Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Description

The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.

Prerequisites

Differential Equations (18.03 or 18.034); Complex Variables with Applications (18.04), Advanced Calculus for Engineers (18.075), or Functions of a Complex Variable (18.112)

Basic theory of one complex variable and ordinary differential equations (ODE). No prior knowledge of partial differential equations theory is assumed.

Course Outline

This is just to give you an idea of the flavor. Some things may be covered in more detail than this implies, or the reverse.

• Introduction. Terminology; boundary and initial value problems; well- and ill-posed problems.
• Linear PDE. Review and classification; the Laplace, wave and diffusion equations; the Klein-Gordon equation; more on characteristics; standard methods: separation of variables, integral transforms, Green's functions; potential scattering; special topics in conformal mapping; dispersion and diffusion; dimensional analysis and self-similarity; regular and singular perturbation theory; asymptotics for complete solutions; geometrical optics and WKB eikonal equation; high-frequency expansions; caustics.
• More on nonlinear PDE. Equations that convert into linear PDE; some exactly solvable cases; Burgers' equation; dimensional analysis and similarity; traveling waves; nonlinear diffusion and dispersion; the KdV, nonlinear Schrödinger and Sine-Gordon equations; reaction-diffusion equations; Fisher's equation; singular perturbations: boundary layers, homogenization, weakly nonlinear geometrical optics, etc.; Solitons; Backlund transformations; Painlevé conjecture.
• Variational Methods. First and second variation; Euler-Lagrange equation; constraints.
• Free-boundary value problems. Formulation; perturbation theory; more on water waves; method of extended gradient; materials surface evolution; some open problems.

Textbook

There is no required textbook for this course. A list of a few recommended textbooks can be found in the Readings section.

Exams and Assignments

There will be two exams, one at midterm and one during the last week of class. There is no final exam. There will be 5 problem sets (one every 1-2 weeks).

Grading

MATLAB®

I strongly urge you to become proficient in MATLAB® It is a very useful tool, and I may use it from time to time to illustrate points in the course. In addition, the problem sets may require some computations, easily done if you know MATLAB. See the Related Resources section for more information about MATLAB.