Dynamic models of engineering systems

Dynamic models of engineering systems; 1. (a) Solve the following wave equation: 31:: - 43/11: : 0 on the infinite domain: -00 < l? < 00 with initial conditions: 31(130) = e"). ML 0) = we”: [15 marks] (b) Repeat part (a) with the same wave equation and initial conditions, but with the semi-infinite solution domain: 0 5 J: < 00 and the boundary condition y(0, t) = 0. [10 marks] 2. Consider the wave equation 31:: = yr: with initial conditions: (330)" 1 (US$31) '(mo)-0 y ’ - 0 (otherwise) ’ y‘ Sketch the solution of this wave equation for 5 representative values of t, when the solution of the wave equation is considered on: a) the infinite domain -00 < .r < oo; [10 marks] (b) the semi-infinite domain 0 S .‘L‘ < 00, with boundary condition y(0, t) = 0. [15 marks] 3. (a) Solve the following heat equation: T' = 4TII + 4sin(7rr), 0 S I 31 with boundary conditions: T(0,t) = T(l,t) = 0 and initial condition: T(-1:,0) = sin(2mr) [15 marks] (b) Repeat part (a) with the same heat equation and boundary conditions, but with the initial condition T(1‘,0) = cos(21ra:) [10 marks] 4. (a) Find the Fourier sine series of f = e"I on the interval 0 S .‘L‘ S l. [20 marks] (b) Use the result from part (a) to show that r 00 . .. _ k 27r(e + l) [5 marks] PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)