Course Coordinator: 
Qing Liu
Partial Differential Equations

Driven by rapid developments in science and engineering, the theory of partial differential equations (PDE) has demonstrated its importance in solving practical problems arising in many fields. Through lectures and hands-on assignments, this course introduces a variety of PDEs with emphasis on the theoretical aspects and related techniques to find solutions and understand their analytic properties.  It familiarizes students with basic concepts and modern techniques for the formulation and solution of various PDE problems.  Main topics include the method of characteristics for first order PDE, formulation and solutions to the wave equation, heat equation and Laplace equation, and classical tools to study properties of these PDEs.
Target students For students who intend to learn mathematical details of the theory and use it to understand PDE models with more specific applications.

• State the definitions of basics notions and adopt the terminology in partial differential equations. • Apply the method of characteristics to solve nonlinear first order partial differential equations. • Use formulas to compute explicit solutions to second order equations including the wave equation, heat equation and Laplace equation. • Apply analytic methods appropriately to investigate partial differential equations. • Adopt proof techniques correctly to verify properties of solutions rigorously.
Course Content: 

Basic concepts of PDEs
First-order PDEs and Method of characteristics
General first-order PDEs
Wave equation and D’Alembert’s formula
Conservation of energy and Duhamel’s principle
Heat equation and its fundamental solution
Energy method
Maximum principle for diffusion equations
Laplace equation and harmonic functions
Mean value property and maximum principle
Green's function
Fourier series
Separation of variables

Course Type: 
Exam: 50%, Homework: 50%
Text Book: 
Partial Differential Equations, An Introduction to Theory and Applications, Michael Shearer, Rachel Levy, Princeton University Press, 2015
Reference Book: 
Partial Differential Equations, Lawrence C. Evans, 2nd edition, American Mathematical Society, 2022
Applied Partial Differential Equations: An Introduction, Alan Jeffrey, Academic Press, 2002
Partial Differential Equations: A First Course, Rustum Choksi, American Mathematical Society, 2022
Prior Knowledge: 

Single-variable and multi-variable calculus, Linear algebra, ordinary differential equations,  real analysis, or equivalent knowledge.