Consider the temperature u(x, t) at time at each point of a rod length 5, if...

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Consider the temperature u(x, t) at time at each point of a rod length 5, if one of its end is in contact with ice at 0° and the other end is exposed to warm air at 35°. Assume that initially the rod is submerged in boiling water at 100%. a. Derive the initial-boundary value prob em of the problem described above. Explain all the steps, do not repeat the derivations done i b. Show that the solution of the initial-boundary value problem obtained in part (a) has a unique solution. [Hint: apply the maximum principle to the heat equation.] (10 Marks) c. Deduce that the unique solution u(x, t) obtained in part (b.) is such that u(x, t) ≥ 0.V € [0.5]. Vt 20. Consider the temperature u(x, t) at time at each point of a rod length 5, if one of its end is in contact with ice at 0° and the other end is exposed to warm air at 35°. Assume that initially the rod is submerged in boiling water at 100%. a. Derive the initial-boundary value prob em of the problem described above. Explain all the steps, do not repeat the derivations done i b. Show that the solution of the initial-boundary value problem obtained in part (a) has a unique solution. [Hint: apply the maximum principle to the heat equation.] (10 Marks) c. Deduce that the unique solution u(x, t) obtained in part (b.) is such that u(x, t) ≥ 0.V € [0.5]. Vt 20.