241Final-S11

Vi consider the heat equation u t = u xx with

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Unformatted text preview: VI) Consider the heat equation u t = u xx with boundary values u (0 ,t ) = u (1 ,t ) = 0 and with initial condition u ( x, 0) = sin (6 πx ). Compute u ( 1 4 , 1 4 ): a)- exp (- 5 π 2 ) b) exp (- 7 π 2 ) c)- exp (- 9 π 2 ) d) exp (- 3 π 2 ) e)- exp (- π 2 ) 4 VII) For the PDE: u x + 3 u y = 0, we are interested only in solutions of the form u ( x,y ) = X ( x ) Y ( y ). For such a solution, suppose we know u (0 , 1) = e and u (1 , 0) = e 2 . Compute u (1 , 1): a) exp ( 1 4 ) b) exp ( 3 4 ) c) exp ( 5 4 ) d) exp ( 7 4 ) e) exp ( 9 4 ) VIII) A certain function u ( x,y ) is defined on the square whose vertices are (0,0), (1,0), (0,1) and (1,1). The function u satisfies the two PDE’s: u xx = 0 and u yy = 0 and has values at the corners of the square given by u (0 , 0) = 0 , u (1 , 0) = 3 , u (0 , 1) =- 2 , u (1 , 1) = 1. For this function u , the value u ( 1 2 , 1 2 ) is: a) 1 b)-2 c) d)-3/2 e) 1/2 5 IX) We expand the function f ( z ) = e z ( z- 1) 2 in a Laurent Series valid in the annulus 1 < | ( z- 1) | < 2. This series has the form ∑ ∞ m =1 b m 1 ( z- 1) m + ∑ ∞ k = o a k ( z- 1) k . Then the value of the product b 2 b 1 a o a 1 a 2 is: a) e 5 b) e 5 / 288 c) e 5 / 120 d) e 5 / 84 e) e 5 / 316. X) For the function f ( z ) = 2 z- 1 z ( z- 5) 2 , find its residue at z = 5. a) b) i/2 c) 2/5 d) π /4 e) 1/25 THE NEXT 5 PROBLEMS ARE LONG ANSWER–WORK IS REQUIRED TO BE SHOWN. 6 XI) Compute the integral R ∞-∞ sin 2 ( x ) 1+ x 2 dx . 7 XII) If u ( x,y ) is a harmonic function (that is, it satisfies Laplace’s DE: Δ u = 0) defined on the whole plane, then it is known that there is another...
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VI Consider the heat equation u t = u xx with boundary...

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