# HW10 - f ( x ) , u t ( x, 0) = g ( x ) ( IC ) when (i) f (...

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Math 401 (Sec. 502) Spring, 2009 Homework 10 (due April 9) Q1. Prove that the solution to the heat equation u t = ku xx , 0 < x < L, t > 0 u (0 , t ) = T 0 , u x ( L, t ) = 0 , u ( x, 0) = f ( x ) is given by u ( x, t ) = T 0 + s n =1 b n sin (2 n - 1) πx 2 L exp[ - (2 n - 1) 2 π 2 kt 4 L 2 ] , where b n = 2 L i L 0 ( f ( x ) - T 0 )sin (2 n - 1) πx 2 L dx, n = 1 , 2 , 3 , . . . [ Hint. First, use the substitution v ( x, t ) = u ( x, t ) - T 0 and then apply separation of variables.] Q2. Use separation of variables to solve the IBVP u t = u xx , 0 < x < 1 , t > 0 u (0 , t ) = 0 , u x (1 , t ) = 0 , u ( x, 0) = f ( x ) when (i) f ( x ) = sin (2 πx ) - 3 sin (6 πx ). (ii) f ( x ) = - 2. Q3. Use separation of variables to solve the IBVP u tt = c 2 u xx , 0 < x < L, t > 0 u x (0 , t ) = 0 , u x ( L, t ) = 0 ( BC )

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u ( x, 0) = f ( x ) , u t ( x, 0) = g ( x ) . ( IC ) Q4. Use Q3. to solve the IBVP: u tt = u xx , 0 < x < 1 , t > 0 u x (0 , t ) = 0 , u x (1 , t ) = 0 ( BC ) u ( x, 0) =
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Unformatted text preview: f ( x ) , u t ( x, 0) = g ( x ) ( IC ) when (i) f ( x ) = 2-3cos (4 πx ) , g ( x ) = 2 cos (3 πx ), (ii) f ( x ) = x-1 , g ( x ) = 2-cos ( πx ). Q5. Use eigenfunction expansion to solve the heat equation u t = u xx + q, < x < 1 , t > , and q ≡ q ( x, t ) u (0 , t ) = 0 , u (1 , t ) = 0 , u ( x, 0) = f ( x ) when (i) q ( x, t ) = 2 t sin(2 πx ) , f ( x ) = sin (2 πx )-5 sin (4 πx ). (ii) q ( x, t ) = e-t sin(3 πx )-sin(5 πx ) , f ( x ) = sin ( πx ) + 2 sin (3 πx )....
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## This note was uploaded on 04/27/2009 for the course MATH 401 taught by Professor Sarkar during the Spring '09 term at Texas A&M.

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HW10 - f ( x ) , u t ( x, 0) = g ( x ) ( IC ) when (i) f (...

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