Classical methods in ordinary differential equations

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