Basic Partial Differential Equations

Quantities which depend on space and/or time variables are often governed by differential equations which are based on underlying physical principles. Partial differential equations (PDEs) not only accurately express these principles, but also help to predict the behavior of a system from an initial state of the system and from given external influences. Thus, it is hard to overestimate the relevance of PDEs in all forms of science and engineering, or any endeavor which involves reasonably smooth, predictable changes of measurable quantities. Having taught from the material in this book for ten years with much feedback from students, we have found that the book serves as a very readable introduction to the subject for undergraduates with a year and a half of calculus, but not necessarily any more. In particular, one need not have had a linear algebra course or even a course in ordinary differential equations to understand the material. As the title suggests, we have concentrated only on what we feel are the absolutely essential aspects of the subject, and there are some crucial topics such as systems of PDEs which we only touch on. Yet the book certainly contains more material than can be covered in a single semester, even with an exceptional class. Given the broad relevance of the subject, we suspect that a demand for a second semester surely exists, but has been largely unmet, partly due to the lack of books which take the time and space to be readable by sophomores.

Table of Contents:
1. Review and Introduction.- 1.1 A Review of Ordinary Differential Equations.- 1.2 Generalities About PDEs.- 1.3 General Solutions and Elementary Techniques.- 2. First-Order PDEs.- 2.1 First-Order Linear PDE s (Constant Coefficients).- 2.2 Variable Coefficients.- 2.3 Higher Dimensions, Quasi-linearity, Applications.- 2.4 Supplement on Nonlinear First-Order PDEs (Optional).- 3. The Heat Equation.- 3.1 Derivation of the Heat Equation and Solutions of the Standard Initial/Boundary-Value problems.- 3.2 Uniqueness and the Maximum Principle.- 3.3 Time-Independent Boundary Conditions.- 3.4 Time-Dependent Boundary Conditions.- 4. Fourier Series and Sturm-Liouville Theory.- 4.1 Orthogonality and the Definition of Fourier Series.- 4.2 Convergence Theorems for Fourier Series.- 4.3 Sine and Cosine Series and Applications.- 4.4 Sturm-Liouville Problems.- 5. The Wave Equation.- 5.1 The Wave Equation - Derivation and Uniqueness.- 5.2 The D'Alembert Solution of the wave equation.- 5.3 Inhomogeneous Boundary Conditions and Wave Equations.- 6. Laplace's Equation.- 6.1 General Orientation.- 6.2 The Dirichlet Problem for the rectangle.- 6.3 The Dirichlet Problem for Annuli and Disks.- 6.4 The Maximum Principle and Uniqueness for the Dirichlet Problem.- 6.5 Complex Variable Theory with Applications.- 7. Fourier Transforms.- 7.1 Complex Fourier Series.- 7.2 Basic Properties of Fourier Transforms.- 7.3 The Inversion Theorem and Parseval's Equality.- 7.4 Fourier Transform Methods for PDE s.- 7.5 Applications to Problems on Finite and Semi-Infinite Intervals.- 8. Numerical Solutions. An Introduction.- 8.1 The O Symbol and Approximation of Derivatives.- 8.2 The Explicit Difference Method and the Heat Equation.- 8.3 Difference Equations and Round-off Errors.- 8.4 An Overview of Some Other Numerical Methods.- 9. PDEs in Higher Dimensions.- 9.1 Higher-Dimensional PDEs - Rectangular Coordinates.- 9.2 The Eigenfunction Viewpoint.- 9.3 PDEs in Spherical Coordinates.- 9.4 Spherical Harmonics, Laplace Series and Applications.- 9.5 Special Functions and Applications.- 9.6 Solving PDEs on Manifolds.- A.1 The Classification Theorem.- A.2 Fubini's Theorem.- A.3 Leibniz's Rule.- A.4 The Maximum/Minimum Theorem.- A.5 Table of Fourier Transforms.- A.6 Bessel Functions.- References.- Selected Answers.- Index of Notation.