APM3701 Assignment 1 (DETAILED ANSWERS) 1 2025 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references ,... QUESTION 1 Solve the following (initial)-boundary value problem, a. uxy (x, y) = xy3, x, y 0. u (x, 0) = f (x) , and uy (0, y) = g (y) . Determine u (x, y) , if f (x) = cosx and g (y) = y+sin y. (Check your answer by substituting, and explain all the steps clearly) (15 Marks) b. xux + yuy = yu u (2x2, x) = x2 − 1. (1) (Check your answer by substituting before applying the boundary condition, and explain all the steps clearly) (15 Marks) [30 Marks] QUESTION 2 Consider the heat flow in a rod of length a unit and heat conductivity c. a. If initially the rod was submerged in a fluid which is kept at a temperature K◦C, write down the initial boundary value problem that is satisfied by the rod, if the heat flux is of k unit at the left end and A units at the right end. (Explain the meaning of every constant and variable). (5 Marks) b. Determine the temperature of the rod at any point x of the rod at time t 0. (Explain all the steps). (20 Marks) [25 Marks] QUESTION 3 Consider the temperature distribution in a rectangular plate as described in the figure below. Determine the steady-state temperature distribution u (x, y) in the plate. Calculate the steady-sate temperature at the centre of the plate. [25 Marks] W L Ambiant temperature Ice at 00 Ice at 00 Ice at 00 5 Downloaded by Corona Virus () lOMoARcPSD| QUESTION 4 A homogeneous and infinitely long string is stretched horizontally and vibrates vertically. If its vertical vibration at position x and at time t of is determined by the function u = u (x, y). Suppose that when the string is straight, it has a linear density of 9 and the tension at any given point of the string is 3. a. Derive the partial differential equation satisfied by u . [Do not repeat the derivation done in the Study Guide, you may use the appropriate formula and explain the constants and variables used.] (3 Marks) b. If the initial position is x − x2 and the initial velocity is 9x write down the initial–value problem which models the above situation. (3 Marks) c. Use the appropriate formula to solve the initial value problem found in (b). (14 Marks) [20 Marks]
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