M257-316Notes_Lecture11

# M257-316Notes_Lecture11 - 8.2. SEPARATION OF VARIABLES:...

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8.2. SEPARATION OF VARIABLES: Lecture 11 How do we fnd the b n ? Observe that we have a new type o± eigenvalue problem subject to X (0) = 0 X ( L ) = 0. Just as in the case with matrices we obtain sequence o± eigenvalues which in this case is infnite: λ n = ³ L ´ n =1 , 2 ,... (8.11) and corresponding eigen±unctions x n ( x ) = sin λ n x = sin ³ nπx L ´ ½ sin ³ πx L ´ , sin µ 2 L , sin µ 3 L ¾ . (8.12) Recall that ±or symmetric matrices the eigenvectors ±orm a basis. Aside: How do we expand a vector? Express f in terms o± the basis vectors © v 1 , v 2 , v 3 ª f = α 1 v 1 + α 2 v 2 + α 3 v 3 f · v k = α 1 v 1 · v k + α 2 v 2 · v k + α 3 v 3 · v k v 1 · v 1 v 1 · v 2 v 1 · v 3 v 1 · v 2 v 2 · v 2 v 2 · v 3 v 1 · v 3 v 2 · v 3 v 3 · v 3 α 1 α 2 α 3 = f · v 1 f · v 2 f · v 3 (8.13) v k v ` , k 6 = ` i.e. the v k are orthogonal α k = f · v k v k · v k (8.14) But ±unctions are just infnite dimensional vectors: 53

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Separation of Variables f ' [ f 1 ,f 2 ,...,f N ] g ' [ g 1 ,g 2 ,...,g N ] f · g
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## This note was uploaded on 05/04/2011 for the course MATH 25 taught by Professor Lo during the Spring '11 term at BC.

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M257-316Notes_Lecture11 - 8.2. SEPARATION OF VARIABLES:...

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