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| Preface | |
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| Introduction to Partial Differential Equations | |
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| Partial Differential Equations | |
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| Equations and Solutions | |
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| Classification | |
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| Linear versus Nonlinear | |
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| Linear Equations | |
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| Conservation Laws | |
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| One Dimension | |
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| Higher Dimensions | |
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| Constitutive Relations | |
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| Initial and Boundary Value Problems | |
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| Waves | |
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| Traveling Waves | |
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| Plane Waves | |
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| Plane Waves and Transforms | |
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| Nonlinear Dispersion | |
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| First-Order Equations and Characteristics | |
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| Linear First-Order Equations | |
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| Advection Equation | |
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| Variable Coefficients | |
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| Nonlinear Equations | |
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| Quasilinear Equations | |
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| The General Solution | |
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| Propagation of Singularities | |
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| General First-Order Equation | |
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| Complete Integral | |
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| A Uniqueness Result | |
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| Models in Biology | |
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| Age Structure | |
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| Structured Predator-Prey Model | |
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| Chemotherapy | |
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| Mass Structure | |
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| Size-Dependent Predation | |
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| Weak Solutions to Hyperbolic Equations | |
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| Discontinuous Solutions | |
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| Jump Conditions | |
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| Rarefaction Waves | |
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| Shock Propagation | |
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| Shock Formation | |
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| Applications | |
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| Traffic Flow | |
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| Plug Flow Chemical Reactors | |
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| Weak Solutions: A Formal Approach | |
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| Asymptotic Behavior of Shocks | |
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| Equal-Area Principle | |
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| Shock Fitting | |
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| Asymptotic Behavior | |
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| Hyperbolic Systems | |
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| Shallow-Water Waves; Gas Dynamics | |
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| Shallow-Water Waves | |
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| Small-Amplitude Approximation | |
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| Gas Dynamics | |
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| Hyperbolic Systems and Characteristics | |
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| Classification | |
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| The Riemann Method | |
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| Jump Conditions for Systems | |
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| Breaking Dam Problem | |
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| Receding Wall Problem | |
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| Formation of a Bore | |
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| Gas Dynamics | |
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| Hodographs and Wavefronts | |
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| Hodograph Transformation | |
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| Wavefront Expansions | |
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| Weakly Nonlinear Approximations | |
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| Derivation of Burgers' Equation | |
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| Diffusion Processes | |
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| Diffusion and Random Motion | |
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| Similarity Methods | |
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| Nonlinear Diffusion Models | |
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| Reaction-Diffusion; Fisher's Equation | |
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| Traveling Wave Solutions | |
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| Perturbation Solution | |
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| Stability of Traveling Waves | |
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| Nagumo's Equation | |
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| Advection-Diffusion; Burgers' Equation | |
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| Traveling Wave Solution | |
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| Initial Value Problem | |
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| Asymptotic Solution to Burgers' Equation | |
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| Evolution of a Point Source | |
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| Dynamical Systems | |
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| Reaction-Diffusion Systems | |
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| Reaction-Diffusion Models | |
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| Predator-Prey Model | |
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| Combustion | |
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| Chemotaxis | |
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| Traveling Wave Solutions | |
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| Model for the Spread of a Disease | |
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| Contaminant Transport in Groundwater | |
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| Existence of Solutions | |
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| Fixed-Point Iteration | |
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| Semilinear Equations | |
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| Normed Linear Spaces | |
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| General Existence Theorem | |
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| Maximum Principles and Comparison Theorems | |
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| Maximum Principles | |
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| Comparison Theorems | |
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| Energy Estimates and Asymptotic Behavior | |
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| Calculus Inequalities | |
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| Energy Estimates | |
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| Invariant Sets | |
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| Pattern Formation | |
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| Equilibrium Models | |
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| Elliptic Models | |
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| Theoretical Results | |
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| Maximum Principle | |
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| Existence Theorem | |
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| Eigenvalue Problems | |
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| Linear Eigenvalue Problems | |
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| Nonlinear Eigenvalue Problems | |
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| Stability and Bifurcation | |
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| Ordinary Differential Equations | |
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| Partial Differential Equations | |
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| References | |
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| Index | |