An Excursion Through Partial Differential Equations

✍ Scribed by Svetlin G. Georgiev

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 425
• Series: Problem Books in Mathematics
• Category: Library

✦ 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