AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
CONTENTS INTRODUCTION PRELIMINARIES CLASSI?CATION DIFFERENTIAL OPERATORS AND THE SUPERPOSITION PRINCIPLE DIFFERENTIAL EQUATIONS AS MATHEMATICAL MODELS ASSOCIATED CONDITIONS SIMPLE EXAMPLES EXERCISES FIRST-ORDER EQUATIONS 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 SECOND-ORDER LINEAR EQUATIONS IN TWO INDENPENDENT VARIABLES CANONICAL FORM OF HYPERBOLIC EQUATIONS CANONICAL FORM OF PARABOLIC EQUATIONS CANONICAL FORM OF ELLIPTIC EQUATIONS THE ONE-DIMENSIONAL WAVE EQUATION CANONICAL FORM AND GENERAL SOLUTION THE CAUCHY PROBLEM AND D’ALEMBERT’S FORMULA DOMAIN OF DEPENDENCE AND REGION OF IN?UENCE THE CAUCHY PROBLEM FOR THE NONHOMOGENEOUS WAVE EQUATION THE METHOD OF SEPARATION OF VARIABLES 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 STURM–LIOUVILLE PROBLEMS AND EIGENFUNCTION EXPANSIONS THE STURM–LIOUVILLE PROBLEM INNER PRODUCT SPACES AND ORTHONORMAL SYSTEMS THE BASIC PROPERTIES OF STURM–LIOUVILLE EIGENFUNCTIONS AND EIGENVALUES NONHOMOGENEOUS EQUATIONS NONHOMOGENEOUS BOUNDARY CONDITIONS ELLIPTIC EQUATIONS 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 GREEN’S FUNCTIONS AND INTEGRAL REPRESENTATIONS GREEN’S FUNCTION FOR DIRICHLET PROBLEM IN THE PLANE NEUMANN’S FUNCTION IN THE PLANE THE HEAT KERNEL EQUATIONS IN HIGH DIMENSIONS CLASSI?CATION OF SECOND-ORDER EQUATIONS THE EIGENVALUE PROBLEM FOR THE LAPLACE EQUATION SEPARATION OF VARIABLES FOR THE HEAT EQUATION SEPARATION OF VARIABLES FOR THE LAPLACE EQUATION SCHR¨ODINGER EQUATION FOR THE HYDROGEN ATOM MUSICAL INSTRUMENTS GREEN’S FUNCTIONS IN HIGHER DIMENSIONS HEAT KERNEL IN HIGHER DIMENSIONS 10 VARIATIONAL METHODS CALCULUS OF VARIATIONS FUNCTION SPACES AND WEAK FORMULATION 11 NUMERICAL METHODS 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 ?NITE ELEMENTS METHOD 12 SOLUTIONS OF ODD-NUMBERED PROBLEMS TRIGONOMETRIC FORMULAS INTEGRATION FORMULAS ELEMENTARY ODES DIFFERENTIAL OPERATORS IN POLAR COORDINATES DIFFERENTIAL OPERATORS IN SPHERICAL COORDINATES