Calculus Basic Partial Differential Equations by D. Bleecker and G. Csordas

 Preface
 Review and Introduction
 A Review of Ordinary Differential Equations
 Generalities about PDEs
 General Solutions and Elementary Techniques

 First-Order PDEs
 First-Order Linear PDEs (Constant Coefficients)
 Variable Coefficients
 Higher Dimensions, Quasi-linearity, Applications
 Supplement on General Nonlinear First-Order PDEs (Optional

 The Heat Equation
 Derivation of the Heat Equation and Solutions of the Standard Initial/Boundary-Value Problems
 Uniqueness and the Maximum Principle
 Time-Independent Boundary Conditions
 Time-Dependent Boundary Conditions and Duhamel's Principle for Inhomogeneous Heat Equations

 Fourier Series and Sturm-Liouville Theory
 Orthogonality and the Definition of Fourier Series
 Convergence Theorems for Fourier Series
 Sine and Cosine Series and Applications
 Sturm-Liouville Theory

 The Wave Equation
 The Wave Equation - Derivation and Uniqueness
 D'Alambert's Solution of Wave Problems
 Other Boundary Conditions and Inhomogeneous Wave Equations

 Laplace's Equation
 General Orientation
 The Dirichlet Problem for a Rectangle
 The Dirichlet Problem for Annuli and Disks
 The Maximum Principle and Uniqueness for the Dirichlet Problem
 Complex Variable Theory with Applications

 Fourier Transforms
 Complex Fourier Series
 Basic Properties of Fourier Transforms
 The Inversion Theorem and Parseval's Equality
 Fourier Transform Methods for PDEs
 Applications to Problems on Finite and Semi-Finite Intervals

 Numerical Solutions of PDEs - An Introduction
 The O Symbol and Approximations of Derivatives
 The Explicit Difference Method and the Heat Equation
 Difference Equations and Round-off Errors
 An Overview of Some Other Numerical Methods for PDEs (Optional)

 PDEs in Higher Dimensions
 Higher-Dimensional PDEs - Rectangular Coordinates
 The Eigenfunction Viewpoint
 PDEs in Spherical Coordinates
 Spherical Harmonics, Laplace Series and Applications
 Special Functions and Applications
 Solving PDEs on Manifolds

 Appendix
 The Classification Theorem
 Fubini's Theorem
 Leibniz's Rule
 The Maximum/Minimum Theorem
 A Table of Fourier Transforms
 Bessel Functions

 References
 Selected Answers
 Index of Notation
 Notation