Partial Differential Equations and Applications: A Bridge for Students and Researchers in Applied Sciences offers a unique approach to this key subject by connecting mathematical principles to the latest research advances in select topics. Beginning with very elementary PDEs, such as classical heat equations, wave equations and Laplace equations, the book focuses on concrete examples. It gives students basic skills and techniques to find explicit solutions for partial differential equations.
As it progresses, the book covers more advanced topics such as the maximum principle and applications, Green’s representation, Schauder’s theory, finite-time blowup, and shock waves. By exploring these topics, students gain the necessary tools to deal with research topics in their own fields, whether proceeding in math or engineering areas.
Key Features
- Class tested over multiple years with advanced undergraduate and graduate courses
- Features many concrete examples and chapter exercises
- Appropriate for advanced undergraduate and graduate courses geared to math and engineering students
- Requires minimal background beyond advanced calculus and differential equations
Preface
CHAPTER 1 Basics of partial differential equations
1.1 Introduction
1.2 Fundamental PDE questions: Well-posedness and ill-posedness
1.3 Some important equations and systems
1.4 Classification of second-order partial differential equations
1.5 Some elementary formulas and inequalities
1.6 Notes and remarks
1.7 Exercises
CHAPTER 2 Function spaces and the Fredholm Alternative
2.1 Banach spaces and Hilbert spaces
2.2 Function spaces
2.3 Completeness and series representation
2.4 Fourier series
2.5 The contraction mapping principle and applications
2.6 The continuitymethod
2.7 The Fredholm Alternative and applications
2.8 The Riesz representation and the Lax–Milgram theorem
2.9 Notes and remarks
2.10 Exercises
CHAPTER 3 Eigenvalue problems and eigenfunction expansions
3.1 The method of separation of variables
3.2 The Sturm–Liouville theory
3.3 The main theorem in the Sturm–Liouville theory
3.4 Eigenvalues problems in several space dimensions
3.5 Boundary eigenvalue problems
3.6 Convergence of function series
3.7 Notes and remarks
3.8 Exercises
CHAPTER 4 The heat equation
4.1 The mathematical model of heat conduction
4.2 Solution of the heat equation in one-space dimension
4.3 Solution of the heat equation in higher-space dimension
4.4 The well-posedness and energy estimates
4.5 A qualitative property: The maximum principle
4.6 Long-time behaviors of solutions
4.7 The comparison principle
4.8 Notes and remarks
4.9 Exercises
CHAPTER 5 The wave equation
5.1 The mathematical model of a vibrating string
5.2 Solutions of the wave equation in one-space dimension
5.3 Wave propagation in several space dimensions
5.4 Solution of the wave equation in higher space dimensions
5.5 The well-posedness and energy estimates
5.6 A qualitative property: The finite propagation speed
5.7 The Cauchy problem for the wave equation
5.8 Notes and remarks
5.9 Exercises
CHAPTER 6 The Laplace equation
6.1 Some mathematical models of the Laplace equation
6.2 Series solution for the Laplace equation
6.3 The well-posedness of the Laplace equation
6.4 A qualitative property: The mean-value formula
6.5 Themaximumprinciple and applications
6.6 An L8(Q)-estimate: Moser’s iteration method
6.7 Notes and remarks
6.8 Exercises
CHAPTER 7 The Fourier transform and applications
7.1 Definition of the Fourier transform
7.2 Properties of the Fourier transform
7.3 Applications to the Laplace equation
7.4 Applications to the heat equation .
7.5 Application to the wave equation
7.6 Generalized Fourier transform
7.7 Notes and remarks
7.8 Exercises
CHAPTER 8 The fundamental solution and Green’s representation
8.1 Introduction to the fundamental solution
8.2 The fundamental solution of the Laplace equation
8.3 Green’s functions for the Laplace operator
8.4 The fundamental solution of the heat equation
8.5 Green’s function of the heat equation
8.6 Finite-time blowup and extinction .
8.7 Notes and remarks
8.8 Exercises
CHAPTER 9 Systems of first-order partial differential equations
9.1 Some first-order PDE models in physical sciences
9.2 The characteristic method
9.3 Well-posedness for a system of first-order PDEs
9.4 The solution of Maxwell’s equations
9.5 The conservation law and shock waves
9.6 A class of semilinear wave equations
9.7 Remarks and notes
9.8 Exercises
APPENDIX A Some essential results in ordinary differential equations
A.1 The solution of a first-order differential equation and stability
A.2 Basic results for second-order differential equations
A.3 Dynamics of the solution of a linear system of ODEs
APPENDIX B Sobolev spaces
B.1 Weak derivatives and Sobolev spaces
B.2 Smooth approximation and Sobolev embedding
B.3 Sobolev spaces with t-variable
