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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 | |