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.

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

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