Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.
This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing the mathematical model) and how to solve the equation (along with initial and boundary conditions). Written for advanced undergraduate and graduate students, as well as professionals working in the applied sciences, this clearly written book offers realistic, practical coverage of diffusion-type problems, hyperbolic-type problems, elliptic-type problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.
Author(s): Stanley J. Farlow
Series: Dover Books on Mathematics
Edition: Reprint
Publisher: Dover Publications
Year: 1993
Language: English
Pages: C, XI, 414, B
PART 1 Introduction
LESSON 1 Introduction to Partial Differential Equations
PURPOSE OF LESSON
What Are PDEs?
Why Are PDEs Useful?
How Do You Solve a Partlal Dlfferentlal Equation?
Kinds of PDEs
NOTES
PROBLEMS
OTHER READING
PART 2 Diffusion-Type Problems
LESSON 2 Diffusion-Type Problems (Parabolic Equations)
PURPOSE OF LESSON
A Simple Heat-Flow Experiment
The Mathematlcal Model of the Heat-Flow Experiment
The Heat Equation
Boundary Conditions
Initial Conditions
More Diffusion-Type Equations
Internal Heat Source
Diffusion-convection Equation
NOTES
PROBLEMS
OTHER READING
LESSON 3 Boundary Conditions for Diffusion Type Problems
PURPOSE OF LESSON
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
NOTES
PROBLEMS
OTHER READING
LESSON 4 Derivation of the Heat Equation
PURPOSE OF LESSON
Derivation of the Heat Equation
Mean Value Theorem
NOTES
PROBLEMS
OTHER READING
LESSON 5 Separation of Variables
PURPOSE OF LESSON
Overview of Separation of Variables
Separation of Variables
NOTES
PROBLEMS
OTHER READING
LESSON 6 Transforming Nonhomogeneous BCs into Homogeneous Ones
PURPOSE OF LESSON
Transforming Nonhomogeneous BCs to Homogeneous Ones
Transforming Time Varying BCs to Zero BCs
NOTES
PROBLEMS
OTHER READING
LESSON 7 Solving More Complicated Problems by Separation of Variables
PURPOSE OF LESSON
Heat-Flow Problem with Derivative BC
NOTES
PROBLEMS
OTHER READING
LESSON 8 Transforming Hard Equations into Easier Ones
PURPOSE OF LESSON
Transforming a Heat-Flow Problem with Lateral Heat Loss Into an Insulated Problem
NOTES
PROBLEMS
OTHER READING
LESSON 9 Solving Nonhomogeneous PDEs (Eigenfunction Expansions)
PURPOSE OF LESSON
Solution by the Eigenfunction Expansion Method
Solution of a Problem by the Eigenfunction-Expansion Method
NOTES
PROBLEMS
OTHER READING
LESSON 10 lntegral Transforms (Sine and Cosine Transforms)
PURPOSE OF LESSON
The Spectrum of a Function
Solution of an Infinite-Diffusion Problem via the Sine Transform
Interpretation of the Solution
PROBLEMS
OTHER READING
LESSON 11 The Fourier Series and Transform
PURPOSE OF LESSON
Discrete Frequency Spectrum of a Periodic Function
The Fourier Transform
NOTES
PROBLEMS
OTHER READING
LESSON 12 The Fourier Transform and Its Application to PDEs
PURPOSE OF LESSON
Useful Properties of the Fourier Transform
Property 1 (Fourier Transform Pair)
Property 2 (Linear Transformation)
Property 3 (Transformation of Parti'OI Derivatives)
Property 4 (Convolution Property)
Example of a Convolutlon of Two Functions
Solution of an lnltlal-Value Problem
NOTES
PIOILEMS
OTHER READING
LESSON 13 The Laplace Transform
PURPOSE OF LESSON
Properties of the Laplace Transform
Property 1 (Transform Pair)
Sufficient Conditions to Insure the Existence of a Laplace Transform
Property 2 (Transforms of Partial Derivatives)
Property 3 (Convolution Property)
Deflnttlon of the Finite Convolutlon
Heat Conduction In a Semi lnflnHe Medium
MOTii
PROBLEMS
OTHER READING
LESSON 14 Duhamel's Principle
PURPOSE OF LESSON
Heat Flow within a Rod with Temperature Fixed on the Boundaries
The Importance of Duhamel's Prlnclple
NOTES
PROBLEMS
OTHER READING
LESSON 15 The Convection Term Ux in the Diffusion Problems
PURPOSE OF LESSON
Laplace Transform Solution to the Convection Problem
NOTES
PROBLEMS
PART 3 Hyperbolic-Type Problems
LESSON 16 The One-Dimensional Wave Equation (Hyperbolic Equations)
PURPOSE OF LESSON
Vibrating-String Problem
Intuitive Interpretation of the Wave Equatton
NOTES
PROBLEMS
OTHER READING
LESSON 17 The D'Alembert Solution of the Wave Equation
PURPOSE OF LESSON
D´Alembert's Solution to the One-Dlmenslonal Wave Equation
Examples of the D'Alembert Solution
1. Motion of an Initial Sine Wave
2. Motion of a Simple Square Wave
3. Initial Velocity Given
NOTES
PROBLEMS
OTHER READING
LESSON 18 More on the D'Alembert Solution
PURPOSE OF LESSON
The Space-Time Interpretation of D' Alembert's Solution
Solutlon of the Seml-lnflntte String via the D'Alembert Formula
NOTES
PROBLEMS
OTHER READING
LESSON 19 Boundary Conditions Associated with the Wave Equation
PURPOSE OF LESSON
1. Controlled End Points
2. Force Given on the Boundaries
3. Elastic Attachment on the Boundaries
NOTES
PROBLEMS
OTHIR READING
LESSON 20 The Finite Vibrating String (Standing Waves)
PURPOSE OF LESSON
Separation-of-Varlables Solutlon to the Finite Vibrating String
NOTES
PROBLEMS
OTHER READING
LESSON 21 The Vibrating Beam (Fourth-Order PDE)
PURPOSE OF LESSON
The Simply Supported Beam
Sample Vibrating Beam
NOTES
PROBLEMS
OTHE11 READING
LESSON 22 Dimensionless Problems
PURPOSE OF LESSON
Converting a Diffusion Problem to Dimensionless Form
Transforming the Dependent Variable u --> U
Transforming the Space Variable x -->E
Transforming the Time Variable t --> T
Example of Transforming a Hyperbolic Problem to Dimenslonless Form
NOTES
PROBLEMS
OTHER READING
LESSON 23 Classification of PDEs (Canonical Form of the Hyperbolic Equation)
PURPOSE OF LESSON
Examples of Hyperbolic, Parabolic, and Elllptlc Equations
The Canonical Form of the Hyperbolic Equation
NOTES
PROBLEMS
LESSON 24 The Wave Equation in Two and Three Dimensions (Free Space)
PURPOSE OF LESSON
Waves In Three Dimensions
Two-Dlmenslonal Wave Equation
NOTES
PROBLEMS
OTHER READING
LESSON 25 The Finite Fourier Transforms (Sine and Cosine Transforms)
PURPOSE OF LESSON
Examples of the Sine Transform
Properties of the Transforms
Solvlng Problems via Finite Transforms
NOTES
PROBLEMS
OTHER READING
LESSON 26 Superposition (The Backbone of Linear Systems)
PURPOSE OF LESSON
Superposition Used to Break an IBVP Into Two Simpler Problems
Separation of Varlables and Integral Transforms as Superpositions
NOTES
PROBLEMS
OTHER READING
LESSON 27 First-Order Equations (Method of Characteristics)
PURPOSE OF LESSON
General Strategy for Solving the First-Order Equation
NOTE
PROBLEMS
OTHER READING
LESSON 28 Nonlinear First-Order Equations (Conservation Equations)
PURPOSE OF LESSON
Derivation of the Conservation Equation
Conservation Equation Applled to the Traffic Problem
The Nonlinear lnltlal-Value Problem
NOTES
PROBLEMS
OTHER READING
LESSON 29 Systems of PDEs
PURPOSE OF LESSON
Solution of the Linear System u, + Aux = 0
NOTES
PROBLEMS
OTHER READING
LESSON 30 The Vibrating Drumhead (Wave Equation in Polar Coordinates)
PURPOSE OF LESSON
Solution of the Helmholtz Eigenvalue Problem (Subproblem)
Interpretation of Jo(ko,r) ...
NOTES
PROBLEMS
OTHER READING
PART 4 ElIiptic-Type Problems
LESSON 31 The Laplacian (an Intuitive Description)
PURPOSE OF LESSON
Interpretations of v2 in Two Dimensions
Intuitive Meanings of Some Basic Laws of Physics
Changing Coordinates
NOTES
PROBLEMS
OTHER READING
LESSON 32 General Nature of Boundary Value Problems
PURPOSE OF LESSON
Steady-State Problems
Factoring out the Time Component in Hyperbolic and Parabolic Problems
The Three Main Types of BCs In Boundary-Value Problems
PROBLEMS
OTHER READING
LESSON 33 Interior Dirichlet Problem for a Circle
PURPOSE OF LESSON
Observations on the Dlrlchlet Solution
Poisson Integral Formula
NOTES
PROBLEMS
OTHER READING
LESSON 34 The Dirichlet Problem in an Annulus
PURPOSE OF LESSON
Product Solutions to Laplace's Equation
Worked Problems for the Dirichlet Problem in an Annulus
Exterior Dlrlchlet Problem
NOTES
PROBLEMS
OTHER READING
LESSON 35 Laplace's Equation in Spherical Coordinates (Spherical Harmonics)
PURPOSE OF LESSON
Speclal Cases of the Dlrlchlet Problem
NOTES
PROBLEMS
OTHER READING
LESSON 36 A Nonhomogeneous Dirichlet Problem (Green's Function)
PURPOSE OF LESSON
Potentials from Point Sources and Sinks
Poisson's Equation Inside a Clrcle
Finding the Potential Response G
Steps for Finding The Solution
NOTES
PROBLEMS
OTHER READING
PART 5 Numerical and Approximate Methods
LESSON 37 Numerical Solutions (Elliptic Problems)
PURPOSE OF LESSON
Finite-Difference Approximations
Dirichlet Problem Solved by the Finite-Difference Method
Numerical Algorithm for Solving the Dirichlet Problem (liebmann's method)
NOTES
PROBLEMS
OTHER READING
LESSON 38 An Explicit Finite-Difference Method
PURPOSE OF LESSON
The Expllclt Method for Parabolic Equations
Algorithm for the Explicit Method
NOTES
PROBLEMS
OTHER READING
LESSON 39 An Implicit Finite-Difference Method (Crank-Nicolson Method)
PURPOSE OF LESSON
The Heat-Flow Problem Solved by an lmpllclt Method
Implicit Algorithm for Heat Problem (39.2)
PROBLEMS
OTHER READING
LESSON 40 Analytic versus Numerical Solutions
PURPOSE OF LESSON
Meaning of Analytlc Solutlons
Meaning of Numerlcal Solutlons
Comparing Numerlcal and Analytlc Solutions
Advantages of the Analytic Solution
Advantages of Numerical Solutions
Parameter Identification (In Biology)
PROBLEMS
OTHER READING
LESSON 41 Classification of PDEs (Parabolic and Elliptic Equations)
PURPOSE OF LESSON
Reducing Parabolic Equations to Canonical Form
Transforming the Parabolic Equation uxx + 2uxv + uvv =0 Into Canonical Form
Reducing ElllpHc Equations to Canonical Form
Changing the Equation y2uxx + x2Uyy = 0 to Canonical Form
NOTES
PROBLEMS
OTHER READING
LESSON 42 Monte Carlo Methods (an Introduction)
PURPOSE OF LESSON
Evaluatlng an Integral
Random Numbers
NOTES
PROBLEMS
OTHER READING
LESSON 43 Monte Carlo Solution of Partial Differential Equations
PURPOSE OF LESSON
How Tour du Wino Is Played
Reason for Playing Tour du Wino
Solution of Laplace's Equation by the Monte Carlo Method
Solution to a Dirichlet Problem with Variable Coefficients
NOTES
PROBLEMS
OTHER READING
LESSON 44 Calculus of Variations (Euler-Lagrange Equations)
PURPOSE OF LESSON:
Minimizing the General Functional J[y)
NOTES
PROBLEMS
OTHER READING
LESSON 45 Variational Methods for Solving PDEs (Method of Ritz)
PURPOSE OF LESSON
Method of Ritz for Minimizing Functionals
NOTES
PROBLEMS
OTHER READING
LESSON 46 Perturbation Methods for Solving PDEs
PURPOSE OF LESSON
A Perturbation Solution of the Nonlinear Equation \Delta^2 u + u^2 = o
NOTES
PROBLEMS
OTHER READING
LESSON 47 Conformal-Mapping Solutions of PDEs
PURPOSE OF LESSON
Conformal Mappings and Complex Functions
Definition of a Conformal Mapping
Laplace's Equation in the Upper-Half Plane
Dirichlet Problem Between Two Nonconcentric Circles
NOTES
PROBLEMS
OTHER READING
ANSWERS TO SELECTED PROBLEMS
APPENDIX 1 Integral Transform Tables
Definitions of Functions in Tables
APPENDIX 2 PDE Crossword Puzzle
APPENDIX 3 Laplacian in Different Coordinate Systems
APPENDIX 4 Types of Partial Differential Equations
Index
