Uniquely provides fully solved problems for linear partial differential equations and boundary value problems
Partial Differential Equations: Theory and Completely Solved Problems utilizes real-world physical models alongside essential theoretical concepts. With extensive examples, the book guides readers through the use of Partial Differential Equations (PDEs) for successfully solving and modeling phenomena in engineering, biology, and the applied sciences.
The book focuses exclusively on linear PDEs and how they can be solved using the separation of variables technique. The authors begin by describing functions and their partial derivatives while also defining the concepts of elliptic, parabolic, and hyperbolic PDEs. Following an introduction to basic theory, subsequent chapters explore key topics including:
• Classification of second-order linear PDEs
• Derivation of heat, wave, and Laplace’s equations
• Fourier series
• Separation of variables
• Sturm-Liouville theory
• Fourier transforms
Each chapter concludes with summaries that outline key concepts. Readers are provided the opportunity to test their comprehension of the presented material through numerous problems, ranked by their level of complexity, and a related website features supplemental data and resources.
Extensively class-tested to ensure an accessible presentation, Partial Differential Equations is an excellent book for engineering, mathematics, and applied science courses on the topic at the upper-undergraduate and graduate levels
Author(s): Thomas Hillen, I. Ed Leonard, Henry van Roessel
Edition: 1
Publisher: Wiley
Year: 2012
Language: English
Pages: C, XIV, 678
Cover
S Title
PARTIAL DIFFERENTIAL EQUATIONS: Theory and Completely Solved Problems
Copyright
© 2012 by John Wiley & Sons. Inc.
ISBN 978-1-118-06330-9 (hardback)
QA377.H55 2012 515' .353-dc23
LCCN 2012017382
Contents
Preface
PART I: THEORY
Chapter 1: Introduction
1.1 Partial Differential Equations
11.2 Classification of Second-order Linear Pdes
1.3 Side Conditions
1.3.1 Boundary Conditions on an Interval
1.4 Linear Pdes
1.4.1 Principle of Superposition
1.5 Steady-state and Equilibrium Solutions
1.6 First Example for Separation of Variables
1.7 Derivation of the Diffusion Equation
1.7.1 Boundary Conditions
1.8 Derivation of the Heat Equation
1.9 Derivation of the Wave Equation
1.10 Examples of Laplace's Equation
1.11 Summary
1.11.1 Problems and Notes
Chapter 2: Fourier Series
2.1 Piecewise Continuous Functions
2.2 Even, Odd, and Periodic Functions
2.3 Orthogonal Functions
2.4 Fourier Series
2.4.1 Fourier Sine and Cosine Series
2.5 Convergence of Fourier Series
2.5.1 Gibbs' Phenomenon
2.6 Operations on Fourier Series
2.7 Mean Square Error
2.8 Complex Fourier Series
2.9 Summary
2.9.1 Problems and Notes
Chapter 3: Separation of Variables
3.1 Homogeneous Equations
3.1.1 General Linear Homogeneous Equations
3.1.2 Limitations of the Method of Separation of Variables
3.2 Nonhomogeneous Equations
3.2.1 Method of Eigenfunction Expansions
3.3 Summary
3.3.1 Problems and Notes
Chapter 4: Sturm Liouville Theory
4.1 Formulation
4.2 Properties of Sturm-liouville Problems
4.3 Eigenfunction Expansions
4.4 Rayleigh Quotient
4.5 Summary
4.5.1 Problems and Notes
Chapter 5: Heat, Wave, and Laplace Equations
5.1 One-dimensional Heat Equation
5.2 Two-dimensional Heat Equation
5.3 One-dimensional Wave Equation
5.3.1 d' Alembert's Solution
5.4 Laplace's Equation
5.4.1 Potential in a Rectangle
5.5 Maximum Principle
5.6 Two-dimensional Wave Equation
5.7 Eigenfunctions in Two Dimensions
5.8 Summary
5.8.1 Problems and Notes
Chapter 6: Polar Coordinates
6.1 Interior Dirichlet Problem for a Disk
6.1.1 Poisson Integral Formula
6.2 Vibrating Circular Membrane
6.3 Bessel's Equation
6.3.1 Series Solutions of Odes
6.4 Bessel Functions
6.4.1 Properties of Bessel Functions
6.4.2 Integral Representation of Bessel Functions
6.5 Fourier-bessel Series
6.6 Solution to the Vibrating Membrane Problem
6.7 Summary
6.7.1 Problems and Notes
Chapter 7: Spherical Coordinates
7.1 Spherical Coordinates
7.1.1 Derivation of the Laplacian
7.2 Legendre's Equation
7.3 Legendre Functions
7.3.1 Legendre Polynomials
7.3.2 Fourier-legendre Series
7.3.3 Legendre Functions of the Second Kind
7.3.4 Associated Legendre Functions
7.4 Spherical Bessel Functions
7.5 Interior Dirichlet Problem for a Sphere
7.6 Summary
7.6.1 Problems and Notes
Chapter 8: Fourier Transforms
8.1 Fourier Integrals
8.1.1 Fourier Integral Representation
8.1.2 Examples
8.1.3 Fourier Sine and Cosine Integral Representations
8.1.4 Proof of Fourier's Theorem
8.2 Fourier Transforms
8.2.1 Operational Properties of the Fourier Transform
8.2.2 Fourier Sine and Cosine Transforms
8.2.3 Operational Properties of the Fourier Sine and Cosine Transforms
8.2.4 Fourier Transforms and Convolutions
8.2.5 Fourier Transform of a Gaussian Function
8.3 Summary
8.3.1 Problems and Notes
Chapter 9: Fourier Transform Methods in Pdes
9.1 The Wave Equation
9.1.1 D' Alembert's Solution to the One-dimensional Wave Equation
9.2 The Heat Equation
9.2.1 Heat Flow in an Infinite Rod
9.2.2 Fundamental Solution to the Heat Equation
9.2.3 Error Function
9.2.4 Heat Flow in a Semi-infinite Rod: Dirichlet Condition
9.2.5 Heat Flow in a Semi-infinite Rod: Neumann Condition
9.3 Laplace's Equation
9.3.1 Laplace's Equation in a Half-plane
9.3.2 Laplace's Equation in a Semi-infinite Strip
9.4 Summary
9.4.1 Problems and Notes
Chapter 10: Method of Characteristics
10.1 Introduction to the Method of Characteristics
10.2 Geometric Interpretation
10.3 D' Alembert's Solution
10.4 Extension to Quasilinear Equations
10.5 Summary
10.5.1 Problems and Notes
PART II: EXPLICITLY SOLVED PROBLEMS
Chapter 11: Fourier Series Problems
Chapter 12: Sturm-liouville Problems
Chapter 13: Heat Equation Problems
Chapter 14: Wave Equation Problems
Chapter 15: Laplace Equation Problems
Chapter 16: Fourier Transform Problems
Chapter 17: Method of Characteristics Problems
Chapter 18: Four Sample Midterm Examinations
18.1 Midterm Exam 1
18.2 Midterm Exam 2
18.3 Midterm Exam 3
18.4 Midterm Exam 4
Chapter 19: Four Sample Final Examinations
19.1 Final Exam 1
19.2 Final Exam 2
19.3 Final Exam 3
19.4 Final Exam 4
Appendix A: Gamma Function
Bibliography
Index
