Boundary Value Problems for Linear Partial Differential Equations

✍ Scribed by Manuel Mañas, Luis Martínez Alonso

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 452
• Edition: 1
• Category: Library

✦ Synopsis

Boundary value problems play a significant role in modeling systems characterized by established conditions at their boundaries. On the other hand, initial value problems hold paramount importance in comprehending dynamic processes and foreseeing future behaviors. The fusion of these two types of problems yields profound insights into the intricacies of the conduct exhibited by many physical and mathematical systems regulated by linear partial differential equations.

Boundary Value Problems for Linear Partial Differential Equations provides students with the opportunity to understand and exercise the benefits of this fusion, equipping them with realistic, practical tools to study solvable linear models of electromagnetism, fluid dynamics, geophysics, optics, thermodynamics and specifically, quantum mechanics. Emphasis is devoted to motivating the use of these methods by means of concrete examples taken from physical models.

Features

• No prerequisites apart from knowledge of differential and integral calculus and ordinary differential equations.
• Provides students with practical tools and applications
• Contains numerous examples and exercises to help readers understand the concepts discussed in the book.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
About the authors
Chapter 1: Introduction
1.1. Partial Differential Equations
1.1.1. PDEs
1.1.2. Linear PDEs in Physics
1.1.3. Change of Variables
1.2. Boundary and Initial Conditions
1.2.1. Domains and Boundaries
1.2.2. Boundary Conditions
1.2.3. Initial Conditions
1.2.4. Smooth Functions
1.3. Local Solvability
1.4. Characteristics
1.4.1. First-Order PDEs
1.4.2. Characteristics and Discontinuities I
1.4.3. Second-Order PDEs
1.4.4. Characteristics and Discontinuities II
1.5. General Solutions
1.5.1. First-Order PDEs
1.5.2. The Hodograph Method
1.6. Remarkable Lives and Achievements
Chapter 2: Linear PDEs
2.1. Linear Differential Operators
2.1.1. Linear Differential Operators
2.1.2. Initial Condition and Boundary Operators
2.1.3. Green Functions
2.1.4. Eigenvalues and Eigenfunctions
2.2. One-Dimensional Eigenvalue Problems
2.2.1. Constant Coefficients Operators
2.3. Linear BVPs in Physics
2.3.1. Uniqueness
2.4. Exercises
2.4.1. Exercises with Solutions
Chapter 3: Separation of Variables Method
3.1. Separable Homogeneous Linear PDEs
3.1.1. Separable Differential Operators
3.1.2. Separable Homogeneous Linear PDEs
3.2. Separable Homogeneous BVP
3.3. Equations of Mathematical Physics
3.4. Helmholtz Equation
3.5. Remarkable Lives and Achievements
3.6. Exercises
3.6.1. Exercises with Solutions
3.6.2. Exercises
Chapter 4: Symmetric Differential Operators
4.1. Hilbert Spaces
4.1.1. Scalar Product of Functions
4.1.2. Change of Variables
4.2. Orthogonal Sets of Functions
4.2.1. Orthogonal Function Series Expansions
4.3. Green Functions
4.3.1. Inverting Differential Operators
4.4. Symmetric Differential Operators
4.4.1. Eigenvalues and Eigenfunctions
4.4.2. Spectral Representation of Green Functions
4.5. Sturm-Liouville Differential Operators
4.5.1. One-Dimensional Sturm-Liouville Operators
4.5.2. Singular SturmŁLiouville Operators
4.5.3. Three-Dimensional Sturm-Liouville Operators
4.6. Remarkable Lives and Achievements
4.7. Exercises
4.7.1. Exercises with Solutions
4.7.2. Exercises
Chapter 5: Fourier Analysis
5.1. Fourier Trigonometric Bases
5.2. Fourier Series
5.3. Convergence of Fourier Series
5.3.1. Advanced Results on Convergence
5.4. Fourier Transform
5.4.1. Fourier Transform Properties
5.5. Remarkable Lives and Achievements
5.6. Exercises
5.6.1. Exercises with Solutions
5.6.2. Exercises
Chapter 6: Eigenfunction Expansion Method
6.1. Preliminary Discussion: Restricted Inhomogeneities
6.1.1. Restricted Inhomogeneous Problems
6.1.2. Well- and Ill-Posed Problems
6.2. Application to Evolution Equations
6.3. General Discussion: Full Inhomogeneities
6.3.1. Illustrative Examples
6.4. Unbounded Domains and Fourier Transform
6.5. Exercises
6.5.1. Exercises with Solutions
6.5.2. Exercises
Chapter 7: Special Functions
7.1. Frobenius Series
7.2. Ordinary Points
7.2.1. Hermite Equation
7.2.2. Legendre Equation
7.3. Regular Singular Points
7.3.1. Euler Equation
7.3.2. Frobenius Series
7.4. Bessel Equation
7.4.1. The solution u1 and its Frobenius Series
7.4.2. The solution u2
7.4.3. Bessel Meets Sturm and Liouville
7.5. Euler Gamma Function
7.5.1. EulerŁs Definition as an Infinite Product
7.5.2. LegendreŁs Definition as an Integral
7.5.3. Weierstrass Infinite Product
7.6. Remarkable Lives and Achievements
7.7. Exercises
7.7.1. Exercises with Solutions
7.7.2. Exercises
Chapter 8: Cylindrical and Spherical BVPs
8.1. Cylindrical BVPs
8.1.1. Polar Coordinates
8.1.2. Quantum Particle in a Wedge
8.1.3. Quantum Particle in a Cylinder
8.1.4. Quantum Particle Trapped in a Corral
8.1.5. Quantum Particle in an Annulus
8.1.6. Fluid in a Pipe
8.1.7. Playing the Drum
8.1.8. Playing the Clarinet
8.2. Spherical BVPs
8.2.1. Separation of Variables for the Helmholtz Equation
8.2.2. Associated Legendre Functions
8.2.3. Spherical Harmonics
8.2.4. Cosmic Microwave Background
8.2.5. Periodic Table
8.2.6. Addition Formulas and Multipolar Expansion
8.2.7. Radial Equation
8.2.8. Quantum Particle in a Sphere
8.2.9. Fluid Inside a Sphere
8.2.10. Problems of Electrostatics and Fluid Mechanics
8.2.11. Charged Conducting Sphere
8.2.12. Fluid Around a Ball
8.3. Beyond Cylindrical and Spherical
8.3.1. EisenhartŁs Coordinate Systems
8.3.2. Stäckel Determinant and Separation of Variables
8.4. Exercises
8.4.1. Exercises with Solutions
8.4.2. Exercises
Bibliography
Subject Index
Index of Capsule Biographies