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please do both questions!

Using separation of variables, solve the one-dimensional heat equation Ju(x, t) Ət Pu(x,t) k = მე2 0 for the temperature u at position x and time t along a thin metal rod that sits between x = 0 and x = a. The ends of the rod are in contact with an ice water (0° Celsius) reservoir and at time zero, the middle of the rod from x = a/4 to x = 3a/4 is heated to 100° C. Make plots of the temperature versus distance for a few times or a single three- dimensional plot of (x,t,u) for the problem above.
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Transcribed Image Text:Using separation of variables, solve the one-dimensional heat equation Ju(x, t) Ət Pu(x,t) k = მე2 0 for the temperature u at position x and time t along a thin metal rod that sits between x = 0 and x = a. The ends of the rod are in contact with an ice water (0° Celsius) reservoir and at time zero, the middle of the rod from x = a/4 to x = 3a/4 is heated to 100° C. Make plots of the temperature versus distance for a few times or a single three- dimensional plot of (x,t,u) for the problem above.
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