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**Unformatted text preview: **Math 212 Introductory Partial Differential Equations Course notes , Fall 2007 Instructor: Dr. Friedemann Brock Chapter 1: Introduction 1.1. Notation IN natural numbers IR real numbers IC complex numbers IR n Euclidean space of dimension n , = { x = ( x 1 , . . . , x n ) : x i ∈ IR , i = 1 , . . . , n } , if n = 2 , 3 we often write ( x, y ), ( x, y, z ) for points in IR n , respectively, A × B := { ( a, b ) : a ∈ A, b ∈ B } A set D ⊂ IR n is called a domain if it is open and connected. By ∂D we denote the boundary of D . f : D → IR denotes a function from D into IR. If u : D → IR then ∂ k u ∂x i 1 1 ··· ∂x i n n , ( k, i 1 , . . . , i n ∈ IN , n X j =1 i j = k ) , denotes a partial derivative of order k of u . We often abbreviate partial derivatives using subscripts. For instance, if u = u ( x, y, t ), then ( ∂u/∂x ) = u x , ( ∂u/∂t ) = u t are partial derivatives of first order, and ( ∂ 2 u/∂y∂t ) = u yt , ( ∂ 2 u/∂x∂y ) = u xy are partial derivatives of second order. If k ∈ IN, we set C ( D ) := { u : D → IR : u is continuous on D } , C k ( D ) := { u : D → IR : u together with its partial derivatives of order ≤ k are continuous } . If u ∈ C 2 ( D ), then u x i x j = u x j x i , ( i, j = 1 , . . . n ). 1.2. Some linear PDE (PDE = P artial D ifferential E quation ). Assume that D is a domain in IR n , and u : D → IR. We set Lu := au + n X i =1 b i u x i + n X i,j =1 c ij u x i x j + ··· , (1) 1 where the a, b i , c ij ’s are given functions on D , and the dots ··· in (1) stand for further terms involving higher-order derivatives of u which enter linearly . The operator L is then a linear DOP (DOP= D ifferential Op erator), since we have for any functions u i : D → IR, and numbers c i ∈ IR, ( i = 1 , . . . , k ), L k X i =1 c i u i = k X i =1 c i Lu i . (2) An equation of the form Lu = 0 is called a linear homogeneous PDE. If F : D → IR, then an equation of the form Lu = F is called a linear...

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