APM3701 Assignment 2 Memo | Due 8 August 2025

APM3701 Assignment 2 Memo | Due 8 August 2025. Step-by-Step Calculations Provided. QUESTION 1 Consider the heat flow in an horizontal rod of length L units and heat conductivity 1. We assume that initially the rod was submerged in a meduim where the temperature at each point x of the rod is described by the function f (x) . We also suppose that the left and the right ends of the rod are in contact with media which temperatures change with time and are described by the functions g1 (t) and g2 (t) respectively. (a) Write down the initial-boundary problem satisfied by the temperature distribution u (x, t) in the rod at any point x and time t (Explain all the meaning of the variables and parameters used). (5 Marks) (b) Suppose that f, g1, g2 are bounded, there exist constants m and M such that for all x in the domain of g1 and g2, and all t ≥ 0, we have m ≤ f (x) ≤ M;m ≤ g1 (x) ≤ M;m ≤ g2 (x) ≤ M; and the temperature u (x, t) solution of the IBVP described above satisfies the inequalities m ≤ u (x, t) ≤ M; for all x and t ≥ 0. Show that the solution u (x, t) of the heat problem described above is unique. (Explain clearly all the steps (10 Marks) (c) Suppose that u1 (x, t) and u2 (x, t) are solutions of the heat problem above (with different initial and boundary conditions) are such that u1 (0, t) ≤ u2 (0, t) , u1 (L, t) ≤ u2 (L, t) , and u1 (x, 0) ≤ u2 (x, 0) . Show that u1 (x, t) ≤ u2 (x, t) for all 0 ≤ x ≤ L and all t ≥ 0. (10 Marks) [25 Marks] QUESTION 2 Find the displacement u (x, t) of a semi–infinite vibrating string, if the finite end is fixed, the initial velocity is zero and the initial displacement is xex at every point x of the string. (Explain all the details) [25 Marks] QUESTION 3 When there is heat transfer from the lateral side of an infinite cylinder of radius a into a surrounding medium, the temperature inside the rod depends upon the time t and the distance r from its longitudinal axis (i.e. the axis through the centre and parallel to the lateral side). (a) Write down the partial differential equation that models this problem. (4 Marks) (b) Suppose that the surrounding medium is ice (at temperature zero) and the initial temperature is constant at every point. Derive the initial and boundary conditions. [Hint: For the boundary condition use Newton’s law of cooling.] (7 Marks) (c) Solve the initial boundary value problem obtained in (a) and (b). (14 Marks) [25 Marks] QUESTION 4 Find the displacement u (r, t) of a circular membrane of radius c clamped along its circumference if its initial displacement is zero and the circular membrane is given an constant initial velocity v in the upward direction. [25 Marks] TOTAL: [100 Marks]