دانلود کتاب Partial Differential Equations: An Introduction to Theory and Applications

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Preface ix 1. Introduction 1 1.1. Linear PDE 2 1.2. Solutions; Initial and Boundary Conditions 3 1.3. Nonlinear PDE 4 1.4. Beginning Examples with Explicit Wave-like Solutions 6 Problems 8 2. Beginnings 11 2.1. Four Fundamental Issues in PDE Theory 11 2.2. Classification of Second-Order PDE 12 2.3. Initial Value Problems and the Cauchy-Kovalevskaya Theorem 17 2.4. PDE from Balance Laws 21 Problems 26 3. First-OrderPDE 29 3.1. The Method of Characteristics for Initial Value Problems 29 3.2. The Method of Characteristics for Cauchy Problems in Two Variables 32 3.3. The Method of Characteristics in R n 35 3.4. Scalar Conservation Laws and the Formation of Shocks 38 Problems 40 4. TheWaveEquation 43 4.1. The Wave Equation in Elasticity 43 4.2. D’Alembert’s Solution 48 4.3. The Energy E(t) and Uniqueness of Solutions 56 4.4. Duhamel’s Principle for the Inhomogeneous Wave Equation 57 4.5. The Wave Equation on R 2 and R 3 59 Problems 61 5. TheHeatEquation 65 5.1. The Fundamental Solution 66 5.2. The Cauchy Problem for the Heat Equation 68 5.3. The Energy Method 73 5.4. The Maximum Principle 75 5.5. Duhamel’s Principle for the Inhomogeneous Heat Equation 77 Problems 78 6. SeparationofVariablesandFourierSeries 81 6.1. Fourier Series 81 6.2. Separation of Variables for the Heat Equation 82 6.3. Separation of Variables for the Wave Equation 91 6.4. Separation of Variables for a Nonlinear Heat Equation 93 6.5. The Beam Equation 94 Problems 96 7. EigenfunctionsandConvergenceofFourierSeries 99 7.1. Eigenfunctions for ODE 99 7.2. Convergence and Completeness 102 7.3. Pointwise Convergence of Fourier Series 105 7.4. Uniform Convergence of Fourier Series 108 7.5. Convergence in L 2 110 7.6. Fourier Transform 114 Problems 117 8. Laplace’sEquationandPoisson’sEquation 119 8.1. The Fundamental Solution 119 8.2. Solving Poisson’s Equation in R n 120 8.3. Properties of Harmonic Functions 122 8.4. Separation of Variables for Laplace’s Equation 125 Problems 130 9. Green’sFunctionsandDistributions 133 9.1. Boundary Value Problems 133 9.2. Test Functions and Distributions 136 9.3. Green’s Functions 144 Problems 149 10. FunctionSpaces 153 10.1. Basic Inequalities and Definitions 153 10.2. Multi-Index Notation 157 10.3. Sobolev Spaces W k,p (U) 158 Problems 159 11. EllipticTheorywithSobolevSpaces 161 11.1. Poisson’s Equation 161 11.2. Linear Second-Order Elliptic Equations 167 Problems 173 12. TravelingWaveSolutionsofPDE 175 12.1. Burgers’ Equation 175 12.2. The Korteweg-deVries Equation 176 12.3. Fisher’s Equation 179 12.4. The Bistable Equation 181 Problems 186 13. ScalarConservationLaws 189 13.1. The Inviscid Burgers Equation 189 13.2. Scalar Conservation Laws 196 13.3. The Lax Entropy Condition Revisited 201 13.4. Undercompressive Shocks 204 13.5. The (Viscous) Burgers Equation 206 13.6. Multidimensional Conservation Laws 208 Problems 211 14. SystemsofFirst-OrderHyperbolicPDE 215 14.1. Linear Systems of First-Order PDE 215 14.2. Systems of Hyperbolic Conservation Laws 219 14.3. The Dam-Break Problem Using Shallow Water Equations 239 14.4. Discussion 241 Problems 242 15. TheEquationsofFluidMechanics 245 15.1. The Navier-Stokes and Stokes Equations 245 15.2. The Euler Equations 247 Problems 250 AppendixA.Multivariable Calculus 253 AppendixB.Analysis 259 AppendixC.Systems of Ordinary Differential Equations 263 References 265 Index 269