Test 3 - which satis¯es the boundary[12 conditions w(0 t =...

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MATH 3705A Test 3 Solutions March 6, 2009 Questions 1-3 are multiple choice. Circle the correct answer. Only the answer will be marked. [Marks] 1. Let f ( x ) = ½ 1 ; 0 x 1 2 ; 1 < x 2 ¾ . At x = 59, the Fourier sine series of f on [0 ; 2] con- [4] verges to (a) 0 (b) 3 2 (c) ¡ 3 2 (d) 1 (e) ¡ 2 Solution: (c) 2. Let f ( x ) be as in Question 1. At x = 59 : 5, the Fourier cosine series of f on [0 ; 2] [4] converges to (a) 0 (b) 3 2 (c) ¡ 3 2 (d) 1 (e) 2 Solution: (d) 3. Let f ( x ) = x on [0 ; 1] and let f 1 ( x ) = j x j on [ ¡ 1 ; 1]. Then the Fourier series of f 1 on [4] [ ¡ 1 ; 1] is identical to (a) The Fourier sine series of f on [0 ; 1] (b) The Fourier cosine series of f on [0 ; 1] (c) The Fourier series of f on [0 ; 1] (d) The Fourier series of the 1-periodic extension of f to R = ( ¡ 1 ; 1 ) (e) None of the above Solution: (b) 4. Find the Fourier cosine series of f ( x ) = 4 x on [0 ; 1]. [6] Solution: L = 1, a 0 = 2 Z 1 0 4 x dx = 4, and for n ¸ 1, a n = 2 Z 1 0 4 x cos( n¼x ) dx = 8 x sin( n¼x ) ¯ ¯ ¯ ¯ 1 0 ¡ Z 1 0 8 sin( n¼x ) dx = 8 n 2 ¼ 2 cos( n¼x ) ¯ ¯ ¯ ¯ 1 0 = 8 n 2 ¼ 2 [( ¡ 1) n ¡ 1]. Thus, the Fourier cosine series of f on [0 ; 1] is 2 + 1 X n =1 8[( ¡ 1) n ¡ 1] n 2 ¼ 2 cos( n¼x ). 5. The solution of the heat equation w xx = 1 ® 2 w t ; 0 < x < L
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Unformatted text preview: , which satis¯es the boundary [12] conditions w (0 ; t ) = w ( L; t ) = 0, has the form w ( x; t ) = 1 X n =1 b n sin ³ n¼x L ´ e ¡ ® 2 n 2 ¼ 2 L 2 t : 2 Find the solution u ( x; t ) of u xx = 1 9 u t ; < x < 3 ; which satis¯es the boundary condi-tions u (0 ; t ) = 1 ; u (3 ; t ) = 4, and the initial condition u ( x; 0) = 3 x + 1. Write down the complete solution u ( x; t ). Solution: u ( x; t ) = v ( x ) + w ( x; t ), where v ( x ) = x + 1 and w ( x; t ) = 1 X n =1 b n sin ³ n¼x 3 ´ e ¡ n 2 ¼ 2 t , with w ( x; 0) = u ( x; 0) ¡ v ( x ) = 2 x ) b n = 2 3 Z 3 2 x sin ³ n¼x 3 ´ dx = ¡ 4 n¼ x cos ³ n¼x 3 ´ ¯ ¯ ¯ ¯ 3 + Z 3 4 n¼ cos ³ n¼x 3 ´ dx = ¡ 12 n¼ cos( n¼ ). Thus, u ( x; t ) = x + 1 + 1 X n =1 12( ¡ 1) n ¡ 1 n¼ sin ³ n¼x 3 ´ e ¡ n 2 ¼ 2 t ....
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