Galileo has been quoted as saying that "nature is an open book, but the language in which it is written is mathematics." The fact that mathematics, one of the most abstract creations of the human mind, is so effective a tool for understanding our surroundings, has puzzled many great minds over the centuries. Galileo made his statement shortly before mathematics was about to go through a sea change with the invention of calculus and the work of such thinkers as Fermat, Descartes, Pascal, Leibniz and, above all, Newton and Euler. Leonhard Euler was the most prolific mathematician of all times, so far. His work fills some eighty volumes and there is no area to which he did not contribute. Among Euler's many achievements, a very important one may have been the introduction of differential equations as a means of modelling nature; setting Newton's laws of motion into the form of differential equations. It is always hard to attribute priorities, especially if you are not a historian. Euler's immediate predecessors (and teachers), the Bernoulli brothers^(**), already dabbled in differential equations and their solutions to the brachistochrone and tautochrone problems are seen as the origins of the calculus of variations.

At a very superficial level, differential equations come in two flavors: Ordinary and Partial. Ordinary differential equations involve functions of a single variable, partial differential equations involve functions of several variables. There is some relation between the two "flavors" and, in a sense, ordinary differential equations are seen as the more basic object. This does not mean that ordinary differential equations are "easier" than partial differential equations, but one can do things with ordinary differential equations that would be very hard or impossible to do with the partial ones. For example, ordinary differential equations have a unified existence theory; no good analogue exists in the case of partial differential equations. Both fields are vast and an introductory course such as this one has to make serious choices on what material to cover. As the decider of the course, I have decided for the following syllabus:

1. Introduction: What are partial differential equations, where do they show up; well posed problems.
2. Classical solutions and weak solutions. Regularity.
3. Classification of partial differential equations: Elliptic equations, hyperbolic, parabolic equations and systems.
4. Four important linear partial differential equations: transport equation, Laplace's equation, heat equation, wave equation.
5. The inequalities of Sobolev, Gagliardo-Nirenberg, Morrey, Poincaré.
6. Second order elliptic equations.
Prerequisites: I want to keep the prerequisites at a minimum but, of course, some prerequisites are needed. It goes without saying (but I'll say it anyway) that you must have had a decent multi-variable calculus course, and feel comfortable working with partial derivatives. Material that should be covered in such a course, but usually isn't at FAU (and perhaps not covered in any FAU course) will be briefly developed as needed. Concerning ordinary differential equations, we may have occasion to use the standard existence and uniqueness theorem for initial value problems. It is something that can easily be learned. Anybody who took Professor Kalies' Fall course on Ordinary Differential Equations is much more than adequately prepared for this course.

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Note on the portraits: The portraits represent some mathematicians who made fundamental contributions to the theory of differential equations and who had biographies in the so called MacTutor pages published by the University of Saint Andrews in Scotland.. They are arranged chronologically by date of birth, going clockwise and beginning at the upper left-hand corner. Clicking on any portrait should take you to a biography of the portrayed mathematician.