First-Order Differential Equations: (Dover Books on Mathematics)

Table of Contents:
Preface 0. Mathematical Preliminaries 0.1 Functions and Their Derivatives 0.2 Functions of Functions and their Derivatives 0.3 Implicit Functions 0.4 Sets of Functions 0.5 Differentiation of Implicit Functions 0.6 Surfaces 0.7 Tangents and Normals 0.8 Direction Cosines and Space Curves 0.9 Directional Derivatives 0.10 Envelopes 0.11 Differential Equations 0.12 Strips References 1. Mathematical Models That Give First-Order Partial Differential Equations 1.1 Introduction 1.2 Chromatography of a Single Solute 1.3 Chromatography of Several Solutes 1.4 Chromatography with Heat Effects 1.5 Countercurrent Adsorber 1.6 Heat Exchanger 1.7 Polymerization in a Batch Reactor 1.8 Other Problems in Chemical Kinetics 1.9 Tubular Reactor 1.10 Enhanced Oil Recovery 1.11 Kinematic Waves in General 1.12 Equations of Compressible Fluid Flow 1.13 Flow of Electricity and Heat and Propagation of Light 1.14 Two Problems in Optimization 1.15 An Estimation Problem 1.16 Geometrical Origins 1.17 Cauchy-Riemann Equations References 2. Motivations, Classifications, and Some Methods of Solution 2.1 Comparisons Between Ordinary and Partial Differential Equations 2.2 Classification of Equations 2.3 When Has an Equation Been Solved? 2.4 Special Methods for Certain Equations 2.5 Method of Characteristics for Quasi-linear Equations 2.6 Alternative Treatment of the Quasi-linear Equations References 3. Linear and Semilinear Equations 3.1 Linear and Semilinear Equations with Constant Coefficients 3.2 Examples of Linear and Semilinear Equations 3.3 Homogeneous Equations 3.4 Equilibrium Theory of the Parametric Pump 3.5 Linear Equations with Variable Coefficients 3.6 Linear Equations with n Independent Variables References 4. Chromatographic Equations with Finite Rate Expressions 4.1 Solution by the Laplace Transformation 4.2 Linear Chromatography 4.3 Laplace Transformation as a Moment-generating Function 4.4 Chromatography with a Langmuir Isotherm 4.5 Fixed-bed Adsorption with Recycle 4.6 Poisoning in Fixed-bed Reactors References 5. Homogeneous Quasi-linear Equations 5.1 Reducible Equations 5.2 Simple Waves 5.3 Equilibrium Chromatography of a Single Solute 5.4 Discontinuities in Solutions 5.5 Discontinuous Solutions in Equilibrium Chromatography 5.6 Water Flooding 5.7 Quasi-linear Equations with n Independent Variables References 6. Formation and Propagation of Shocks 6.1 Formation of a Shock 6.2 Saturation of a Column 6.3 Development of a Finite Chromatogram 6.4 Propagation of a Pulse 6.5 Analysis of a Countercurrent Adsorber 6.6 Analysis of Traffic Flow 6.7 Theory of Sedimentation References 7. Conservation Equations, Weak Solutions, and Shock Layers 7.1 Chromatographic Equations and Initial Data 7.2 Conservation Equations and the Jump Condition 7.3 Intermezzo on Convex Function and the Legendre Transformation 7.4 Weak Solutions and the Entropy Condition 7.5 Lax's Solution for the Quasi-linear Conservation Law 7.6 Some Additional Properties of Weak Solutions 7.7 Sound Waves of Finite Amplitude 7.8 Some General Properties of Chromatograms 7.9 Asymptotic Behavior 7.10 Shock-layer Analysis References 8. Nonhomogeneous Quasi-linear Equations 8.1 Nonhomogeneous Equations with Two Independent Variables 8.2 Analysis of Transient Volumetric Pool Boiling 8.3 Black-box Steady State 8.4 Countercurrent Adsorber under Nonequilibrium Conditions 8.5 Countercurrent Adsorber with Reaction References 9. Nonlinear Equations 9.1 Nonlinear Equations with Two Independent Variables 9.2 Geometry of the Solution Surface 9.3 Nonlinear Equations with n Independent Variables 9.4 Some Questions of Existence and Continuity 9.5 A Problem in Optimization References 10. Variational Problems 10.1 Basic Problem of the Calculus of Variations 10.2 Canonical form of the Euler Equations 10.3 Hamilton-Jacobi Equation 10.4 Equivalence of First-order Partial Differential Equations and Variational Problems 10.5 Principles of Fermat and Huygens References Author Index; Subject Index