Homework4 - to solve problem when F x = R R> 0 constant 2...

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Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 26 October–2 November 2007 1. Consider the inhomogeneous heat problem ( * ) u t = ku xx + F ( x ) , (0 < x < l, t > 0) , u ( x, 0) = 0 , (0 < x < l ) , u (0 , t ) = u ( l, t ) = 0 , ( t > 0) , where l > 0 , k > 0, and F is some given function. One can interpret this as a model for the temperature distribution u ( x, t ) of a metalic rod of length l , with heat sources along the rod given by F ( x ). (a) Suppose that u 0 is the solution of the steady state problem ku 00 0 + F ( x ) = 0 , u 0 (0) = u 0 ( l ) = 0 . Then show that v ( x, t ) := u ( x, t ) - u 0 ( x ) solves the homogeneous heat problem v t = kv xx , (0 < x < l, t > 0) , v ( x, 0) = - u 0 ( x ) , (0 < x < l ) , v (0 , t ) = v ( l, t ) = 0 , ( t > 0) . (b) Use (a)
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Unformatted text preview: to solve problem (*) when F ( x ) = R , ( R > 0, constant). 2. Consider the following problem for the inhomogeneous one-dimensional wave equation, ( ** ) u tt = ku xx + F ( x ) , (0 < x < l, t > 0) , u ( x, 0) = f ( x ) , u t ( x ) = g ( x ) , (0 < x < l ) , u (0 , t ) = u ( l, t ) = 0 , ( t > 0) , where F, f and g are given functions. (a) Solve this problem using the technique of exercise 1. (b) Find the solution if g ( x ) = 0, f ( x ) = M , and F ( x ) = R , ( M, R positive constants)....
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