Boundary Value Problems: And Partial Differential Equations

✍ Scribed by David L. Powers

• Publisher: Academic Press
• Year: 2009
• Tongue: English
• Leaves: 518
• Edition: 6
• Category: Library

✦ Synopsis

Boundary Value Problems, Sixth Edition, is the leading text on boundary value problems and Fourier series for professionals and students in engineering, science, and mathematics who work with partial differential equations. In this updated edition, author David Powers provides a thorough overview of solving boundary value problems involving partial differential equations by the methods of separation of variables. Additional techniques used include Laplace transform and numerical methods.

The book contains nearly 900 exercises ranging in difficulty from basic drills to advanced problem solving exercises.

Professors and students agree that Powers is a master at creating examples and exercises that skillfully illustrate the techniques used to solve science and engineering problems.

✦ Table of Contents

Cover
Table of Contents
Preface
Chapter 0: Ordinary Differential Equations
0.1 Homogeneous Linear Equations
0.2 Nonhomogeneous Linear Equations
0.3 Boundary Value Problems
0.4 Singular Boundary Value Problems
Chapter 1: Fourier Series and Integrals
1.1 Periodic Functions and Fourier Series
1.2 Arbitrary Period and Half-Range Expansions
1.3 Convergence of Fourier Series
1.4 Uniform Convergence
1.5 Operations on Fourier Series
1.6 Mean Error and Convergence in Mean
1.7 Proof of Convergence
1.8 Numerical Determination of Fourier Coefficients
1.9 Fourier Integral
1.10 Complex Methods
1.11 Applications of Fourier Series and Integrals
1.12 Comments and References
Chapter Review
Miscellaneous Exercises
Chapter 2: The Heat Equation
2.1 Derivation and Boundary Conditions
2.2 Steady-State Temperatures
2.3 Example: Fixed End Temperatures
2.4 Example: Insulated Bar
2.5 Example: Different Boundary Conditions
2.6 Example: Convection
2.7 Sturm-Liouville Problems
2.8 Expansion in Series of Eigenfunctions
2.9 Generalities on the Heat Conduction Problem
2.10 Semi-Infinite Rod
2.11 Infinite Rod
2.12 The Error Function
2.13 Comments and References
Chapter Review
Miscellaneous Exercises
Chapter 3: The Wave Equation
3.1 The Vibrating String
3.2 Solution of the Vibrating String Problem
3.3 D'Alembert's Solution
3.4 One-Dimensional Wave Equation: Generalities
3.5 Estimation of Eigenvalues
3.6 Wave Equation in Unbounded Regions
3.7 Comments and References
Chapter Review
Miscellaneous Exercises
Chapter 4: The Potential Equation
4.1 Potential Equation
4.2 Potential in a Rectangle
4.3 Further Examples for a Rectangle
4.4 Potential in Unbounded Regions
4.5 Potential in a Disk
4.6 Classification and Limitations
4.7 Comments and References
Chapter Review
Miscellaneous Exercises
Chapter 5: Higher Dimensions and Other Coordinates
5.1 Two-Dimensional Wave Equation: Derivation
5.2 Three-Dimensional Heat Equation: Vector Derivation
5.3 Two-Dimensional Heat Equation: Double Series Solution
5.4 Problems in Polar Coordinates
5.5 Bessel's Equation
5.6 Temperature in a Cylinder
5.7 Vibration of a Circular Membrane
5.8 Some Applications of Bessel Functions
5.9 Spherical Coordinates; Legendre Polynomials
5.10 Some Applications of Legendre Polynomials
5.11 Comments and References
Chapter Review
Miscellaneous Exercises
Chapter 6: Laplace Transform
6.1 Definition and Elementary Properties
6.2 Partial Fractions and Convolutions
6.3 Partial Differential Equations
6.4 Some Nontrivial Examples
6.5 Comments and References
Miscellaneous Exercises
Chapter 7: Numerical Methods
7.1 Boundary Value Problems
7.2 Heat Problems
7.3 Wave Equation
7.4 Potential Equation
7.5 Comments and References
Miscellaneous Exercises
Bibliography
Appendix: Mathematical References
Trigonometric Functions
Hyperbolic Functions
Calculus
Table of Integrals
Answers to Odd-Numbered Exercises
Chapter 0
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Index