Syllabus and Course Details

This page constitutes the official syllabus and course policy.

Course Information

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Title: Partial Differential Equations
Code:: Math 7530 - Fall 2024
Lecture: MWF 2:00pm - 2:50pm - Richardson 200

Instructor Information

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Instructor: Sam Punshon-Smith
Office: Gibson 423
Email: spunshonsmith@tulane.edu
Office Hours:

• Wednesday: 10:00 AM - 11:00 AM
• Friday: 3:00 PM - 4:00 PM

Description

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This course delves into the core principles and techniques for analyzing partial differential equations (PDEs). Students will gain a strong foundation in the theory of PDEs, including the study of classical weak and strong maximum principles, Sobolev spaces, and weak solutions. The course will cover the application of these concepts to fundamental linear PDEs, such as elliptic, parabolic, and hyperbolic equations.

For a detailed outline of the course topics by week, see the course schedule page.

Course Goals

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By the end of the course, students will

• Feel comfortable with basic analysis of fundamental linear partial differential equations
• Develop a foundation of skills that will allow to study more advanced topics in partial differential equations
• Know how to solve certain partial differential equations when exact formulas exist
• Develop techniques for analyzing partial differential equations when exact formulas do not exist
• Learn fundamental techniques and strategies for proving well-posedness of various linear partial differential equations

Learning Objectives

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After successfully completing this class, students will be able to:

• Solve basic transport equations via the method of characteristics
• Use Green's functions to analyze and represent solutions to elliptic/parabolic equations on general domains
• Use analytical techniques and principles, like the weak/strong maximum principle, Hopf's lemma, the mean value property, and Harnack's inequality, the Dirichlet's principle and energy methods to study and linear elliptic/ parabolic equations and their uniqueness and regularity properties.
• Use D'Alembert's formula, Kirchoff and Poisson's formula, energy methods and the cone of dependence to represent and analyze solutions to the wave equation.
• Understand the fundamentals of Sobolev spaces and their embeddings and compactness properties
• Use Sobolev spaces to obtain a priori estimates and existence of weak solutions to certain linear PDE via Lax-Milgram and Fredholm alternative

Prerequisites

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Must have a solid background in analysis and vector calculus. Some familiarity with basic functional analysis of Hilbert and Banach spaces and weak topology is needed to treat the material in Sobolev spaces.

Course Materials

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The only material required for the course is the following textbook by L.C Evans:

• Required Text Evans, Lawrence C. Partial Differential Equations (Second Edition). United States, American Mathematical Society, 2010. Amazon Link

This book is a classic text for learning graduate PDE and a highly recommended addition to your library. You can find copies online if you look (not very hard). Make sure you are using the second edition!

The material for the course will closely follow this text and problems will be assigned out of it.

Communication and Help

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I will be available during regular business hours on weekdays via email and during my office hours listed above. Please give me at least 24 hrs to respond to emails. If you cannot attend my office hours, you may request to schedule a one-on-one appointment with me over email.

Assignments and Exams

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Problem Sets

There will be weekly problem sets on most weeks. They will be assigned on the course schedule one week or more in advance. Problems must be written clearly and legibly with sufficient exposition and explanation. Illegible solutions or solutions without sufficient explanation will be penalized. It is highly recommended (though not required) for you to write your assignments in LaTeX. All problems sets are due in lecture at the beginning (due dates are listed on the schedule) in paper form. Assignments are listed in the course schedule.

Late Assignments

Late work will be deducted 10% per twenty-four hour period that elapses after the due date (this includes weekends). You may submit electronic versions of your assignment of your assignment over the weekend to avoid further penalty, but must submit an identical physical copy later. If foreseen or unforeseen circumstances prevent you from completing an assignment on time, you may request an extension. Extensions must be requested in advance of the due date. If the situation warrants an extension, we will determine a new due date for the assignment based on your individual circumstances.

Exams

There will also be two exams, one midterm exam and one final exam. The first exam will in person and second will be take home. The exam dates listed on the course schedule. Topics covered are listed on the course schedule. The dates will be fixed at least two weeks before the exam. The final will be cumulative.

Grading

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The breakdown and percentages for all graded materials are described in the following table:

Grades will tentatively be assigned standard cutoffs. Grade modifiers (+-) may be added to grades within ~1% of the boundary points on a case by case basis. Eg: 89% = B+

Code of Academic Conduct

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The Code of Academic Conduct applies to all undergraduate students, full-time and part-time, in Tulane University. Tulane University expects and requires behavior compatible with its high standards of scholarship. By accepting admission to the university, a student accepts its regulations (i.e., Code of Academic Conduct and Code of Student Conduct) and acknowledges the right of the university to take disciplinary action, including suspension or expulsion, for conduct judged unsatisfactory or disruptive.

ADA/Accessibility

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Tulane University is committed to offering classes that are accessible. If you anticipate or encounter disability-related barriers in a course, please contact the Goldman Center for Student Accessibility to establish reasonable accommodations. If approved by Goldman, make arrangements with me as soon as possible to discuss your accommodations so that they may be implemented in a timely fashion. I will never ask for medical documentation from you to support potential accommodation needs. Goldman Center contact information: Email: goldman@tulane.edu; Phone (504) 862-8433; Website: accessibility.tulane.edu

Diversity

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This course is designed to support an inclusive learning environment where diverse perspectives are recognized, respected and seen as a source of strength. It is our intent to provide materials and activities that are respectful of various levels of diversity: mathematical background, gender, sexuality, disability, age, socioeconomic status, ethnicity, race, and culture.