Preface Acknowledgments 1 Introduction 1.1 Model Equations 1.1.1 Advection Equation 1.1.2 Diffusion Equation 1.1.3 Laplace's Equation 1.1.4 Wave Equation 1.2 PDEs Are Everywhere 1.3 Exercise 2 First-Order PDEs 2.1 Constant Coefficient Equations 2.2 Linear Equations 2.3 Method of Characteristics 2.4 Quasilinear Equations 2.5 Higher-Dimensional Equations 2.6 Fully Nonlinear First-Order Equations 2.6.1 Method of Characteristics 2.6.2 Charpit's Method 2.7 Exercises 3 Second-Order Linear PDEs 3.1 Introduction 3.2 Standard Forms 3.2.1 Parabolic Standard Form 3.2.2 Hyperbolic Standard Form 3.2.3 Modified Hyperbolic Form 3.2.4 Regular Hyperbolic Form 3.2.5 Elliptic Standard Form 3.3 The Wave Equation 3.4 Exercises 4 Fourier Series 4.1 Fourier Series 4.2 Fourier Series on [-π,π] 4.3 Fourier Series on [-L,L] 4.4 Odd and Even Extensions 4.4.1 Sine Series 4.4.2 Cosine Series 4.5 Exercises 5 Separation of Variables 5.1 The Heat Equation 5.1.1 Nonhomogeneous Boundary Conditions 5.1.2 Nonhomogeneous Equations 5.1.3 Equations with a Solution-Dependent Source Term 5.1.4 Equations with a Solution-Dependent Convective Term 5.2 Laplace's Equation 5.2.1 Laplace's Equation on an Arbitrary Rectangular Domain 5.3 The Wave Equation 5.4 Exercises 6 Fourier Transform 6.1 Fourier Transform 6.2 Fourier Sine and Cosine Transforms 6.3 Exercises 7 Solutions Author's Biography 編輯手記
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