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| Preface | |
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| A Preview of Applications and Techniques | |
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| What Is a Partial Differential Equation? | |
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| Solving and Interpreting a Partial Differential Equation | |
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| Fourier Series | |
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| Periodic Functions | |
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| Fourier Series | |
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| Fourier Series of Functions with Arbitrary Periods | |
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| Half-Range Expansions: The Cosine, and Sine Series | |
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| Mean Square Approximation and Parseval's Identity | |
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| Complex Form of Fourier Series | |
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| Supplement on Convergence | |
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| Uniform Convergence of Sequences and Series of Functions | |
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| Dirichlet Test and Convergence of Fourier Series | |
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| Partial Differential Equations in Rectangular Coordinates | |
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| Partial Differential Equations in Physics and Engineering | |
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| Modeling: Vibrating Strings and the Wave Equation | |
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| Solution of the One Dimensional Wave Equation: The Method of Separation of Variables | |
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| D'Alembert's Method | |
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| The One Dimensional Heat Equation | |
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| Heat Conduction in Bars: Varying the Boundary Conditions | |
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| The Two Dimensional Wave and Heat Equations | |
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| Laplace's Equation in Rectangular Coordinates | |
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| Poisson's Equation: The Method of Eigenfunction Expansions | |
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| The Maximum Principle | |
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| Partial Differential Equations in Polar and Cylindrical Coordinates | |
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| The Laplacian in Various Coordinate Systems | |
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| Vibrations of a Circular Membrane: Symmetric Case | |
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| Vibrations of a Circular Membrane: General Case | |
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| Steady-State Temperature in a Disk | |
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| Steady-State Temperature in a Cylinder | |
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| The Helmholtz and Poisson Equations | |
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| Supplement on Bessel Functions | |
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| Bessel's Equation and Bessel Functions | |
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| Bessel Series Expansions | |
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| Partial Differential Equations in Spherical Coordinates | |
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| Preview of Problems and Methods | |
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| Dirichlet Problems with Symmetry | |
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| Spherical Harmonics and the General Dirichlet Problem | |
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| The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations | |
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| Supplement on Legendre Functions | |
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| Legendre's Differential Equation | |
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| Legendre Polynomials and Legendre Series Expansions | |
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| Associated Legendre Functions and Series Expansions | |
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| Sturm-Liouville Theory with Engineering Applications | |
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| Orthogonal Functions | |
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| Sturm-Liouville Theory | |
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| The Hanging Chain | |
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| Fourth Order Sturm-Liouville Theory | |
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| Elastic Vibrations and Buckling of Beams | |
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| The Fourier Transform and its Applications | |
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| The Fourier Integral Representation | |
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| The Fourier Transform | |
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| The Fourier Transform Method | |
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| The Heat Equation and Gauss's Kernel | |
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| A Dirichlet Problem and the Poisson Integral Formula | |
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| The Fourier Cosine and Sine Transforms | |
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| Problems Involving Semi-Infinite Intervals | |
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| The Laplace and Hankel Transforms with Applications | |
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| The Laplace Transform | |
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| Further Properties of the Laplace Transform | |
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| The Laplace Transform Method | |
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| The Hankel Transform with Applications | |
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| Finite Difference Numerical Methods | |
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| The Finite Difference Method for the Heat Equation | |
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| The Finite Difference Method for the Wave Equation | |
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| The Finite Difference Method for Laplace's Equation | |
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| Interation Methods for Laplace's Equation | |
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| Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations | |
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| The Sampling Theorem | |
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| Partial Differential Equations and the Sampling Theorem | |
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| The Discrete and Fast Fourier Transforms | |
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| The Fourier and Discrete Fourier Transforms | |
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| An Introduction to Quantum Mechanics | |
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| Schrodinger's Equation | |
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| The Hydrogen Atom | |
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| Heisenberg's Uncertainty Principle | |
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| Supplement on Orthogonal Polynomials | |
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| Hermite and Laguerre Polynomials | |
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| Appendixes | |
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| Ordinary Differential Equations: Review of Concepts and Methods | |
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| Linear Ordinary Differential Equations | |
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| Linear Ordinary Differential Equations with Constant Coefficients | |
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| Methods for Solving Ordinary Differential Equations | |
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| The Method of Power Series | |
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| The Method of Frobenius | |
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| Tables of Transforms | |
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| Fourier Transforms | |
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| Fourier Cosine Transforms | |
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| Fourier Sine Transforms | |
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| Laplace Transforms | |
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| References | |
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| Answers to Selected Exercises | |
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| Index | |