problem and a basic lemma
5.2. Uniqueness
5.3. Existence using Schauder's fixed point theorem
5.4. Existence using a continuation method
5.5. Existence using linear algebra and finite dimensional continuation
5.6. A fourth proof
5.7. Exercises
Chapter 6 Homoclinic orbits of the FitzHugh-Nagumo equations
6.1. Introduction
6.1.1. Preliminary results.
6.2. Existence of two bounded solutions
6.3. Existence of homoclinic orbits using geometric perturbation theory
6.3.1. The singular solution.
6.3.2. The first transverse intersection
6.3.3. Example of the exchange lemma
6.3.4. Completion of the proof (outline).
6.4. Existence of homoclinic orbits by shooting
6.4.1. Existence of the fast wave.
6.4.2. Existence of the slow wave.
6.5. Advantages of the two methods
6.6. Exercises
Chapter 7 Singular perturbation problems-rigorous matching
7.1. Introduction to the method of matched asymptotic expansions
7.2. A problem of Kaplun and Lagerstrom
7.2.1. The case n = 3.
7.2.2. The case n = 2.
7.3. A geometric approach
7.4. A classical approach
7.4.1. Existence and uniqueness
7.4.2. Asymptotic expansion
7.5. The case n = 3
7.6. The case n = 2
7.7. A second application of the method
7.7.1. Introduction.
7.8. A brief discussion of blow-up in two dimensions
7.9. Exercises
Chapter 8 Asymptotics beyond all orders
8.1. Introduction
8.2. Proof of nonexistence
8.3. Exercises
Chapter 9 Some solutions of the Falkner-Skan equation
9.1. Introduction
9.2. Periodic solutions
9.3. Further periodic and other oscillatory solutions
9.4. Exercises
Chapter 10 Poiseuille flow: Perturbation and decay
10.1. Introduction
10.2. Solutions for small data
10.2.1. Orr-Sommerfeld operators.
10.3. Some details
10.3.1. Relevant Sobolev spaces.
10.3.2. Application to the operator D
10.4. A classical eigenvalue approach
10.5. On the spectrum of for D large R
10.6. Exercises
Chapter 11 Bending of a tapered rod; variational methods and shooting
11.1. Introduction
11.2. A calculus of variations approach in Hilbert space
11.2.1. Results for p = 2.
11.2.2. Proof of (b) assuming (a).
11.2.3. Remarks on the proof of (a).
11.2.3.1. Minimax principles
11.3. Existence by shooting for p> 2
11.4. Proof using Nehari's method
11.5. More about the case p = 2
11.6. Exercises
Chapter 12 Uniqueness and multiplicity
12.1. Introduction
12.1.1. An application of contraction mapping in a Banach space
12.2. Uniqueness for a third order problem
12.3. A problem with exactly two solutions
12.3.1. One-dimensional case; introduction of the time map
12.3.2. The one-dimensional Gelfand equation.
12.4. A problem with exactly three solutions
12.4.1. The perturbed Gelfand equation in one dimension
12.4.2. The two-dimensional case.
12.5. The Gelfand and perturbed Gelfand equations in three dimensions
12.5.1. Gelfand equation in three dimensions.
12.5.2. Perturbed Gelfand equation in three dimensions
12.6. Uniqueness of the ground state for \Delta u - u + u3 = 0
12.6.1. Coffman's uniqueness proof.
12.7. Exercises
Chapter 13 Shooting with more parameters
13.1. A problem from the theory of compressible flow
13.1.1. Existence of a solution
13.2. A result of Y.-H. Wan
13.3. Exercise
13.4. Appendix: Proof of Wan's theorem
Chapter 14 Some problems of A. C. Lazer
14.1. Introduction
14.2. First Lazer-Leach problem
14.2.1. Proof of Theorem 14.2 using the Schauder fixed point theorem.
14.2.2. Proof using winding number.
14.2.2.1. Proof of Lemma 14.3.
14.3. The pde result of Landesman and Lazer
14.4. Second Lazer-Leach problem
14.5. Second Landesman-Later problem
14.6. A problem of Littlewood, and the Moser twist technique
14.7. Exercises
Chapter 15 Chaotic motion of a pendulum
15.1. Introduction
15.2. Dynamical systems
15.2.1. Continuous and discrete dynamical systems
15.2.2. Poincare maps
15.2.3. Horseshoe maps
15.2.4. Finding horseshoes more generally
15.3. Melnikov's method
15.3.1. A forced Duffing equation.
15.4. Application to a forced pendulum
15.5. Proof of Theorem 15.3 when \delta = 0
15.6. Damped pendulum with nonperiodic forcing
15.6.1. Outline of proof.
15.6.2. Proofs of Lemmas 15.6 and 15.7.
15.7. Final remarks
15.8. Exercises
Chapter 16 Layers and spikes in reaction-diffusion equations, I
16.1. Introduction
16.2. A model of shallow water sloshing
16.3. Proofs
16.3.1. Proof of existence (Theorem 16.2).
16.3.2. Proof of asymptotic behavior (Theorem 16.4).
16.3.3. Proofs of uniqueness.
16.4. Complicated solutions ("chaos")
16.5. Other approaches
16.6. Exercises
Chapter 17 Uniform expansions for a class of second order problems
17.1. Introduction
17.2. Motivation
17.2.1. Carrier's problem
17.2.2. Shallow water sloshing.
17.3. Asymptotic expansion
17.4. Exercise
Chapter 18 Layers and spikes in reaction-diffusione quations, II
18.1. A basic existence result
18.2. Variational approach to layers
18.3. Three different existence proofs for a single layer in a simple case
18.3.1. Existence using subsolutions and supersolutions
18.3.2. Existence by a variational method.
18.3.3. Existence using shooting.
18.4. Uniqueness and stability of a single layer
18.4.1. Stability.
18.5. Further stable and unstable solutions, including multiple layers
18.5.1. Orientation of layers
18.6. Single and multiple spikes
18.6.1. Combining layers and spikes.
18.7. A different type of result for the layer model
18.8. Exercises
Chapter 19 Three unsolved problems
Statements of Problems
19.1. Homoclinic orbit for the equation of a suspension bridge
19.2. The nonlinear Schrodinger equation
19.3. Uniqueness of radial solutions for an elliptic problem
References and some background
19.4. Comments on the suspension bridge problem
19.5. Comments on the nonlinear Schrodinger equation
19.6. Comments on the elliptic problem and a new existence proof
19.6.1. Existence and uniqueness of solutions.
19.6.2. Extensions
19.6.3. Existence of bound states for (19.4).
19.7. Exercises
Bibliography
Index
