Course Objectives

Partial differential equations are an exciting tool: they help us learn more about the world around us and gain insights into many natural and technologies processes. Real-world applications of partial differential equations range from cancer research, climate modeling, and morphogenesis to aircraft design, option pricing, and visual-effects animations in movie productions.

Learning Goals

By the end of the course, you will be able to

• formulate questions about real-world problems and create partial differential equations models to answer them

• solve partial differential equations using analytical, numerical, and qualitative techniques

• identify similarities between waves in real-world problems and solutions of partial differential equations

• draw conclusions about real-world problems from partial differential equation models

• formulate and interpret mathematical statements

• support arguments using the theoretical foundations of partial differential equations 

Topics

We will cover the following content in APMA 0365 (with applications, programming, and theory topics highlighted in different colors):

• Definition of solutions and their visualization

• Overview of PDEs and their dynamics

• Advection + convection equations: method of characteristics

• Heat equation on R

• Fourier transform

• Wave equation on R

• Heat equation on bounded intervals

• Sturm-Liouville eigenvalue problems

• Heat and wave equation on bounded intervals (continued)

• Nonlinear PDEs: fronts + patterns

• Laplace's equation + maximum principles

• Method of characteristics (continued)

• Applications: fluids, wealth+income distributions, epidemiology