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Index
Title Page
Copyright Page
Preface
Table of Contents
PART 1 - Introduction
LESSON 1 - Introduction to Partial Differential Equations
What Are PDEs?
Why Are PDEs Useful?
How Do You Solve a Partial Differential Equation?
Kinds of PDEs
PART 2 - Diffusion-Type Problems
LESSON 2 - Diffusion-Type Problems (Parabolic Equations)
A Simple Heat-Flow Experiment
The Mathematical Model of the Heat-Flow Experiment
More Diffusion-Type Equations
LESSON 3 - Boundary Conditions for Diffusion-Type Problems
Type 1 BC (Temperature specified on the boundary)
Type 2 BC (Temperature of the surrounding medium specified)
Type 3 BC (Flux specified—including the special case of insulated boundaries)
Typical BCs for One-Dimensional Heat Flow
LESSON 4 - Derivation of the Heat Equation
Derivation of the Heat Equation
LESSON 5 - Separation of Variables
Overview of Separation of Variables
Separation of Variables
LESSON 6 - Transforming Nonhomogeneous BCs into Homogeneous Ones
Transforming Nonhomogeneous BCs to Homogeneous Ones
Transforming Time Varying BCs to Zero BCs
LESSON 7 - Solving More Complicated Problems by Separation of Variables
Heat-Flow Problem with Derivative BC
LESSON 8 - Transforming Hard Equations into Easier Ones
Transforming a Heat-Flow Problem with Lateral Heat Loss into an Insulated Problem
LESSON 9 - Solving Nonhomogeneous PDEs (Eigenfunction Expansions)
Solution by the Eigenfunction Expansion Method
Solution of a Problem by the Eigenfunction-Expansion Method
LESSON 10 - Integral Transforms (Sine and Cosine Transforms)
The Spectrum of a Function
Solution of an Infinite-Diffusion Problem via the Sine Transform
Interpretation of the Solution
LESSON 11 - The Fourier Series and Transform
Discrete Frequency Spectrum of a Periodic Function
The Fourier Transform
LESSON 12 - The Fourier Transform and Its Application to PDEs
Useful Properties of the Fourier Transform
Example of a Convolution of Two Functions
Solution of an Initial-Value Problem
LESSON 13 - The Laplace Transform
Properties of the Laplace Transform
Sufficient Conditions to Insure the Existence of a Laplace Transform
Definition of the Finite Convolution
Heat Conduction in a Semi Infinite Medium
LESSON 14 - Duhamel’s Principle
Heat Flow within a Rod with Temperature Fixed on the Boundaries
The Importance of Duhamel’s Principle
LESSON 15 - The Convection Term ux in the Diffusion Problems
Laplace Transform Solution to the Convection Problem
PART 3 - Hyperbolic-Type Problems
LESSON 16 - The One-Dimensional Wave Equation (Hyperbolic Equations)
Vibrating-String Problem
Intuitive Interpretation of the Wave Equation
LESSON 17 - The D’Alembert Solution of the Wave Equation
D’Alembert’s Solution to the One-Dimensional Wave Equation
Examples of the D’Alembert Solution
LESSON 18 - More on the D’Alembert Solution
The Space-Time Interpretation of D’Alembert’s Solution
Solution of the Semi-infinite String via the D’Alembert Formula
LESSON 19 - Boundary Conditions Associated with the Wave Equation
LESSON 20 - The Finite Vibrating String (Standing Waves)
Separation-of-Variables Solution to the Finite Vibrating String
LESSON 21 - The Vibrating Beam (Fourth-Order PDE)
The Simply Supported Beam
LESSON 22 - Dimensionless Problems
Converting a Diffusion Problem to Dimensionless Form
Example of Transforming a Hyperbolic Problem to Dimensionless Form
LESSON 23 - Classification of PDEs (Canonical Form of the Hyperbolic Equation)
Examples of Hyperbolic, Parabolic, and Elliptic Equations
The Canonical Form of the Hyperbolic Equation
LESSON 24 - The Wave Equation in Two and Three Dimensions (Free Space)
Waves in Three Dimensions
Two-Dimensional Wave Equation
LESSON 25 - The Finite Fourier Transforms (Sine and Cosine Transforms)
Examples of the Sine Transform
Properties of the Transforms
Solving Problems via Finite Transforms
LESSON 26 - Superposition (The Backbone of Linear Systems)
Superposition Used to Break an IBVP into Two Simpler Problems
Separation of Variables and Integral Transforms as Superpositions
LESSON 27 - First-Order Equations (Method of Characteristics)
General Strategy for Solving the First-Order Equation
LESSON 28 - Nonlinear First-Order Equations (Conservation Equations)
Derivation of the Conservation Equation
Conservation Equation Applied to the Traffic Problem
The Nonlinear Initial-Value Problem
LESSON 29 - Systems of PDEs
Solution of the Linear System u, + Aux = 0
LESSON 30 - The Vibrating Drumhead (Wave Equation in Polar Coordinates)
Solution of the Helmholtz Eigenvalue Problem (Subproblem)
PART 4 - Elliptic-Type Problems
LESSON 31 - The Laplacian (an Intuitive Description)
Interpretations of ∇2 in Two Dimensions
Intuitive Meanings of Some Basic Laws of Physics
Changing Coordinates
LESSON 32 - General Nature of Boundary-Value Problems
Steady-State Problems
Factoring out the Time Component in Hyperbolic and Parabolic Problems
The Three Main Types of BCs in Boundary-Value Problems
LESSON 33 - Interior Dirichlet Problem for a Circle
Observations on the Dirichlet Solution
Poisson Integral Formula
LESSON 34 - The Dirichlet Problem in an Annulus
Product Solutions to Laplace’s Equation
Worked Problems for the Dirichlet Problem in an Annulus
Exterior Dirichlet Problem
LESSON 35 - Laplace’s Equation in Spherical Coordinates (Spherical Harmonics)
Special Cases of the Dirichlet Problem
LESSON 36 - A Nonhomogeneous Dirichlet Problem (Green’s Function)
Potentials from Point Sources and Sinks
Poisson’s Equation inside a Circle
Finding the Potential Response G(r,θ,p,φ)
Steps for Finding The Solution
PART 5 - Numerical and Approximate Methods
LESSON 37 - Numerical Solutions (Elliptic Problems)
Finite-Difference Approximations
LESSON 38 - An Explicit Finite-Difference Method
The Explicit Method for Parabolic Equations
LESSON 39 - An Implicit Finite-Difference Method (Crank-Nicolson Method)
The Heat-Flow Problem Solved by an Implicit Method
LESSON 40 - Analytic versus Numerical Solutions
Meaning of Analytic Solutions
Meaning of Numerical Solutions
Comparing Numerical and Analytic Solutions
Parameter Identification (in Biology)
LESSON 41 - Classification of PDEs (Parabolic and Elliptic Equations)
Reducing Parabolic Equations to Canonical Form
Transforming the Parabolic Equation uxx + 2uxy + uyy = 0 into Canonical Form
Reducing Elliptic Equations to Canonical Form
Changing the Equation y2uxx + x2uyy = 0 to Canonical Form
LESSON 42 - Monte Carlo Methods (an Introduction)
Evaluating an Integral
Random Numbers
LESSON 43 - Monte Carlo Solution of Partial Differential Equations
How Tour du Wino is Played
Reason for Playing Tour du Wino
LESSON 44 - Calculus of Variations (Euler-Lagrange Equations)
Minimizing the General Functional J[y] = F(x,y,y′) dx
LESSON 45 - Variational Methods for Solving PDEs (Method of Ritz)
Method of Ritz for Minimizing Functionals
LESSON 46 - Perturbation Methods for Solving PDEs
A Perturbation Solution of the Nonlinear Equation ∇2u + u2 = 0
LESSON 47 - Conformal-Mapping Solutions of PDEs
Conformal Mappings and Complex Functions
Definition of a Conformal Mapping
ANSWERS TO SELECTED PROBLEMS
APPENDIX 1 - Integral Transform Tables
APPENDIX 2 - PDE Crossword Puzzle
APPENDIX 3 - Laplacian in Different Coordinate Systems
APPENDIX 4 - Types of Partial Differential Equations
Index
A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS
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