APM3701 Assignment 2 (DETAILED ANSWER) 2026 - DISTINCTION GUARANTEED

APM3701 Assignment 2 (DETAILED ANSWER) 2026 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references.. QUESTION 1 Consider the heat flow in an homogeneous rod of length L with heat conductivity k and constant source of energy A. 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 1 − sin x. We also suppose that the heat flux is e−t units at the left end and cos (t − π) units at the right end. (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). (10 Marks) (b) If we define the energy by E (t) := Z L 0 (w (x, t))2 dx. (1) Use the energy method to show that the solution of the initial-boundary value problem find in a) is unique. (15 Marks) [25 Marks] QUESTION 2 (a) Assume that p is a piecewise continuous absolutely integrable function on R, with F (p) (0) = 0, if q (x) = Z x −∞ p (α) dα, x ∈ (−∞,∞) . Show that (Explain clearly all the steps) F (q) (α) = F (p) (α) iα , for α ̸= 0. (10 Marks) 8 Downloaded by Polar magnats () lOMoARcPSD| APM3701/101/0/2026 (b) Consider the vibrations u (x, t) of an infinite homogeneous string with wave speed c, which initially has the shape f (x) and has initial velocity g (x). Use an appropriate Fourier transform method to derive d’Alembert’s solution for the wave equation u (x, t) = 1 2 (f (x + ct) + f (x − ct)) + 1 2c Z x+ct x−ct g (τ ) dτ. . (15 Marks) [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.