18.152 - Introduction to PDEs (Spring 2017)

## COURSE DESCRIPTION

The two primary goals of many pure and applied scientific disciplines can be summarized as follows: i) formulate/devise a collection of mathematical laws (i.e., equations) that model the phenomena of interest; ii) analyze solutions to these equations in order to extract information and make predictions. The end result of i) is often a system of partial differential equations (PDEs). Thus, ii) often entails the analysis of a system of PDEs. This course will provide an application-motivated introduction to some fundamental aspects of both i) and ii).

In order to provide a broad overview of PDEs, our investigation of i) will touch upon a diverse array of equations including a) the Laplace and Poisson equations of electrostatics; b) the diffusion equation, which models e.g. the spreading out of heat energy and chemical diffusion processes; c) the SchrÃ¶dinger equation, which governs the evolution of quantum-mechanical wave functions; d) the wave equation, which models e.g. the propagation of sound waves in the linear acoustical approximation; e) the Maxwell equations of electrodynamics; and other topics as time permits.

In our introduction to ii), we will study three important classes of PDEs that differ markedly in their qualitative and quantitative properties: elliptic, diffusive, and hyperbolic. In each case, we will discuss some fundamental analytical tools that will allow us to probe the nature of the corresponding solutions.

Prerequisites: 18.100; and one of 18.06, 18.700, or 18.701

Textbook: Partial Differential Equations in Action by Sandro Salsa (3rd Edition)

## RECENT UPDATE

[02.07.2017] Welcome to the start of the semester! Our first lecture falls on this date.

[02.10.2017] Problem Set #1 has been posted.

[02.15.2017] Problem Set #2 has been posted.

[02.22.2017] Problem Set #3 has been posted. Solutions to Problem Set #1 have been posted.

[03.1.2017] Problem Set #4 has been posted.

[03.15.2017] Problem Set #5 has been posted. Solutions to Problem Set #2 have been posted.

[03.21.2017] Solutions to Problem Set #3 have been posted.

[03.22.2017] Problem Set #6 has been posted.

[03.23.2017] Solutions to Problem Set #4 have been posted.

[03.24.2017] Practice Midterm and Formula Sheet have been posted (click on the "Exams" link at the top).

[03.24.2017] Note: Your midterm will take place in class on Thursday April 6.

[04.03.2017] Solutions to the Practice Midterm have been posted (click on the "Exams" link at the top).

[04.06.2017] Solutions to Problem Set #5 have been posted.

[04.12.2017] Problem Set #7 has been posted.

[04.20.2017] Problem Set #8 has been posted. Solutions to Problem Set #6 have been posted.

[04.26.2017] Problem Set #9 has been posted.

[04.27.2017] Solutions to Problem Set #7 have been posted.

[05.03.2017] Problem Set #10 (the last one!) has been posted.

[05.04.2017] Solutions to Problem Set #8 have been posted.

[05.06.2017] Solutions to the in-class Midterm have been posted.

[05.11.2017] Solutions to Problem Set #9 have been posted.

[05.15.2017] Practice final exam with solutions has been posted.

[05.17.2017] Solutions to Problem Set #10 have been posted.