final - Math 415 Review Ch10.5 10.8 1 background(Look at ch...

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Math 415 Review Ch10.5 - 10.8 1. background (Look at ch 10.1-10.4) (1)eigenvalues and eigenfunctions for boundary value problem. X 00 + λX = 0 (a) BC : X (0) = X ( L ) = 0 Then λ n = n 2 π 2 L 2 and X n = sin nπx L ( n = 1 , 2 , 3 , ... ) (b)BC : X 0 (0) = X 0 ( L ) = 0 Then λ 0 = 1 , X 0 = 1 and λ n = n 2 π 2 L 2 and X n = cos nπx L ( n = 1 , 2 , 3 , ... ) (2)Fourier series : check formula in the previous review note or textbook. Especially, we need Fourier sine or cosine series. 2. PDE Problems There are three important problems - u t = α 2 u xx , u tt = a 2 u xx , u xx + u yy = 0. 1) separation of variable : u ( x, t ) = X ( x ) T ( t ) or u ( x, y ) = X ( x ) Y ( y ) 2) Find eigenvalue problem and then find λ n , X n , T n or λ n , X n , Y n 3) (i) u ( x, t ) = n =1 c n X n T n or u ( x, t ) = c 0 2 X 0 T 0 + n =1 c n X n T n (ii) u ( x, y ) = n =1 c n X n Y n or u ( x, y ) = c 0 2 X 0 Y 0 + n =1 c n X n Y n 4)Find c n using Fourier sine or cosine series. ex) 10.5.7, 10.7.1, 10.8.8(a) (Warning) Above method can be used for homogeneous boundary condition. For non-homogeneous cases, check the textbook.
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