11_3 - 11/3 hw 10.6 1 9(graded November 4 2010 10.6.1 and...

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11/3 hw 10.6 1, 9(graded) November 4, 2010 10.6.1 Find the steady state solution of the heat conduction equation, α 2 u xx = u t that satisfies the following set of B.C.s, u (0 , t ) = 10 , u (50 , t ) = 40 (1) After a long time the steady state temperature dist. v ( x ) will be reached, which is independent of time. For v ( x ) to satisfy the heat conduction equation, v = 0 (2) The B.C.s imply, u (0 , t ) = 10 v (0) = 10 (3) u (50 , t ) = 40 v (50) = 40 (4) Solving (2) and applying the B.C.s v (0) = 10 and v (50) = 40 give the steady state solution as, v ( x ) = 3 5 x + 10 (5) 10.6.9 Let an aluminum rod of length 20 cm be initially at the uniform temperature of 25 C. Supposed that at time t = 0 the end x = 0 is cooled to 0 C while the end x = 20 is heated to 60 C and both are thereafter maintained at those temps. a) Find the temp. dist. in the rod at any time t . b) Plot the initial final temp. dist. α 2 u xx = u t , 0 < x < 20 (6) u (0 , t ) = 20 , u (20 , 60) , t > 0 (7) u ( x, 0) = 25 , 0 < x < 20 (8) α = (9) This is a nonhomogeneous boundary value problem and we decompose the solution into a steady state solution, v ( x ), and a transient solution, w ( x, t ).
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