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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 find the b n ? Observe that we have a new type of eigenvalue problem subject to X (0) = 0 X ( L ) = 0. Just as in the case with matrices we obtain sequence of eigenvalues which in this case is infinite: λ n = L n = 1 , 2 , . . . (8.11) and corresponding eigenfunctions x n ( x ) = sin λ n x = sin nπx L sin πx L , sin 2 πx L , sin 3 πx L , . . . . (8.12) Recall that for symmetric matrices the eigenvectors form a basis. Aside: How do we expand a vector? Express f in terms of the basis vectors v 1 , v 2 , v 3 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) If v k v , k = i.e. the v k are orthogonal α k = f · v k v k · v k (8.14) But functions are just infinite 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 = f 1 g 1 + f 2 g 2 + · · · + f N g N Δ x = L
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M257-316Notes_Lecture11 - 8.2 SEPARATION OF VARIABLES...

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