A 2m rod is cooled on both ends (to ensure that the temperature at the ends...

 

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A 2m rod is cooled on both ends (to ensure that the temperature at the ends is always 0). The middle of the rod is heated for a period after which the temperature on the rod is H(x) = 200x - 100x. The heating is then immediately removed from the rod, and the rod is left to cool. In this scenario: Let u(x, t) be the function describing the temperature of the rod over time. The rod has a thermal conductivity of k = 1/10. You can suppose that t = 0 corresponds to the instant the heat is removed (so that u(x, 0) = H(x)). (a) (3 Points) Calculate a Fourier sine series for H(x) on the interval 0 x 2. (b) (3 Points) Solve the heat equation for the rod described in this question. (c) (1 Point) How long does it take (to the nearest second) before the temperature over the entire rod uniformly equal to 0 within 0.05? A 2m rod is cooled on both ends (to ensure that the temperature at the ends is always 0). The middle of the rod is heated for a period after which the temperature on the rod is H(x) = 200x - 100x. The heating is then immediately removed from the rod, and the rod is left to cool. In this scenario: Let u(x, t) be the function describing the temperature of the rod over time. The rod has a thermal conductivity of k = 1/10. You can suppose that t = 0 corresponds to the instant the heat is removed (so that u(x, 0) = H(x)). (a) (3 Points) Calculate a Fourier sine series for H(x) on the interval 0 x 2. (b) (3 Points) Solve the heat equation for the rod described in this question. (c) (1 Point) How long does it take (to the nearest second) before the temperature over the entire rod uniformly equal to 0 within 0.05?