Introduction to partial differential equations and Hilbert space methods

목차 (Table of Contents)

• CONTENTS
• 1 THE USUAL TRINITIES : The basics, and wherein the problems lie = 1
• 1.1 The Usual Three Operators and Classes of Equations = 5
• The Potential Operator.
• The-Diffusion Operator.
• The Wave Operator.
• 1.2 The Usual Three Types of Problems = 9
• Boundary Value Problems.
• Initial Value Problems.
• Eigenvalue
• Problems.
• 1.3 The Usual Three Questions (and the Other Three) = 14
• Existence.
• Uniqueness.
• Stability.
• Construction.
• Regularity.
• Approximation.
• 1.4 The Usual Three Types of "Boundary Conditions" = 18
• Dirichlet boundary condition.
• Neumann boundary condition.
• Robin boundary condition.
• 1.5 The Usual Three Solution Methods (with Historical Remarks) = 22
• 1.5.1. Separation of Variables = 22
• 1.5.2. Green's Function Method = 28
• 1.5.3. Variational Methods = 34
• FIRST PAUSE : EXAMPLES, EXPLANATIONS, EXERCISES = 38
• Separation of Variables with General (e.g., physical) Lengths.
• Green's Functions by Variation of Parameters.
• Variational Formulation of Higher-Order (e.g., clamped rod) Problems.
• 1.6 Three Important Mathematical Tools = 45
• 1.6.1. Divergence Theorem Fundamental Theorem of Calculus. Green's Three Identities. Remarks on Dimension, the General Dirichlet-Poisson Problem, and Mean Value Theorems. = 46
• 1.6.2. Inequalities = 58
• 1. Schwarz's Inequality. The Arithmetic-Geometric Mean Inequality. Stability and Approximation for the Dirichlet-Poisson Problem.
• 2. The Triangle Inequality. Inverse Triangle Inequality. Holder Inequality.
• 3. The Sobolev Inequality. Why the Atom Does Not Collapse. Proofs of the inequalities.
• 1.6.3. Convergence Theorems : Pointwise Convergence. Uniform Convergence. Mean Square (L2) Convergence. Nature of the Limit Function. Implications Between Types of Convergence. Differentiation Term by Term. Differentiation under the Integral. Integration Term by Term. = 69
• SECOND PAUSE : EXAMPLES, EXPLANATIONS, EXERCISES = 77
• Interplay of Divergence Theorem, Inequalities, and Variational Methods.
• Clamped Plate Example.
• Interplay of Inequalities and Fourier Methods.
• Riemann-Lebesgue Lemma.
• Isoperimetric Inequality.
• 1.7 Some Physical Considerations (and Examples) = 83
• 1.7.1. Three Physical Techniques : Conservation Principles. Linearization Assumptions-Perturbation Methods. Derivation of Nonlinear and Linear Heat Equations. = 83
• 1.7.2. Three Physical Setting's (and Examples) : Continuum Mechanics. Statistical Mechanics. Quantum Mechanics. Examples. = 88
• 1.7.3. Some Unsolved Problems = 93
• 1.8 Elements of Bifurcation Theory The Bifurcation Diagram. Three Physical Settings: (a) Buckling of a (Yard, Meter) Stick; (b) A Rotating String ; (c) Chemical Kinetics. How Does Turbulence Occur? : = 94
• 1.9 Supplementary Discussions and Problems = 103
• 1.9.1. Classification = 104
• 1.9.2. Characteristics = 110
• 1.9.3. Maximum Principles = 112
• 1.9.4 Picard-Cauchy-Kowalewski Theorem = 113
• 1.9.5. (1) Some Historical Considerations = 116
• (2) Nice and Nonnice Domains = 116
• (3) Variational Principles and Euler Equations = 117
• 1.9.6. (1) Some Historical Remarks = 120
• (2) Dini Tests and the Gibbs-Effect = 121
• (3) Lebesgue Dominated Convergence Theorem = 125
• 1.9.7. (1) Self-Adjoint Operators = 126
• (2) Distributional Derivatives = 127
• (3) Heat Equation Rigorously = 128
• 1.9.8. Nonlinear Oscillations and the Van der Pol Equation = 131
• 1.9.9. Confirmation Exercises = 134
• 2 FOURIER SERIES AND HILBERT SPACE : One and the same = l4l
• 2.1 Lots of Separation of Variables = 143
• Vibrating String Problem.
• Dirichlet Problems.
• Eigenvalue Problem.
• Poisson Problem.
• Heat Conduction Problem.
• Vibrating Membrane Problems.
• Higher Dimensional Problems.
• Rectangular, Spherical, and Other Geometries.
• Some Special Functions, and Special Equations : Bessel's, Legendre's, Jacobi's, Tchebycheff's, Hermite's, Harmonic Oscillator, Laguerre's, Whittaker's, Banner's.
• 2.2 Mathematical Justifications of the Method = 158
• Uniform Convergence and Term by Term Differentiation.
• Via Maximum Principle.
• Via Overregularity.
• 2.3 Fourier Series and Hilbert Space = 167
• Notions and Examples.
• Maximal Orthonormal Sets.
• The Main Theorem H. Proof of the Theorem.
• Bessel's Inequality.
• 2.4 Fourier Series and Sturm-LiouviIle Equations = 174
• The Hilbert spaces $$L^2$$(a, b, r). Regular Sturm-LiouviIle Equations and their Eigenfunctions as Maximal Orthonormal Sets.
• Other Methods for Showing Maximality. o
• 2.5 Fourier Series and Green*s Functions = 182
• D'Alembert's Formula for the Wave Equation.
• Poisson's Formula for the Dirichlet Problem. Eigenfunction Expansions of Green's Functions.
• THIRD PAUSE : EXAMPLES, EXPLANATIONS, EXERCISES = 191
• Fourier Series and Best Least Squares Fit.
• ParsevaPs Equation.
• Special, Functions and Mathematical Physics.
• Harmonic Oscillator and Hermite Polynomials.
• Haar Orthonormal Bases.
• 2.6 Fourier Series and Variational and Numerical Methods = 194
• Finite Difference Method.
• Finite Element Method.
• Dirichlet Variational Principle.
• Method of Best Least Squares Approximation.
• Galerkin Method.
• Convergence, Consistency, and Stability.
• Rayleigh-Ritz Method.
• Trefftz Method.
• FOURTH PAUSE : EXAMPLES, EXPLANATIONS, EXERCISES = 210
• Further Numerical Examples.
• Mesh Refinement and Accuracy.
• Fortran Code Examples.
• Examples of Sobolev Lemma.
• Weak Solutions.
• 2.7 Some Unbounded Domain Considerations (and Continuous Spectra) = 217 217
• 2.7.1. Recapitulation and Initial Observations : Exterior Domains. Whole-space Domains. Improper Eigenfunctions. = 219
• 2.7.2. Continuous Spectrum Spectrum. : Point Spectrum. Continuous Spectrum. Hydrogen Equation. Helmhottz Equation. = 222
• 2.7.3. Fourier Transform : F and $$F^-1$$. Parseval's Theorem. Convolution Method. = 226
• 2.8 Elements of Scattering Theory : Three Physical Settings: (a) Classical Scattering; (b) Quantum Scattering ; (c) Inverse Scattering. Rayleigh Scattering off a Small Sphere. Why the Sky Is Blue. = 230
• 2.9 Supplementary Discussions and Problems = 238
• 2.9.1. Separation of Variables and Tensor Products = 238
• 2.9.2. Fourier Series Convergence and Closed Operators = 240
• 2.9.3. Separability and Test Functions = 242
• 2.9.4. Limit-Point and Limit-Circle = 246
• 2.9.5. Green's Functions and Continuous Operators = 249
• 2.9.6. Additional Numerical Considerations = 250
• (1) Finite Element Convergence = 251
• (2) Marching Methods for Initial Value Problems = 252
• (3) The Buckley-Leverett Equation in Secondary. Oil Recovery = 253
• 2.9.7. Additional Analytical Considerations = 255
• (1) Two-Dimensional Domains and Riemann Mapping Theorem = 255
• (2) Resolvent .Operator and Resolvent Equation = 258
• (3) Kirchoff's Formula and Huygen's Principle = 260
• 2.9.8. Nonlinear Waves and Solitons = 264
• Hyperbolic Equations.
• Dispersive Equations.
• Dissipative Equations.
• Traffic Flow Equations.
• Burger's Equation.
• Bom-Infeld Equation.
• K-orteweg-deVries Equations.
• Sine-Gordon Equation.
• Fermi-Pasta-Ulam Equation.
• Hodgkin-Huxley Equations.
• Nagumo Equations.
• Concentration-Diffusion, Reaction-Diffusion, Porous-media Equations.
• 2.9.9. Confirmation Exercises = 271
• 3 APPENDICES : Past, present, future = 277
• A. First-Order Equations = 278
• A.1 PDE to ODE : Geometrical Considerations. Characteristic Curves and Surfaces. Tangent Conditions and Tangent Planes. = 278
• A.2 Self-Similar Solutions : Change of Variables. Dimensional Analysis. Scaling Methods. Prandtl Boundary Layer Equations. Atomic Bomb Blasts. = 287
• A.3 Local Transformation Groups : Exact Equations. Invariance Under Infinitesimal Changes of Variable. Characteristic Equation. Transonic Plow Equation. Prolongations. Symmetry Generators. Conservation Laws. = 294
• B. Computational Methods = 302
• B.1 Finite Difference Methods : Notion of Stencil. Forward Euler Scheme. Fourier (Von Neumann) Analysis of Stability. Amplification Matrix Analysis of Stability. Crank-Nicolson Scheme. Leapfrog Scheme. = 303
• B.2 Finite Element Methods : Example Comparison with Finite Difference and Finite Spectral Methods. Example of Quadratic Element Sparse Basis Matrix-Galerkin Problems. = 309
• B.3 Finite Spectral Methods : Connections to Fourier and Galerkin Methods. Collocation Method. Tau (Lanczo's) Method. Weighted Residual Methods. Linear Algebraic Considerations. Graphics Outputs of Numerical Solutions of Previous Problems in the Book. = 310
• C. Advanced Fluid Dynamics = 319
• C.1 Navier-Stokes Equations : Momentum Equation. Continuity Equation. Importance of Dimension. Pressure Equation. Helmholtz Decomposition of a Vector Field. Weak Formulation. Stokes Problem. = 320
• C.2 Turbulence and Attractors : Kolmogorov Postulate. Dimension of Turbulence. Final Attracting Sets. Taylor Cylinders Model. Hopf Bifurcation. Chaotic State Models. = 323
• C.3 Computational Fluid Dynamics : Primitive Variables. Stream Function-Vorticity Variables. Rayleigh Cavity Model. Infinite Sequence of Similarity Corner Vortices. Dynamics of the Cavity Flow. Wall bursting. Separation. Vortex Fission-Fusion. Parity Rules. = 325
• Selected Answers, Hints, and Solutions = 333
• Author Index = 399
• Subject Index = 403