Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (A special case are ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.

Quotes

 * One morning early in my (Hersh's) years as a thesis student of Peter Lax, I entered my mentor's office to find him glowing in smiles. “Louis is back!” he cried out to me. Louis, I wondered? Oh yes, Louis Nirenberg, also one of the partial differential specialists on the faculty of NYU's Courant Institute. He had been on leave in England; now he was back home! At the time. I didn't get it. Louis Nirenberg and Peter Lax were grad students together at NYU. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations, and Peter, a world master of hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success.


 * Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods.
 * J Lindenstrauss, L C Evans, A Douady, A Shalev and N Pippenger, Fields Medals and Nevanlinna Prize presented at ICM-94 in Zürich, Notices Amer. Math. Soc. 41 (9) (1994), 1103-1111.


 * The first systematic attack on a problem involving a partial differential equation was carried out in a sequence of 1746 papers by Jean Le Rond d'Alembert (1717-1783), who sought the fundamental modes of vibration of a vibrating string.


 * The development of the theory of P.D.E. is closely linked with advances in complex analysis; in fact, Riemann’s approach to the study of conformal mapping via the Dirichlet principle led to the systematic development of the theory of elliptic P.D.E. and associated variational problems. The application of these methods to the theory of several complex variables was initiated by Hodge in his theory of harmonic integrals on compact manifolds. It is this work that led H. Weyl to prove the fundamental hypoellipticity theorem, known as Weyl’s lemma, which in turn led to the development of the general theory of elliptic P.D.E.


 * If the original work of Calderón and Zygmund was related to elliptic PDE's, later developments allowed applications of their theory to parabolic equations and to general hypoelliptic operators, and the more recent explosion of interest in the theory of oscillatory integrals and in problems involving curvature has much to do with hyperbolic equations.