Quiz7.Solutions - MATH 2930 Summer 2009 Quiz 7 NAME:

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Unformatted text preview: MATH 2930 Summer 2009 Quiz 7 NAME: ____________________________________________ -_ Cornell NetID: ________________ __ l. (6 Points) Consider the following heat equation ut(m,t) 2 um(x,t); u(0,t) = ut(7r, t) = 0, u($,0) = sin(a:). (1) Consider a separation of variables of “the form u(:1:,t) = X and set up the eigenvalue problem for the variable X and the differential equation for the variable T(t) by following the indicated steps (You are NOT being asked to solve the full problem): 0 Substitute u(:r, t) = X (a2)T(t) in the given equation and obtain uncoupled equations for X and T(t). 0 Use the boundary conditions of the problem to find the BCs on X State the eigenvalue problem for X (at). t 0 Write the differentail equation for T(t) for each eigenvalue of X ‘ i, W” \ FTC (: v-~> m , a "‘2’ if: b m if r7“: Sia- ' :1. CL one) *2 x be) xx at / ,t g Ml‘a’g H "cle l”; : LL :1 Liyiix H ‘M OM. l . 0 W ‘7 « ’° i ran, ‘ u ‘7‘“ t P f ‘T’ Mi“? fury“ 2"“ trunk) :20 21> 2L0) l (:0 O W) (U) Q LMWJMYWW yc‘w r w « it: ~ >« r? -~~~ o *i'fmt 0, WM ” thiiit)~v Maggi) Hit.) 0 2501) fig“) igd X/i Mafia UL / C?) ’3 “ it ’ ell/tr. ambiva I ’ g e’ a, L l? 3 {K )7 7:: , . UV 7: (3x “"5; "flag a o ,: to) <2. Ermine , 2. (6 Points) Consider the following wave equation on the domain 0 < a: < L, t > 0 ytt : 029m; y(0it) = y(L9t) : 0a 2 yt(ma0) : 0' Show that y(:c,t) 2 + at) + F(m — at)) is a solution to the above equation Where is the odd extension of period 2L of f Follow the steps listed below: 0 Verify that the given solution satisfies the BCs y(0, t) = y(L, t) = 0. 0 Verify that the given solution satisfies the initial conditions y(ac, 0) = f (3:), yt(a:, 0) = 0. 0 Verify that the given solution satisfies the differential equation y“ : aQym. is.“ m " ff . i ‘- w 'x skit 7W; 9) a g: lF (W t (“Mil <3 i W) "t “M J ‘/ (to 2 E; U: C Lara-e + F um] ‘“ (a LML) T .2: Ufme gFU l3 ‘2 L lab [(WlACg g‘lf' 2» ‘(X j l "firm; Tam] Ftp) Cy”) mm (2me whim, ~ . o W m, a V , . l 7 l ‘ - ()8 £1} 0194 (i: X)?“ i (3;? (>03 VLMO) [at lime) *dttfl} O ’2” H H [w 3 “‘ 0‘ ""7" “tit {wot} 7H; w l‘ (WWW) L “n I ‘ “it . \l’x'x 3: (i two i t (Ma) 2. M t, t ”' “let W (l W ...
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This note was uploaded on 01/21/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).

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Quiz7.Solutions - MATH 2930 Summer 2009 Quiz 7 NAME:

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