| |
| |
Preface | |
| |
| |
| |
Introduction | |
| |
| |
| |
Preliminaries | |
| |
| |
| |
Classification | |
| |
| |
| |
Differential operators and the superposition principle | |
| |
| |
| |
Differential equations as mathematical models | |
| |
| |
| |
Associated conditions | |
| |
| |
| |
Simple examples | |
| |
| |
| |
Exercises | |
| |
| |
| |
First-order equations | |
| |
| |
| |
Introduction | |
| |
| |
| |
Quasilinear equations | |
| |
| |
| |
The method of characteristics | |
| |
| |
| |
Examples of the characteristics method | |
| |
| |
| |
The existence and uniqueness theorem | |
| |
| |
| |
The Lagrange method | |
| |
| |
| |
Conservation laws and shock waves | |
| |
| |
| |
The eikonal equation | |
| |
| |
| |
General nonlinear equations | |
| |
| |
| |
Exercises | |
| |
| |
| |
Second-order linear equations in two indenpendent variables | |
| |
| |
| |
Introduction | |
| |
| |
| |
Classification | |
| |
| |
| |
Canonical form of hyperbolic equations | |
| |
| |
| |
Canonical form of parabolic equations | |
| |
| |
| |
Canonical form of elliptic equations | |
| |
| |
| |
Exercises | |
| |
| |
| |
The one-dimensional wave equation | |
| |
| |
| |
Introduction | |
| |
| |
| |
Canonical form and general solution | |
| |
| |
| |
The Cauchy problem and d' Alembert's formula | |
| |
| |
| |
Domain of dependence and region of influence | |
| |
| |
| |
The Cauchy problem for the nonhomogeneous wave equation | |
| |
| |
| |
Exercises | |
| |
| |
| |
The method of separation of variables | |
| |
| |
| |
Introduction | |
| |
| |
| |
Heat equation: homogeneous boundary condition | |
| |
| |
| |
Separation of variables for the wave equation | |
| |
| |
| |
Separation of variables for nonhomogeneous equations | |
| |
| |
| |
The energy method and uniqueness | |
| |
| |
| |
Further applications of the heat equation | |
| |
| |
| |
Exercises | |
| |
| |
| |
Sturm-Liouville problems and eigenfunction expansions | |
| |
| |
| |
Introduction | |
| |
| |
| |
The Sturm-Liouville problem | |
| |
| |
| |
Inner product spaces and orthonormal systems | |
| |
| |
| |
The basic properties of Sturm-Liouville eigenfunctions and eigenvalues | |
| |
| |
| |
Nonhomogeneous equations | |
| |
| |
| |
Nonhomogeneous boundary conditions | |
| |
| |
| |
Exercises | |
| |
| |
| |
Elliptic equations | |
| |
| |
| |
Introduction | |
| |
| |
| |
Basic properties of elliptic problems | |
| |
| |
| |
The maximum principle | |
| |
| |
| |
Applications of the maximum principle | |
| |
| |
| |
Green's identities | |
| |
| |
| |
The maximum principle for the heat equation | |
| |
| |
| |
Separation of variables for elliptic problems | |
| |
| |
| |
Poisson's formula | |
| |
| |
| |
Exercises | |
| |
| |
| |
Green's functions and integral representations | |
| |
| |
| |
Introduction | |
| |
| |
| |
Green's function for Dirichlet problem in the plane | |
| |
| |
| |
Neumann's function in the plane | |
| |
| |
| |
The heat kernel | |
| |
| |
| |
Exercises | |
| |
| |
| |
Equations in high dimensions | |
| |
| |
| |
Introduction | |
| |
| |
| |
First-order equations | |
| |
| |
| |
Classification of second-order equations | |
| |
| |
| |
The wave equation in R[superscript 2] and R[superscript 3] | |
| |
| |
| |
The eigenvalue problem for the Laplace equation | |
| |
| |
| |
Separation of variables for the heat equation | |
| |
| |
| |
Separation of variables for the wave equation | |
| |
| |
| |
Separation of variables for the Laplace equation | |
| |
| |
| |
Schrodinger equation for the hydrogen atom | |
| |
| |
| |
Musical instruments | |
| |
| |
| |
Green's functions in higher dimensions | |
| |
| |
| |
Heat kernel in higher dimensions | |
| |
| |
| |
Exercises | |
| |
| |
| |
Variational methods | |
| |
| |
| |
Calculus of variations | |
| |
| |
| |
Function spaces and weak formulation | |
| |
| |
| |
Exercises | |
| |
| |
| |
Numerical methods | |
| |
| |
| |
Introduction | |
| |
| |
| |
Finite differences | |
| |
| |
| |
The heat equation: explicit and implicit schemes, stability, consistency and convergence | |
| |
| |
| |
Laplace equation | |
| |
| |
| |
The wave equation | |
| |
| |
| |
Numerical solutions of large linear algebraic systems | |
| |
| |
| |
The finite elements method | |
| |
| |
| |
Exercises | |
| |
| |
| |
Solutions of odd-numbered problems | |
| |
| |
| |
Trigonometric formulas | |
| |
| |
| |
Integration formulas | |
| |
| |
| |
Elementary ODEs | |
| |
| |
| |
Differential operators in polar coordinates | |
| |
| |
| |
Differential operators in spherical coordinates | |
| |
| |
References | |
| |
| |
Index | |