Partial differential equations : an introduction to theory and applications /
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| Author / Creator: | Shearer, Michael. |
|---|---|
| Imprint: | Princeton : Princeton University Press, [2015] |
| Description: | x, 274 pages ; 26 cm |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/10541873 |
Table of Contents:
- Preface
- 1. Introduction
- 1.1. Linear PDE
- 1.2. Solutions; Initial and Boundary Conditions
- 1.3. Nonlinear PDE
- 1.4. Beginning Examples with Explicit Wave-like Solutions
- Problems
- 2. Beginnings
- 2.1. Four Fundamental Issues in PDE Theory
- 2.2. Classification of Second-Order PDE
- 2.3. Initial Value Problems and the Cauchy-Kovalevskaya Theorem
- 2.4. PDE from Balance Laws
- Problems
- 3. First-Order PDE
- 3.1. The Method of Characteristics for Initial Value Problems
- 3.2. The Method of Characteristics for Cauchy Problems in Two Variables
- 3.3. The Method of Characteristics in R n
- 3.4. Scalar Conservation Laws and the Formation of Shocks
- Problems
- 4. The Wave Equation
- 4.1. The Wave Equation in Elasticity
- 4.2. D'Alembert's Solution
- 4.3. The Energy E(t) and Uniqueness of Solutions
- 4.4. Duhamel's Principle for the Inhomogeneous Wave Equation
- 4.5. The Wave Equation on R 2 and M 3
- Problems
- 5. The Heat Equation
- 5.1. The Fundamental Solution
- 5.2. The Cauchy Problem for the Heat Equation
- 5.3. The Energy Method
- 5.4. The Maximum Principle
- 5.5. Duhamel's Principle for the Inhomogeneous Heat Equation
- Problems
- 6. Separation of Variables and Fourier Series
- 6.1. Fourier Series
- 6.2. Separation of Variables for the Heat Equation
- 6.3. Separation of Variables for the Wave Equation
- 6.4. Separation of Variables for a Nonlinear Heat Equation
- 6.5. The Beam Equation
- Problems
- 7. Eigenfunctsons and Convergence of Fourier Series
- 7.1. Eigenfunctions for ODE
- 7.2. Convergence and Completeness
- 7.3. Pointwise Convergence of Fourier Series
- 7.4. Uniform Convergence of Fourier Series
- 7.5. Convergence in L2
- 7.6. Fourier Transform
- Problems
- 8. Laplace's Equation and Poisson's Equation
- 8.1. The Fundamental Solution
- 8.2. Solving Poisson's Equation in W
- 8.3. Properties of Harmonic Functions
- 8.4. Separation of Variables for Laplace's Equation
- Problems
- 9. Green's Functions and Distributions
- 9.1. Boundary Value Problems
- 9.2. Test Functions and Distributions
- 9.3. Greens Functions
- Problems
- 10. Function Spaces
- 10.1. Basic Inequalities and Definitions
- 10.2. Multi-Index Notation
- 10.3. Sobolev Spaces W k'p (U)
- Problems
- 11. Elliptic Theory with Sobolev Spaces
- 11.1. Poisson's Equation
- 11.2. Linear Second-Order Elliptic Equations
- Problems
- 12. Traveling Wave Solutions of PDE
- 12.1. Burgers' Equation
- 12.2. The Korteweg-de Vries Equation
- 12.3. Fishers Equation
- 12.4. The Bistable Equation
- Problems
- 13. Scalar Conservation Laws
- 13.1. The Inviscid Burgers Equation
- 13.2. Scalar Conservation Laws
- 13.3. The Lax Entropy Condition Revisited
- 13.4. Undercompressive Shocks
- 13.5. The (Viscous) Burgers Equation
- 13.6. Multidimensional Conservation Laws
- Problems
- 14. Systems of First-Order Hyperbolic PDE
- 14.1. Linear Systems of First-Order PDE
- 14.2. Systems of Hyperbolic Conservation Laws
- 14.3. The Dam-Break Problem Using Shallow Water Equations
- 14.4. Discussion
- Problems
- 15. The Equations of Fluid Mechanics
- 15.1. The Navier-Stokes and Stokes Equations
- 15.2. The Euler Equations
- Problems
- Appendix A. Multivariate Calculus
- Appendix B. Analysis
- Appendix C. Systems of Ordinary Differential Equations
- References
- Index