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Unformatted text preview: Homework 7 | Due on March 18 (Thursday) | Solution 1 1. Let a 20 centimeter long aluminium rod be initially at the uniform tempera- ture of 25 Celsius. Suppose that at time t = 0 second the left end is cooled to 0 Celsius, while the right end is heated to 60 Celsius, and both are there- after maintained at those temperatures. Thermal diffusivity of aluminium is . 86 cm 2 /sec. (a) [10 pts] Guess the steady-state temperature distribution as t and give an appropriate physical reasoning for your guess. If we let the left end as x = 0 and the right end as x = 20, then the steady-state temperature will be v ( x ) = 3 x . The reason is as follows: If v xx 6 = 0, then from the differential equation v t = v xx we have v t 6 = 0 which violates the definition of the steady- state temperature (Physically speaking, if the graph of v ( x ) is not a straight line, some points will distribute the heat to the neighboring point, so it is not a steady-state temperature). So we must have a linear function v ( x ) = cx + d ( v xx = 0) satisfying v (0) = 0 and v (20) = 60. (b) [15 pts] Find the temperature u ( x,t ) in the rod at any time t 0. The differential equation model of the original problem is u t = 0 . 86 u xx , < x < 20 , t > , u ( x, 0) = 25 , < x < 20 , u (0 ,t ) = 0 , u (20 ,t ) = 60 , t > , From the previous physical reasoning in (a), it is suggested to let u ( x,t ) := v ( x ) + w ( x,t ), where v ( x ) = 3 x is a steady-state temperature obtained in (a). Since u ( x,t ) = v ( x ) + w ( x,t ), we can derive a differential equation for w ( x,t ): u t = 0 . 86 u xx ( v ( x ) + w ( x,t )) t = 0 . 86( v ( x ) + w ( x,t )) xx w t = 0 . 86 w xx , from the fact that v xx = v t = 0 because v ( x ) = 30 x is a linear t- independent function. For the initial and boundary conditions we have 25 = u ( x, 0) = v ( x ) + w ( x, 0) = 3 x + w ( x, 0) w ( x, 0) = 25- 3 x, 0 = u (0 ,t ) = v (0) + w (0 ,t ) = 0 + w (0 ,t ) w (0 ,t ) = 0 , 60 = u (20 ,t ) = v (20) + w (20 ,t ) = 60 + w (20 ,t ) w (20 ,t ) = 0 ....
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