test1_solns - Test 1 Introduction to PDE MATH 3363-25820...

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Solutions to Test 1 Test 1 Introduction to PDE MATH 3363-25820 (Fall 2009) This exam has 4 questions, for a total of 20 points. Please answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Upon finishing PLEASE write and sign your pledge below: On my honor I have neither given nor received any aid on this exam. 1 Rules You may only use pencils, pens, erasers, and straight edges. No calculators, notes, books or other aides are permitted. Scrap paper will be provided. Be sure to show a few key intermediate steps when deriving results - answers only will not get full marks. 2 Given You may assume the eigenvalues of the Sturm-Liouville problem X 00 + λX = 0 , 0 < x < 1 X 0 (0) = 0 , X (1) = 0 . are λ n = ( n - 1 2 ) 2 π 2 and X n ( x ) = cos ( ( n - 1 2 ) πx ) , for n = 1 , 2 , .. . , without derivation. You may also assume the following orthogonality conditions for m , n positive integers: Z 1 0 cos ( ( m - 1 2 ) πx ) cos ( ( n - 1 2 ) πx ) dx = ± 1 / 2 , m = n, 0 , m 6 = n. Page 1 of 5 Please go to the next page. ..
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Solutions to Test 1 3 Questions Consider the following heat problem in dimensionless variables u t = u xx + bx 2 , 0 < x < 1 , t > 0 u x (0 , t ) = 0 , u (1 , t ) = 1 , t > 0 u ( x, 0) = u 0 , 0 < x < 1 , where b > 0 and u 0 > 0 are constants. This is the heat equation with a source, where the rod is insulated at x = 0 and kept at 1 degree at x = 1.
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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test1_solns - Test 1 Introduction to PDE MATH 3363-25820...

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