Rouben Rostamian

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MATH 404: Introduction to Partial Differential Equations I

Spring 2024 course information

Class Time/Place:	MoWe 1:00pm–2:15pm, SOND 207
Office:	MP 402
Phone:	410–455–2458
Email:	rostamian@umbc.edu
Office hours:	MoWe 3:00pm–4:00pm

Course content

This is a first course on Partial Differential Equations (PDEs). It provides an overview of the subject from an elementary point of view and introduces some classical techniques for solving initial and boundary value problem. An integral part of the course is the analysis of physical phenomena such as heat conduction, wave propagation, etc., and their formulations as PDEs.

Textbook

I am writing (not finished yet) a textbook on PDEs which I naturally believe is better than anything else that I have seen. I will share what I have with you as a PDF file which you will be able to read on your computer screen. I will continue adjusting/expanding the textbook throughout the semester.

The subject of PDEs deals largely with phenomena that change/evolve over time. The time-dependence of the solutions does not come across very well in traditional textbooks which are limited to printed media. A novel feature of my textbook is that it has animations embedded inside the PDF! I hope that this makes for a livelier and more attractive presentation of the material. Have a look at this sample animations extracted from the textbook.

If you still feel a need for a traditional textbook, I recommend what I have used in prior semesters in this course:
J. David Logan, Applied Partial Differential Equations, Springer, (3rd edition, 2015). Available at amazon.com, and elsewhere.

Prerequisites

You must have completed MATH 225 (Ordinary Differential Equations) and MATH 251 (Multivariable Calculus) with a grade of "C" or better before you can take this class. You should be comfortable with

• Math 225: Separation of variables, integrating factors, and solving second order linear equations;
• Math 251: The directional derivative, gradient vector, vector fields, the Divergence Theorem, and Green's identities.

Homework and exams

I will enter homework assignments on this web page shortly after each class. You are expected to work on all the assigned problems and make sure that you can solve them. Homework assigned on Monday and Wednesday of each week is due on the Wednesday of the following week unless announced otherwise. Selected problems from each homework set will be graded.

Late homework won't be accepted; please don't ask for exceptions. However, up to two missed homeworks will be discarded in order to accommodate unanticipated events.

There will be two exams during the semester, and a final exam at the end of the semester. Exams 1 and 2 will cover the first third and second third of the course; they will be given during the regularly scheduled class times. The Final Exam will be comprehensive—it will cover the entire course—however it will put much greater emphasis on the material toward the later parts of the course.

Homework:	20%
Exam 1:	25%
Exam 2:	25%
Final Exam:	30%

Overall course score will be calculated according to the weights attached to various components as shown in the adjacent table. Letter grades will be determined according to:

if { grade ≥ 85: A}
else if { grade ≥ 75: B}
else if { grade ≥ 65: C}
else if { grade ≥ 55: D}
else F

I will make and grade the exams in a fair and reasonable way, but sorry, no "curving" in this course.

Bonus points

You may earn bonus points by going beyond the course's minimal expectations. I will set things up so that your homework may count as much as 30% (up from 20%) when padded with bonus points. There are several ways that you can earn bonus points.

Neat presentation of your work

Is your presentation of the solutions to the homework problems self-contained and self-explanatory? If you mail your work to someone who is not in this class, will that person be able to understand the context and follow your line of reasoning at your absence?

Hint: Have a look at any math textbook to see how math is explained. Splattering equations across a page is not an explanation.

Handwritten or typed?

Much writing in the 21st century is done on a computer keyboard. Scribbling on a paper is so passé. Can you write mathematics on a computer? You should be able to. How will you explain it to your future boss that after 16 years of schooling you still don't know how to write math on a computer.

Hint: LaTeX is the universal software for writing mathematics. This course provides you with the opportunity to learn LaTeX or further your skills if you already know it.

Graphics

Solving a typical PDE problem leads to an answer like \[ u(x,t) = \sum_{n=1}^\infty b_n e^{-(n\pi/L)^2 t} \sin \frac{n\pi x}{L}. \] The solution $u(x,t)$ expresses the temperature in a heated metal bar of length $L$ at the point $x$ at time $t$. It'd be a shame arriving at that solution after four pages of calculations and then having no idea of what that temperature distribution looks like. Do you know how to plot such a function on your computer? If so, then you can enrich your presentation by including a plot of that solution.

There are many software tools for producing graphics. Maple, Mathematica, and Matlab are possiblities. Some students have had a great deal of success with the browser-based Desmos, like this.

About Adobe Acrobat Reader

You may read the textbook's PDF file on whatever device that you normally read PDFs. To be able to see the embedded animations, however, you will need to install the Adobe Acrobat Reader on your computer. The textbook's animations are an essential part of the material's presentation; you will be missing a lot if you can't see the animations.

Adobe Acrobat Reader is available for free download for Linux, Mac, and Windows. Follow these links for downloading and installing instructions:

Installing on Debian-based Linux, such as Ubuntu

How to Install Adobe Acrobat Reader on Ubuntu 22.04

Installing on Mac OS and Windows

Go to Adobe's website

About L^aT_eX

Technical writing is easy with the right tool. L^aT_eX is the computer software of choice for technical writing, especially for articles that contain a lot of mathematics. L^aT_eX has been around since 1980s (that's over 40 years!) so it wouldn't be a waste of your time and effort to invest in learning it. What you learn here will stay with you for a lifetime.

There are tons of L^aT_eX tutorials on the web but most of them are trash. I advise you stay clear of them since you are likely to pick up wrong ideas which will be difficult to get rid of later on.

Once you are convinced that L^aT_eX is the right tool for you, you should consider buying its manual and keeping it within an arm's reach at all times. The very readable manual, written by the creator of L^aT_eX, is:

L^aT_eX: A Document Preparation System by Leslie Lamport.

How to get L^aT_eX

L^aT_eX is an open source software; it may be obtained freely and installed on any computer platform.

Linux

All Linux distributions come with L^aT_eX. If you have your own Linux machine, you may install/activate L^aT_eX with a few mouse clicks. Ask me if you don't know how.

Mac

Download and install MacTeX.

Windows

Download and install MiKTeX.

Another possibility would be to write your L^aT_eX documents within your web browser through application called Overleaf. Being a web application, Overleaf does not require installing software on your computer. The downside is that you will be tied to Overleaf forever. (Think of 10 or 20 years from now.) I think you will be better off becoming independent by installing the L^aT_eX on your computer and getting the full possession of it.

Course calendar and activity log

 

Student Disability Services (SDS)

Services for students with disabilities are provided for all students qualified under the Americans with Disabilities Act of 1990, the ADAA of 2009, and Section 504 of the Rehabilitation Act who request and are eligible for accommodations. The Office of Student Disability Services is the UMBC department designated to coordinate accommodations that would allow for students to have equal access and inclusion in their courses.

UMBC Honors Code

By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal.

The PDF document UMBC Policy for Undergraduate Student Academic Conduct spells out the official academic integrity policies for undergraduates.

A message from the Office of Equity and Civil Rights

This Title IX policy is provided by UMBC's Office of Equity and Civil Rights.