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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]

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