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Unformatted text preview: Chapter 1 What are Partial Differential Equations? Let us begin by specifying our object of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d 4 u dx 4 + u 2 du dx = cos x (1 . 1) is a differential equation for the function u ( x ) depending on a single variable x , while u t 2 u x 2 2 u y 2 + u = 0 (1 . 2) is a differential equation involving a function u ( t, x, y ) of three variables. There are two common notations for partial derivatives, and we shall employ them interchangeably. The first, used in (1.1) and (1.2), is the familiar Leibniz notation that employs a d to denote ordinary derivatives of functions of a single variable, and the symbol (usually also pronounced dee) for partial derivatives of functions of more than one variable. An alternative, more compact notation employs subscripts to indicate par- tial derivatives. For example, u t represents u/t , while u xx is used for 2 u/x 2 , and 3 u/x 2 y for u xxy . Thus, in subscript notation, the partial differential equation (1.2) is written u t u xx u yy + u = 0 . (1 . 3) We will similarly abbreviate partial differential operators, sometimes writing /x as x , while 2 /x 2 can be written as either 2 x or xx , and 2 /x 2 y becomes xxy = 2 x y . A differential equation is called ordinary if the function u only depends on a single variable, and partial if it depends on more than one variable. Usually (but not quite always) the dependence of u can be inferred from the derivatives that appear in the differential equation. The order of a differential equation is that of the highest order derivative that appears in the equation. Thus, (1.1) is a fourth order ordinary differential equation, while (1.2) is a second order partial differential equation. To be a true differential equation, the equation must contain at least one derivative of u , and hence its order must be 1. It is worth pointing out that the preponderance of differential equations arising in applications, in science, in engineering, and within mathematics itself, are of either first or second order, with the latter being by far the most prevalent. Third order equations arise when modeling waves in dispersive media, e.g., water waves or plasma waves. Fourth order equations show up in elasticity, particularly plate and beam mechanics. Equations of order 5 are rather rare. 1/19/12 1 c circlecopyrt 2012 Peter J. Olver A basic prerequisite for studying this text is the ability to solve simple ordinary differential equations: first order equations, linear constant coefficient equations, both homogeneous and inhomogeneous, and linear systems. In addition, we shall assume some familiarity with the basic theorems concerning the existence and uniqueness of solutions to initial value problems. There are many good introductory texts, including [initial value problems....
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