Test 4 - MATH 3705A Test 4 Solutions March 20, 2009 [Marks]...

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MATH 3705A Test 4 Solutions March 20, 2009 [Marks] 1. The solution of the heat equation u xx = 1 ® 2 u t ; 0 <x<L , which satis¯es the boundary [10] conditions u x (0 ;t )= u x ( L; t ) = 0, has the form u ( x; t )= a 0 2 + 1 X n =1 a n cos ³ n¼x L ´ e ¡ ® 2 n 2 ¼ 2 L 2 t : Find the solution of u xx = u t ; 0 <x< 1, which satis¯es the boundary conditions u x (0 ;t )= u x (1 ;t ) = 0, and the initial condition u ( x; 0) = x . Write down the complete solution u ( x; t ). Solution: ® =1 ;L =1 ) u ( x; t )= a 0 2 + 1 X n =1 a n cos ( n¼x ) e ¡ n 2 ¼ 2 t : x = u ( x; 0) = a 0 2 + 1 X n =1 a n cos ( n¼x ) ) a n = 2 1 Z 1 0 x cos( n¼x ) dx; n ¸ 0 : a 0 =2 Z 1 0 xdx = 1, and for n ¸ 1 ; a n = 2 x sin( n¼x ) ¯ ¯ ¯ ¯ 1 0 ¡ 2 Z 1 0 sin( n¼x ) dx = 2 n 2 ¼ 2 cos( n¼x ) ¯ ¯ ¯ ¯ 1 0 = 2 n 2 ¼ 2 [( ¡ 1) n ¡ 1]. Thus, u ( x; t )= 1 2 + 1 X n =1 2 n 2 ¼ 2 [( ¡ 1) n ¡ 1] cos ( n¼x ) e ¡ n 2 ¼ 2 t . 2. The solution of the wave equation
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This note was uploaded on 09/21/2009 for the course MATH 3705 taught by Professor Dr.e.devdariani during the Winter '09 term at Carleton.

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Test 4 - MATH 3705A Test 4 Solutions March 20, 2009 [Marks]...

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