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Introduction | |
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Introduction to Partial Differential Equations | |
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Diffusion-Type Problems | |
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Diffusion-Type Problems (Parabolic Equations) | |
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Boundary Conditions for Diffusion-Type Problems | |
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Derivation of the Heat Equation | |
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Separation of Variables | |
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Transforming Nonhomogeneous BCs into Homogeneous Ones | |
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Solving More Complicated Problems by Separation of Variables | |
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Transforming Hard Equations into Easier Ones | |
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Solving Nonhomogeneous PDEs (Eigenfunction Expansions) | |
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Integral Transforms (Sine and Cosine Transforms) | |
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The Fourier Series and Transform | |
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The Fourier Transform and its Application to PDEs | |
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The Laplace Transform | |
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Duhamel's Principle | |
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The Convection Term u subscript x in Diffusion Problems | |
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Hyperbolic-Type Problems | |
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The One Dimensional Wave Equation (Hyperbolic Equations) | |
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The D'Alembert Solution of the Wave Equation | |
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More on the D'Alembert Solution | |
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Boundary Conditions Associated with the Wave Equation | |
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The Finite Vibrating String (Standing Waves) | |
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The Vibrating Beam (Fourth-Order PDE) | |
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Dimensionless Problems | |
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Classification of PDEs (Canonical Form of the Hyperbolic Equation) | |
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The Wave Equation in Two and Three Dimensions (Free Space) | |
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The Finite Fourier Transforms (Sine and Cosine Transforms) | |
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Superposition (The Backbone of Linear Systems) | |
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First-Order Equations (Method of Characteristics) | |
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Nonlinear First-Order Equations (Conservation Equations) | |
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Systems of PDEs | |
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The Vibrating Drumhead (Wave Equation in Polar Coordinates) | |
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Elliptic-Type Problems | |
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The Laplacian (an intuitive description) | |
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General Nature of Boundary-Value Problems | |
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Interior Dirichlet Problem for a Circle | |
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The Dirichlet Problem in an Annulus | |
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Laplace's Equation in Spherical Coordinates (Spherical Harmonics) | |
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A Nonhomogeneous Dirichlet Problem (Green's Functions) | |
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Numerical and Approximate Methods | |
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Numerical Solutions (Elliptic Problems) | |
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An Explicit Finite-Difference Method | |
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An Implicit Finite-Difference Method (Crank-Nicolson Method) | |
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Analytic versus Numerical Solutions | |
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Classification of PDEs (Parabolic and Elliptic Equations) | |
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Monte Carlo Methods (An Introduction) | |
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Monte Carlo Solutions of Partial Differential Equations) | |
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Calculus of Variations (Euler-Lagrange Equations) | |
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Variational Methods for Solving PDEs (Method of Ritz) | |
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Perturbation method for Solving PDEs | |
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Conformal-Mapping Solution of PDEs | |
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Answers to Selected Problems | |
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Integral Transform Tables | |
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PDE Crossword Puzzle | |
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Laplacian in Different Coordinate Systems | |
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Types of Partial Differential Equations | |
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Index | |