Reg. No. :

B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009

Third Semester

Civil Engineering

MA 2211 — TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

(Common to all branches)

(Regulation 2008)

Time : Three hours Maximum : 100 Marks

Answer ALL Questions

PART A — (10 × 2 = 20 Marks)

1. State the sufficient condition for a function ( )xf to be expressed as a Fourier

series.

2. Obtain the first term of the Fourier series for the function

( ) ππ <<−= xxxf ,2 .

3. Find the Fourier transform of

( )

><

<<=

.0

,

bxax

bxaexf

xki

and

4. Find the Fourier sine transform of x

1.

5. Find the partial differential equation of all planes cutting equal intercepts

from the x and y axes.

6. Solve ( ) 02 23 =′− zDDD .

7. Classify the partial differential equation t

u

x

u

∂∂

=∂

∂2

2

4 .

8. Write down all possible solutions of one dimensional wave equation.

9. If ( ) ( )( )( )43

41

21

2

−−−=

zzz

zzF , find ( )0f .

10. Find the Z-transform of

( )

≥=.0

0!

otherwise

for nn

a

nx

n

Question Paper Code : T3048 3

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T 3048 2

PART B — (5 × 16 = 80 Marks)

11. (a) (i) Obtain the Fourier series of the periodic function defined by

( )

<<

<<−−=

πππxx

xxf

0

0.

Deduce that 85

1

3

1

1

1 2

222

π=∞++++ ⋯ . (10)

(ii) Compute upto first harmonics of the Fourier series of ( )xf given by

the following table

x : 0 6T 3T 2T 32T 65T T

( )xf : 1.98 1.30 1.05 1.30 –0.88 –0.25 1.98

(6)

Or

(b) (i) Expand ( ) 2xxxf −= as a Fourier series in LxL <<− and using

this series find the root mean square value of ( )xf in the interval.

(10)

(ii) Find the complex form of the Fourier series of ( ) xexf −= in

11 <<− x . (6)

12. (a) (i) Find the Fourier transform of ( )

≥=

<−=

1if0

1||if1

x

xxxf and hence

find the value of dtt

t∫∞

0

4

4sin. (8)

(ii) Evaluate ( )( )∫∞

++0

22 254 xx

dx using transform methods. (8)

Or

(b) (i) Find the Fourier cosine transform of 2xe− . (8)

(ii) Prove that x

1 is self reciprocal under Fourier sine and cosine

transforms. (8)

321 3

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21

T 3048 3

13. (a) A tightly stretched string with fixed end points 0=x and lx = is

initially at rest in its equilibrium position. If it is set vibrating giving

each point a initial velocity ( )xlx −3 , find the displacement. (16)

Or

(b) A rod, 30 cm long has its ends A and B kept at 20°C and 80°C

respectively, until steady state conditions prevail. The temperature at

each end is then suddenly reduced to 0°C and kept so. Find the resulting

temperature function ( )txu , taking 0=x at A. (16)

14. (a) (i) Find the inverse Z-transform of 23

102 +− zz

z. (8)

(ii) Solve the equation nnnn uuu 296 12 =++ ++ given 010 == uu . (8)

Or

(b) (i) Using convolution theorem, find the 1−Z of

( )( )34

2

−− zz

z. (8)

(ii) Find the inverse Z-transform of ( ) ( )42

203

3

−−

−

zz

zz. (8)

15. (a) (i) Solve 22qpqypxz ++= . (8)

(ii) Solve ( ) ( ) yxeyxzDDDD 222 sinh2 +++=′+′+ . (8)

Or

(b) (i) Solve ( ) ( ) ( )( )yxyxqxyzpxzy −+=−+− . (8)

(ii) Solve ( ) 73322 +=′+−′− xyzDDDD . (8)

———————

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