An Excursion Through Partial Differential Equations (Problem Books in Mathematics)

✍ Scribed by Svetlin G. Georgiev

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 425
• Edition: 1st ed. 2023
• Category: Library

✦ Synopsis

Presenting a rich collection of exercises on partial differential equations, this textbook equips readers with 96 examples, 222 exercises, and 289 problems complete with detailed solutions or hints. It explores a broad spectrum of partial differential equations, fundamental to mathematically oriented scientific fields, from physics and engineering to differential geometry and variational calculus.

Organized thoughtfully into seven chapters, the journey begins with fundamental problems in the realm of PDEs. Readers progress through first and second-order equations, wave and heat equations, and finally, the Laplace equation. The text adopts a highly readable and mathematically solid format, ensuring concepts are introduced with clarity and organization.

Designed to cater to upper undergraduate and graduate students, this book offers a comprehensive understanding of partial differential equations. Researchers and practitioners seeking to strengthen their problem-solvingskills will also find this exercise collection both challenging and beneficial.

✦ Table of Contents

Preface
Contents
1 General Introduction
1.1 Introduction
1.2 Classification
1.3 History and Applications
1.4 Advanced Practical Problems
2 First Order Partial Differential Equations
2.1 Classifications of First Order Partial Differential Equations
2.2 Solvability of Quasilinear First Order PDEs
2.3 The Cauchy Problem for Quasilinear First Order PDEs
2.4 The Pfaff Equation
2.5 Some Special Systems
2.6 Advanced Practical Problems
3 Classifications of Second Order Partial Differential Equations
3.1 Classifications
3.2 Advanced Practical Problems
4 Classifications and Canonical Forms for Linear Second Order Partial Differential Equations
4.1 Classifications and Canonical Forms for Linear Second Order Partial Differential Equations in Two Independent Variables
4.1.1 The Elliptic Case
4.1.2 The Parabolic Case
4.1.3 The Hyperbolic Case
4.2 Classification and Canonical Form of Second Order Linear Partial Differential Equations in n Independent Variables
4.3 Classification of First Order Systems with Two Independent Variables
4.4 Advanced Practical Problems
5 The Laplace Equation
5.1 Basic Properties of Elliptic Problems
5.2 The Fundamental Solution
5.3 Strong Maximum Principle: Uniqueness
5.4 The Green Function of the Dirichlet Problem
5.5 Separation of Variables
5.5.1 Rectangles
5.5.2 Circular Domains
5.6 Advanced Practical Problems
6 The Heat Equation
6.1 The Cauchy Problem
6.2 The Method of Separation of Variables
6.3 The Mean Value Formula
6.4 The Weak and Strong Maximum Principles
6.5 The Maximum Principle for the Cauchy Problem
6.6 Advanced Practical Problems
7 The Wave Equation
7.1 The One Dimensional Wave Equation
7.1.1 The Cauchy Problem and the d'Alambert Formula
7.1.2 The Cauchy Problem for the Nonhomogeneous Wave Equation
7.1.3 Separation of Variables
7.1.3.1 Homogeneous IBVPs
7.1.3.2 Nonhomogeneous IBVP with Homogeneous Boundary Conditions
7.1.3.3 Nonhomogeneous IBVPs with Nonhomogeneous Boundary Conditions
7.1.4 The Energy Method: Uniqueness
7.2 The Wave Equation in R3
7.2.1 Radially Symmetric Solutions
7.2.2 The Cauchy Problem
7.3 The Two Dimensional Wave Equation
7.4 Advanced Practical Problems
8 Solutions, Hints and Answers to the Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
9 Solutions, Hints and Answers to the Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7.
Index