• Four Science Terps Awarded 2025 Goldwater Scholarships

    Four undergraduates in the University of Maryland’s College of Computer, Mathematical, and Natural Sciences (CMNS) have been awarded 2025 scholarships by the Barry Goldwater Scholarship and Excellence in Education Foundation, which encourages students to pursue advanced study and research careers in the sciences, engineering and mathematics.  Over the last 16 years, UMD’s nominations… Read More

  • Announcing the Winners of the Frontiers of Science Awards

    Congratulations to our colleagues who won the 2025 Frontiers of Science Award: - Dan Cristofaro-Gardiner, for his join paper with Humbler and Seyfaddini: “Proof of the simplicity conjecture”, Annals of Mathematics 2024. - Dima Dolgopyat & Adam Kanigowski, for their joint paper with Federico Rodriguez Hertz: “Exponential mixing implies Bernoulli”, Annals of Mathematics… Read More

  • 2024 Putnam Results

    We are very excited to report that our MAryland Putnam team ranked 7th among 477 institutions that participated in the 2024 Putnam math competition. Our team members this year were Daniel Yuan, Isaac Mammel, and Clarence Lam. Daniel Yuan ranked 26th among 3,988 participants. Clarence Lam and Isaac Mammel were recognized for… Read More

  • From Math Olympiads to Diplomacy: Meet Visiting Math Professor Qendrim Gashi

    Maryland Global, published a great interview with our visiting professor (and diplomat), Qendrim Gashi. The interview is available at https://marylandglobal.umd.edu/about/news/math-olympiads-diplomacy-meet-visiting-math-professor-qendrim-gashi Read More

  • Eugenia Brin, Longtime Supporter of Science and Performing Arts at UMD, Dies

    Eugenia Brin, a Russian immigrant and retired NASA scientist who, with her family of accomplished Terps, became an important benefactor of the University of Maryland, died on Dec. 3, 2024. She was 76 years old. The rest of the article can be read here: https://cmns.umd.edu/news-events/news/eugenia-brin-1948-2024 Read More
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Description

Introduction to the subject of partial differential equations: First order equations (linear and nonlinear) and second order equations (heat equation, wave equation and Laplace equation). Method of characteristics for hyperbolic problems. Solution of initial boundary value problems using separation of variables and eigenfunction expansions. Qualitative properties of solutions. 


Prerequisites

1 course with a minimum grade of C- from (MATH241, MATH340); and 1 course with a minimum grade of C- from (MATH246, MATH341).



Level of Rigor

Standard


Sample Textbooks

Introduction to Partial Differential Equations, by Walter Strauss

First Course in Partial Differential Equations, by Weinberger.


Applications

Economics, business, engineering, physics, astronomy, computer science, chemistry


If you like this course, you might also consider the following courses

MATH 463


Additional Notes

Students interested in grad school in AMSC should strongly consider this course

Students interested in grad school in MATH should consider this course


Topics

First order equations (linear and nonlinear): Method of characteristics.


Second order equations: Diffusion (heat) equation, Wave equation and Laplace equation 

Homogeneous and inhomogeneous problems

Qualitative properties of solutions (Energy, mean value property and maximum principle for harmonic functions)

Initial boundary value problems on the whole line and on the half line (Dirichlet and Neumann boundary conditions).

Initial boundary value problems on a finite interval: Method of separation of variables

Fourier Series

Laplace equation in the rectangle and disk

Heat and wave equations in higher dimensions (eigenfunction expansions)



Additional topics:


Nonlinear conservation laws, derivations, shock waves

Linearized equations

Numerical methods, CFL condition, Crank-Nicolson scheme

Finite element method

Derivation from equations of gas dynamics or from equations of the vibrating string

Derivation from Fourier's Law of cooling or Fick's law of diffusion

Traveling wave solutions to a nonlinear heat equation

Series solution and Poisson kernel representation of solution of the Dirichlet problem in the disk

Harnack inequality and Liouville's theorem

Green's function for the Poisson equation in R2 and R3

 

 

 

 

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