Ma-2211-—-transforms-and-partial-differential-equations-3rd-c2a-.doc 616t3d

Reg. No. :

Question Paper Code :

T3048

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

f ( x to be expressed as a Fourier

function series. 2.

Obtain f (x

π 3.

the

)

first

term

of

the

Fourier

series

for

the

function

)= x , −< x <π. 2

Find the Fourier transform of f (x

ikx

) =e

0

, a < x < bx

x rel="nofollow"> b.

and 4. 5.

1 Find the Fourier sine transform of . x Find the partial differential equation of all planes cutting equal intercepts from the x and y axes.

(

3

)

2 − 2D D ′ z = 0 .

6.

Solve D

7.

∂ u ∂u Classify the partial differential equation 4 2 = . ∂t ∂x

8.

Write down all possible solutions of one dimensional wave equation.

9.

If F ( z ) =

2

10.

z

2

( z − 12 )(z − 14 )(z − 34 )

Find the Z-transform of a

, find f (0 ) .

n

x (n

)

 = n 0

for n ≥ 0 otherwise .

PART B — (5 × 16 = 80 Marks) 11. (a)

(i)

Obtain the Fourier series of the periodic function defined by − π

f (x 1

Deduce that (ii)

1

x

1

+

2

)= 

3

1

+

2

5

−π < x < 0 . 0 < x <π

+L +∞ =

2

π

2

.

(10)

8

Compute upto first harmonics of the Fourier

f (x

series of the following table x:

0

T 6

T 3

T 2

2T 3

1.98 1.30 1.05 1.30 f (x ) : 0.25 1.98

) given by

T

5T 6 –0.88 –

(6)

Or (b)

(i)

Expand f ( x 2 x

)= x−

as a Fourier series in

this series find the root mean square value of

− L < x < L and using f ( x in the

)

inte rval . (10 )

(ii) Find the complex form of the Fourier series of

f (x

) = e − x in

−1 < x <1.

12. (a) of

(i)

(6)

Find the Fourier transform

∞

= 1 − x (f )x = 0 

if

| x |< 1

if

x ≥1

and hence

4

find the value of sin t dt . 4 t 0

∫

∞

(ii)

Evaluate

(8)

dx

∫ (4 + x )(25 + x ) using transform methods. 2

2

(8)

0

Or 2

T 3048

(b) e

(i)

Find the Fourier cosine transform of− x . 2

(ii) Prove that transforms.

1

(8)

is self reciprocal under Fourier sine and

cosine x (8)

3

T 3048

13. (a) points

A tightly stretched string with fixed end

x = 0 and x = l

is

initially at rest in its equilibrium position. If it is set vibrating giving each point a initial velocity 3x (l − x ) , 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 u( x, t

14. (a) of

) taking x = 0 at A.

(i)

Find the inverse Z-transform

(ii) u

Solve the equation

(16)

10 z z

2

. − 3z + 2

(8)

+ 6un +1 + 9un = 2 n given u0

n+ 2

= u1 = 0 .

(8)

Or

(b) of

(i)

Using convolution theorem, find the Z

(ii) of

Find the inverse Z-transform

(i)

Solve z = px + qy + p q .

(ii)

Solve D

−1

2

z . ( z − 4 )( z − 3)

(8)

3

15.

(a)

2

(

2

+ 2DD′ + D′

z − . 20z 3 (z − 2) (z − 4 )

2

2

(8)

(8)

) z = sinh ( x + y ) + e x + 2 y .

(8)

Or (b)

( y − xz ) p + ( yz − x ) q = ( x + y )( x − y ) .

(8)

(

(8)

(i)

Solve

(ii)

Solve D

2

2

)

− D ′ − 3D + 3D ′ z = xy + 7 .

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