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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