1CHAPTER 1 Introduction to Partial Differential Equations
4010.1 Introduction
21.1 Definition and Classification
4110.2 Perturbation Methods
31.2 Examples and Applications
4210.3 Asymptotic Expansions
41.3 Basic Concepts and Notation
4310.4 WKB Approximation
5CHAPTER 2 First-Order Partial Differential Equations
4410.5 Applications in Boundary Layer Theory, Shock Waves, and Singular Perturbations
62.1 Characteristics and Canonical Forms
45CHAPTER 11 Multi-dimensional Problems
72.2 Methods of Solution
4611.1 Introduction
82.3 Conservation Laws
4711.2 Applications in Fluid Dynamics, Solid Mechanics, and Electromagnetism
9CHAPTER 3 Second-Order Elliptic Equations
4811.3 Solutions in Cartesian, Cylindrical, and Spherical Coordinates
103.1 Laplace’s Equation
4911.4 Applications in Heat Transfer, Wave Propagation, and Electrostatics
113.2 Boundary Value Problems
50CHAPTER 12 Applications of Partial Differential Equations
123.3 Fourier Series and Eigenvalue Problems
5112.1 Heat Transfer
13CHAPTER 4 Second-Order Parabolic Equations
5212.2 Fluid Dynamics
144.1 Heat Equation
5312.3 Solid Mechanics
154.2 Initial and Boundary Value Problems
5412.4 Electromagnetism
164.3 Separation of Variables
5512.5 Quantum Mechanics
17CHAPTER 5 Second-Order Hyperbolic Equations
5612.6 Relativity Theory
185.1 Wave Equation
5712.7 Applications in Engineering, Physics, and Mathematics
195.2 Characteristics and Solutions
58CHAPTER 13 Applications in Engineering
205.3 Shock Waves
5913.1 Fluid Dynamics and Aerodynamics
21CHAPTER 6 Numerical Methods for Partial Differential Equations
6013.2 Heat Transfer and Thermodynamics
226.1 Finite Difference Methods
6113.3 Electromagnetics and Wave Propagation
236.2 Finite Element Methods
6213.4 Structural Mechanics and Solid Dynamics
246.3 Computational Techniques
6313.5 Geophysics and Environmental Engineering
25CHAPTER 7 Advanced Topics in Applied Partial Differential Equations
64CHAPTER 14 Applications in Physics
267.1 Nonlinear Partial Differential Equations
6514.1 Quantum Mechanics and Wave Functions
277.2 Asymptotic Methods
6614.2 Relativity Theory and Gravitational Waves
287.3 Multi-dimensional Problems
6714.3 Electrodynamics and Quantum Field Theory
29CHAPTER 8 Applications in Science and Engineering
6814.4 Statistical Mechanics and Kinetic Theory
308.1 Fluid Dynamics
69CHAPTER 15 Applications in Biology and Medicine
318.2 Heat Conduction
7015.1 Mathematical Modeling of Biological Systems
328.3 Electromagnetic Fields
7115.2 Diffusion and Reaction-Diffusion Equations
33CHAPTER 9 Green's Functions
7215.3 Biomechanics and Medical Imaging
349.1 Introduction
73CHAPTER 16 Applications in Finance and Economics
359.2 Properties
7416.1 Black-Scholes Equation and Option Pricing
369.3 Derivation and Physical Interpretation
7516.2 Financial Mathematics and Risk Analysis
379.4 Applications in Boundary Value Problems
7616.3 Economic Modeling and Game Theory
389.5 Green's Functions for the Heat Equation, Wave Equation, and Laplace's Equation
77Glossary
39CHAPTER 11 Asymptotic Methods
78Index
