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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,
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“"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
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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).
 Spring '07
 TERRELL,R

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