# MAT 5673

## Course description

Partial differential equations arise in many different areas as one tries to describe the behavior of a system ruled by some law. Typically, this has to do with some physical process such as heat diffusion in a material, vibrations of a bridge, circulation of fluids, the behavior of microscopic particles or the evolution of the universe as a whole. Modeling by means of partial differential equations has been successful in other disciplines as well, like in the case of the Black-Scholes equation for stock options pricing and the Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.

Textbooks:

• P. Olver: Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics) 1st ed. 2014, Corr. 3rd printing 2016
• L.C. Evans: Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition

## Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

Introduction and classification of PDE, Calculus review

Multivariable Calculus, Chain Rule

• Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
Week 2

Applied examples of PDE

Multivariable Calculus, Chain Rule

• Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
Week 3

The method of characteristics for first-order quasilinear equations

Multivariable Calculus, Chain Rule

• Solving quasilinear first-order equations using the method of characteristics
Week 4

The method of characteristics for first-order fully nonlinear equations

Multivariable Calculus, Chain Rule

• Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
Week 5

Heat and wave equation on the whole real line

Differentiation of integrals with respect to a parameter, integration by parts

• Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
Week 6

Initial-boundary value problem for heat and wave equation I

Partial derivatives, chain rule

• Separation of variables method for heat and wave equation
Week 7

Initial-boundary value problem for heat and wave equation II, introduction to Fourier series

Partial derivatives, chain rule

• Forming more general solutions out of infinite superposition of basic solutions
Week 8

Introduction to Fourier series

Infinite series

• Orthonormal systems of functions, spectral method for the wave and heat equation
Week 9

Schroedinger equation

Complex numbers

• Basic properties of Schroedinger equation, particle in a potential well
Week 10

Qualitative properties of PDE's

Differentiation of integrals with respect to parameter

• Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
Week 11

Introduction to numerical methods for PDE (optional)

Derivatives, Calculus, Matrices, Linear Algebra

• Basic finite difference schemes for first-order quasilinear equations, CFL condition
Week 12

Introduction to the Laplace and Poisson equation

• Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
Week 13

Introduction to the Calculus of Variations

Differentiation of an integral with respect to a parameter, parametric surfaces

• Compute the variational derivative of a functional
Week 14