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| Preface | |
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| Prelude to Chapter 1 | |
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| Introduction | |
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| What are Partial Differential Equations? | |
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| PDEs We Can Already Solve | |
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| Initial and Boundary Conditions | |
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| Linear PDEs-Definitions | |
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| Linear PDEs-The Principle of Superposition | |
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| Separation of Variables for Linear, Homogeneous PDEs | |
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| Eigenvalue Problems | |
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| Prelude to Chapter 2 | |
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| The Big Three PDEs | |
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| Second-Order, Linear, Homogeneous PDEs with Constant Coefficients | |
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| The Heat Equation and Diffusion | |
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| The Wave Equation and the Vibrating String | |
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| Initial and Boundary Conditions for the Heat and Wave Equations | |
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| Laplace's Equation-The Potential Equation | |
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| Using Separation of Variables to Solve the Big Three PDEs | |
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| Prelude to Chapter 3 | |
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| Fourier Series | |
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| Introduction | |
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| Properties of Sine and Cosine | |
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| The Fourier Series | |
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| The Fourier Series, Continued | |
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| The Fourier Series-Proof of Pointwise Convergence | |
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| Fourier Sine and Cosine Series | |
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| Completeness | |
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| Prelude to Chapter 4 | |
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| Solving the Big Three PDEs on Finite Domains | |
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| Solving the Homogeneous Heat Equation for a Finite Rod | |
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| Solving the Homogeneous Wave Equation for a Finite String | |
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| Solving the Homogeneous Laplace's Equation on a Rectangular Domain | |
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| Nonhomogeneous Problems | |
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| Prelude to Chapter 5 | |
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| Characteristics | |
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| First-Order PDEs with Constant Coefficients | |
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| First-Order PDEs with Variable Coefficients | |
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| The Infinite String | |
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| Characteristics for Semi-Infinite and Finite String Problems | |
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| General Second-Order Linear PDEs and Characteristics | |
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| Prelude to Chapter 6 | |
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| Integral Transforms | |
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| The Laplace Transform for PDEs | |
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| Fourier Sine and Cosine Transforms | |
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| The Fourier Transform | |
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| The Infinite and Semi-Infinite Heat Equations | |
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| Distributions, the Dirac Delta Function and Generalized Fourier Transforms | |
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| Proof of the Fourier Integral Formula | |
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| Prelude to Chapter 7 | |
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| Special Functions and Orthogonal Polynomials | |
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| The Special Functions and Their Differential Equations | |
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| Ordinary Points and Power Series Solutions; Chebyshev, Her-mite and Legendre Polynomials | |
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| The Method of Frobenius; Laguerre Polynomials | |
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| Interlude: The Gamma Function | |
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| Bessel Functions | |
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| Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials | |
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| Prelude to Chapter 8 | |
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| Sturm-Liouville Theory and Generalized Fourier Series | |
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| Sturm-Liouville Problems | |
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| Regular and Periodic Stunri-Liouville Problems | |
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| Singular Sturm-Liouville Problems; Self-Adjoint Problems | |
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| The Mean-Square or L<sup>2</sup> Norm and Convergence in the Mean | |
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| Generalized Fourier Series; Parseval's Equality and Completeness | |
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| Prelude to Chapter 9 | |
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| PDEs in Higher Dimensions | |
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| PDEs in Higher Dimensions: Examples and Derivations | |
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| The Heat and Wave Equations on a Rectangle; Multiple Fourier Series | |
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| Laplace's Equation in Polar Coordinates: Poisson's Integral Formula | |
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| The Wave and Heat Equations in Polar Coordinates | |
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| Problems in Spherical Coordinates | |
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| The Infinite Wave Equation and Multiple Fourier Transforms | |
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| Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian | |
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| Prelude to Chapter 10 | |
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| Nonhomogeneous Problems and Green's Functions | |
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| Green's Functions for ODEs | |
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| Green's Function and the Dirac Delta Function | |
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| Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions | |
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| Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation | |
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| Green's Functions for Equations of Evolution | |
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| Prelude to Chapter 11 | |
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| Numerical Methods | |
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| Finite Difference Approximations for ODEs | |
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| Finite Difference Approximations for PDEs | |
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| Spectral Methods and the Finite Element Method | |
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| Uniform Convergence; Differentiation and Integration of Fourier Series | |
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| Other Important Theorems | |
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| Existence and Uniqueness Theorems | |
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| A Menagerie of PDEs | |
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| MATLAB Code for Figures and Exercises | |
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| Answers to Selected Exercises | |
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| References | |
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| Index | |