Synopses & Reviews
This superb introduction differs from other elementary texts on partial differential equations in several ways -- most prominently in devoting almost half its pages to numerical methods for solving such equations. Another difference is the author's constant emphasis on the three entities that must be considered in the modeling of physical systems: the system itself, the continuous model, and the discrete model. The heart of the book -- the solution of problems in partial differential equations -- focuses on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data; first-order equations; numerical solutions of hyperbolic equations; and finite-difference methods for parabolic and elliptic equations. Numerous exercises, with solutions for many.
Synopsis
Superb introduction devotes almost half its pages to numerical methods for solving partial differential equations, while the heart of the book focuses on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included, with solutions for many at end of book. For students with little background in linear algebra, a useful appendix covers that subject briefly.
Synopsis
Superb introduction to numerical methods for solving partial differential equations, boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included, with solutions for many at end of book.
Table of Contents
Preface
Chapter 1. Mathematical Modeling and Partial Differential Equations
1.1 Mathematical Modeling of Physical Systems
1.2 Equation of Heat Conduction
1.3 Steady-State Conduction of Heat
1.4 Transmission Line Equations
1.5 Well-Posed Problems
1.6 Classification of Equations
Chapter 2. Fourier Series and Eigenfunction Expansions; Introduction
2.1 Fourier Series
2.2 Generalized Fourier Series
2.3 Sturm-Liouville Problems
2.4 Discrete Fourier Series
2.5 Function Space L superscript 2
2.6 Multiple Fourier Series
2.7 Summary
Chapter 3. Boundary-Value Problems and Initial-Boundary-Value Problems on Spatially Bounded Domains
3.1 Boundary-Value Problems for Laplace and Poisson Equations
3.2 Evolution Equations: Initial-Boundary-Value Problems for Heat Equation
3.3 Evolution Equations: Initial-Boundary-Value Problems for Wave Equation
Chapter 4. Integral Transforms
4.1 Function Space L superscript 2 (a, b) When (a, b) Is Unbounded
4.2 The Fourier Transform
4.3 The Laplace Transform
Chapter 5. Boundary-Value Problems and Initial-boundary Value Problems on Unbounded Domains
5.1 Elementary Examples on (- infinity, infinity)
5.2 Examples on Semibounded Regions
5.3 Inhomogeneous Equations
5.4 Duhamel's Principle
Chapter 6. Uniqueness and Continuous Dependence on Data
6.1 Well-Posed Problems in Partial Differential Equations
6.2 Green's Identities and Energy Inequalities
6.3 Maximum-Minimum Principles
Chapter 7. First-Order Equations
7.1 Constant-Coefficient Advection Equation
7.2 Linera and Quasi-linear Equations
7.3 Conservation Law Equations
7.4 Generalized Solutions
7.5 Applications of Scalar Conservation Laws
7.6 Systems of First-Order Equations
Chapter 8. Finite-Difference Methods for Parabolic Equations
8.1 Difference Formulas
8.2 Finite-Difference Equations for u subscript t - a superscript 2 u subscript xx = S
8.3 Computational Methods
8.4 Fourier's Method for Difference Equations
8.5 Stability of Finite-Difference Methods
8.6 Difference Methods in Two Space Variables
8.7 Conservation Law Difference Equations
8.8 Material Balance Difference Equation in Two Space Variables
Chapter 9. Numerical Solutions of Hyperbolic Equations
9.1 Difference Methods for a Scalar Initial-Value Problem
9.2 Difference Methods for a Scalar Initial-Boundary-Value Problem
9.3 Scalar Conservation Laws
9.4 Dispersion and Dissipation
9.5 Systems of Equations
9.6 Second-Order Equations
9.7 Method of Characteristics
Chapter 10. Finite-Difference Methods for Elliptic Equations
10.1 Difference Equations for Elliptic Equations
10.2 Direct Solution of Linear Equations
10.3 Fourier's Method
10.4 Iterative Methods
10.5 Convergence of Iterative Methods
Appendix Linear Algebra
Solutions to Selected Exercises; Index