Partial differential equation

Inmathematics,apartial differential equation(PDE) is adifferential equationthat contains unknownmultivariable functionsand theirpartial derivatives.(A special case areordinary differential equations(ODEs), which deal withfunctionsof a single variable and theirderivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevantcomputer model.

Quotes

• One morning early in my (Hersh's) years as a thesis student ofPeter 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.
  • Reuben Hersh; Vera John-Steiner (13 December 2010).Loving and Hating Mathematics: Challenging the Myths of Mathematical Life.Princeton University Press. p. 138.ISBN 1-4008-3611-5.

• 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 ofvibrationof a vibratingstring.
  • Peter V. O'Neil (1999).Beginning Partial Differential Equations.John Wiley & Sons. p. 451.ISBN 978-0-471-23887-4.

• The development of the theory of P.D.E. is closely linked with advances incomplex 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 byHodgein his theory of harmonic integrals on compact manifolds. It is this work that ledH. Weylto 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.
  • (January 2004) "Donald C. Spencer (1912–2001)"(PDF).Notices of the American Mathematical Society51(1).

• If the original work ofCalderónandZygmundwas related to elliptic PDE's, later developments allowed applications of their theory to parabolic equations and to generalhypoelliptic operators,and the more recent explosion of interest in the theory ofoscillatory integralsand in problems involving curvature has much to do with hyperbolic equations.
  • Ricci, Fulvio (1999). "Review:Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals,by Elias Stein".Bull. Amer. Math. Soc. (N.S.)36(4): 505–521.DOI:10.1090/s0273-0979-99-00792-2.