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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.12 Partial Differential Equations

Sergiu Klainerman

Introduction

Partial differential equations (or PDEs) are an important class of functional equations: they are equations, or systems of equations, in which the unknowns are functions of more than one variable. As a very crude analogy, PDEs are to functions as polynomial equations (such as x2 + y2 = 1, for example) are to numbers. The distinguishing feature of PDEs, as opposed to more general functional equations, is that they involve not only unknown functions, but also various partial derivatives of those functions, in algebraic combination with each other and with other, fixed, functions. Other important kinds of functional equations are integral equations, which involve various integrals ...

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