APM3701 Assignment 1 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED

APM3701 Assignment 1 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references.. Solve the following (initial)-boundary value problem, (Check your answer by substituting, and explain all the steps clearly) a. ∂2u ∂x∂y∂t (x, y, t) = 2xt; x, y, z ∈ R u (1, y, t) = yt2 2 + t 2 + y 4 + 1, ∂u ∂x (x, y, 0) = xy 2 and ∂2u ∂x∂t (x, 0, t) = 2xt + x − t. (10 Marks) b.  yux − exuy = yu u (x, 0) = (1 + x) ex. (15 Marks) [25 Marks] QUESTION 2 Consider the heat flow in a horizontal rod of length p units and heat conductivity k. a. If initially, the left half of the rod is in contact with ice at 0◦C, and the right half of the rod is at the air temperature at A◦C, write down the initial boundary value problem that is satisfied by the rod, if both ends are isolated. (Explain the meaning of every constant and variable). (10 Marks) b. Determine the temperature of the rod at any point x of the rod at time t 0. (Explain all the steps). (15 Marks) [25 Marks] QUESTION 3 A vibrating homogeneous string with wave speed c, has its ends fixed. Assume that the rod was initially held in the position as described in the figure 1. a. If at time t = 1, the string is released from rest, write down the initial–boundary value problem satisfied by the vibrations of the string. (Explain the meaning of every constant and variable used). (10 Marks) b. Solve the initial-boundary value obtained in (a) (Explain all the steps). (15 Marks) [25 Marks] QUESTION 4 6 Downloaded by Polar magnats () lOMoARcPSD| APM3701/101/0/2026 u x 1 4 1 3 4 1 2 0 Figure 1: Question 3. Initial position of the homogeneous string A B C D Figure 2: Question 4. Steady-state temperature in a square plate. Consider the temperature distribution in a 1 × 1 square plate as described in figure 2. a. Determine the steady-state in the plate if the temperature at each point on the top edge AB is proportional to the distance from that point to the point B and the other three edges are in contact with ice. (20 Marks) b. Calculate the steady-sate temperature at the centre of the plate. (5 Marks)