PART – B

1) Form the partial differential equation by eliminating the arbitrary functions

f and g in ).2()2( 33 yxgyxfz

2) Form the partial differential equation by eliminating the arbitrary functions

f and g in ).()( 22 xgyyfxz

3) Form the partial differential equation by eliminating the arbitrary functions

f and from ).()( zyxyfz

4) Find the singular solution of .1622 qpqypxz

5) Solve .1 22 qpqypxz

6) Find the singular integral of the partial differential equation

.22 qpqypxz

7) Solve .1 22 qpz

8) Solve ).1()1( 2 zqqp

9) Solve .4)(9 22 qzp

10) Solve .)1( qzqp

11) Solve ).)(()()( yxyxqxyzpxzy

12) Solve .2)2()( zxqyxpzy

13) Find the general solution of .32)24()43( xyqzxpyz

14) Solve )()()( yxzqxzypzyx

15) Solve ).2(2 yzxxyqpy

16) Solve .)()( 222 xyzqzxypyzx

17) Solve .)()( 22 yxzqyxzpyx

18) Solve .)2()2( xyqyzpzx

19) Find the general solution of .)( 22 qypxyxz

20) Solve ).()()( 222222 xyzqxzypzyx

21) Solve ).4sin()20( 522 yxezDDDD yx

22) Solve ).2sin(3)54( 222 yxezDDDD yx

23) Solve ).cos()( 23223 yxezDDDDDD yx

24) Solve .)30( 622 yxexyzDDDD

25) Solve .)6( 3222 yxeyxzDDDD

26) Solve .)sinh(22

22

2

2

xyyxy

z

yx

z

x

z

27) Solve .cos62

22

2

2

xyy

z

yx

z

x

z

28) Solve .sin)65( 22 xyzDDDD

29) Solve .)1222( 222 yxezDDDDDD

30) Solve .7)33( 22 xyzDDDD

UNIT – I PARTIAL DIFFERENTIAL EQUATIONS

PART-B

1) Find the Fourier series of period 2 for the function )2,(;2

),0(;1)(xf and

hence find the sum of the series ........5

1

3

1

1

1222

.

2) Obtain the Fourier series for )2,(;2

),0(;)(

x

xxf .

3) Expand 2;0

0;sin)(

x

xxxf as a Fourier series of periodicity 2 and

hence evaluate ..........7.5

1

5.3

1

3.1

1.

4) Determine the Fourier series for the function 2)( xxf of period 2 in

20 x .

5) Obtain the Fourier series for 21)( xxxf in ),( . Deduce that

6.........

3

1

2

1

1

1 2

222.

6) Expand the function xxxf sin)( as a Fourier series in the interval

x .

7) Determine the Fourier expansion of xxf )( in the interval x .

8) Find the Fourier series for xxf cos)( in the interval ),( .

9) Expand xxxf 2)( as Fourier series in ),( .

10) Determine the Fourier series for the function xx

xxxf

0,1

0,1)( .

Hence deduce that 4

.........5

1

3

11 .

11) Find the half range sine series of xxxf cos)( in ),0( .

12) Find the half range cosine series of xxxf sin)( in ),0( .

13) Obtain the half range cosine series for xxf )( in ),0( .

14) Find the half range sine series for )()( xxxf in the interval ),0( .

15) Find the half range sine series of 2)( xxf in ),0( .

Hence find )(xf .

UNIT – II FOURIER SERIES

18. Write down the appropriate solutions of the two dimensional heat

Equations.

19. In two dimensional heat flow, what is the temperature along the

Normal to the xy- plane?

20. If a square plate has its faces and the edge y = 0 insulated, its edges

x = 0 and x = n are kept at zero temperature and its fourth edge is

kept at temperature u, then what are the boundary conditions for this

problem?

PART –B

21. A tightly stretched string with fixed end points x = 0 and x = l is

initially in a position given by y = y0 sin3( π x / l ). If it is released from

rest from this position, find the displacement y( x, t).

22. A tightly stretched string of length l has its ends fastened at x = 0 ,

x = l. The mid-point of the string is then taken to height h and then

released from rest in that position. Find the lateral displacement of a

point of the string at time t from the instant of release.

23. A tightly stretched string with fixed end points x = 0 and x = l. At time

t = 0, the string is given a shape defined by F(x) = μ x ( l - x ), where μ

is constant, and then released . Find the displacement of any point x of

the string at any time t >0.

24. The points of trisection of a string are pulled aside through the same

distance on opposite sides of the position of equilibrium and the string

is released from rest. Derive an expression for the displacement of the

string at subsequent time and show that the mid-point of the string

always remains at rest.

25. A tightly stretched string of length l with fixed ends is initially in

equilibrium position. It is set vibrating by giving each point a velocity

v0 sin3( π x / l ). Find the displacement y(x,t).

26. A tightly stretched string with fixed end points x = 0 and x = l is

initially at rest in its equilibrium position. . It is set vibrating by giving

each point a velocity λ x ( l - x ), find the displacement of the string at

any distance x from one end at any time t.

27. A taut string of length 20cms.fastened at both ends is displaced from its

position of equilibrium , by imparting to each of its points an initial velocity

given by: v = x in 0 < x < 10 and, x being the

20 – x in 10 < x <20 distance from one end.

Determine the displacement at any subsequent time.

28. An insulated rod of length l has its ends A and B maintained at 00C and

1000c respectively until steady state conditions prevail. If B is suddenly

reduced to 00C and maintained at 00C, find the temperature at a distance x

from A at time t.

29. A homogeneous rod of conducting material of length 100cm has its ends

kept at zero temperature and the temperature initially is

u(x,0) = x in 0 < x < 50

100 – x in 50 < x <100 Find the temperature u(x,t) at any time.

30. An insulated rod of length l has its ends A and B maintained at 00C and 1000c

respectively until steady state conditions prevail. If the change consists of

raising the temperature of A to 200c and reducing that of B to 800c ,

find the temperature at a distance x from A at time t.

31. The ends A and B of a rod 20cm.long have the temperature at 300c and 800c

until steady state prevails. The temperature of the ends are changed to 400c

and 600c respectively. Find the temperature distribution in the rod at time t.

32. The ends A and B of a rod 10cm.long have the temperature at 500c and 1000c

until steady state prevails. The temperature of the ends are changed to 900c

and 600c respectively. Find the temperature distribution in the rod at time t.

33. A square plate is bounded by the lines x =0, y = 0 , x =20 and y = 20.

Its faces are insulated. The temperature along the upper horizontal edge

Is given by u ( x,20) = x ( 20 – x) when 0 < x < 20 while the other three edges

are kept at 00c. Find the steady state temperature in the plate.

34. Find the steady state temperature at any point of a square plate whose two

adjacent edges are kept at 00c and the other two edges are kept at the constant

temperature 1000c.

35. Find the steady temperature distribution at points in a rectangular plate

With insulated faces the edges of the plate being the lines x = 0 , x = a ,

y = 0 and y = b. When three of the edges are kept at temperature zero

and the fourth at fixed temperature a0c.

36. Solve the BVP Uxx + Uyy = 0 , 0 < x, y < π with u( 0 ,y ) = u(π , y) =

u( x , π) = 0 and u(x,0) = sin3x.

37. A rectangular plate is bounded by the lines x = 0 , y = 0 , x = a , y = b . Its

Surfaces are insulated. The temperature along x = 0 and y = 0 are kept

at 00 c and the others at 1000 c . Find the steady state temperature at any

point of the plate.

38. A long rectangular plate has its surfaces insulated and the two long sides

as well as one of the short sides are maintained at 00 c . Find an

expression for the steady state temperature u(x,y) if the short side y = 0

is π cm long and is kept at uo0c.

39. An infinitely long rectangular plate with insulated surface is 10cm wide.

The two long edges and one short edge are kept at zero temperature

while the other short edge x = 0 is kept at temperature given by

U = 20y for 0 < y < 5

20 ( 10 – y ) for 5 < y < 10 .Find the steady state temperature

distribution in the plate.

40. An infinitely long – plane uniform plate is bounded by two parallel edges

and an end at right angle to them. The breadth of this edge x =0 is π, this

end is maintained at temperature as u = k (πy – y2) at all points while the

other edges are at zero temperature. Determine the temperature u(x,y)

at any point of the plate in the steady state if u satisfies Laplace equation.

PART –B

1. A function f(x) is defined as f(x) = 1 if |x| < 1

0 , otherwise . Using Fourier

integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in 0 < x< .

2. A function f(x) is defined as f(x) = 1 if |x| < 1

0 , otherwise . Using Fourier

cosine integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in

0 < x < .

3. Find the F.T of f(x) is defined as f(x) = 1 if |x| < 1

0 , otherwise . Hence evaluate

(i) [sin / ] d (ii) [sin

2 /

2]

d in (0 , ).

4. Find the F.T of f(x) is defined as f(x) = a -|x| if |x| < a

0 , otherwise . Hence evaluate

(i) [sin / ] 2

d in (0 , ). (ii) [sin / ] 4 d in (0 , ).

5. Find the F.T of f(x) is defined as f(x) = 1 –x2 if |x| < 1

0 , otherwise . Hence evaluate

(i) [sin t – t cos t/ t 3] dt in (0 , ). (ii) [x cos x - sin x / x

3] cos (x /2)dx in (0 , ).

6. Find the F.T of f(x) is defined as f(x) = e-a2x2

,a >0. Hence S.T e-x2 / 2

is self reciprocal

under F.T.

7. Find the F.T of e-|x|

and hence find the F.T of e-|x|

cos 2x.

8. Obtain the F.S.T of f(x) = x if 0 <x < 1

2 – x if 1<x<2

0 , otherwise

9. Find the F.C.T of f(x) is defined as f(x) = cos x if 0 <x < a

0 , otherwise .

10. State and Prove Parseval’s Identity.

11. Find the F.S.T and F.C.T of x n-1

, where 0 < n< 1, x >0 . Deduce that 1/ x is self-

Unit – IV

FOURIER TRANSFORMS

reciprocal under both F.S.T and F.C.T.

12. Find the F.S.T of e-ax

/ x . Hence find F.S.T of 1 / x.

13. Evaluate [dx / (a2 + x

2 ) (b

2 + x

2) ]

dx in (0 , ).

14. Find F c {f ’(x)}.

15. Solve the integral equation [f(x) cos x] dx in (0 , ) and also [cos x / ( 1 +

2)]

d

in (0 , ).

Z – TRANSFORM PART –B

z2/ ( z -a ) ( z - b )

1. Find Z [ an cos nθ ] and Z [an sin nθ ]

2. Find Z [an n2 ].

3. Find Z [ cos nπ/2 ] and Z [ sin nπ/2 ]

4. Find the Z – transforms of the following (i) ean (ii) n ean

5. Find the Z – transform of (i) cosh nθ (ii) an cosh nθ

6. Find Z [ cos ( nπ/2 + π/4 ) ]

7. Find the Z – transform of (i) ncp (ii) n+p cp

8. Find the Z – transform of unit impulse sequence and unit step sequence.

9. Find the Z – transform of (i) sinh nθ (ii) an sinh nθ

10. Find Z [ et sin2t ] and Z [ e-2t sin3t ].

11. Find the inverse Z – transform of z / ( z + 1 )2 by division method.

12. Find the inverse Z – transform of { 2 z2 + 3z } / ( z + 2) ( z – 4 ) by partial

fractions method.

13. Find the inverse Z – transform of ( z3 – 20 z ) / ( z – 2 ) 3 ( z – 4 ) by partial

fraction method.

14. Find the inverse Z – transform of 10 z / ( z-1) ( z-2) by inversion integral

method.

15. Find the inverse Z – transform of 2z / ( z -1 ) ( z - i ) ( z + I )

by inversi on intergral method

16. Using convolution theorem , evaluate the inverse Z – transform of

17. Using convolution theorem , evaluate the inverse Z – transform of

z2/ ( z –a) 2

18. Show that ( 1/ n! ) * (1/ n! ) = 2n / n!

19. Solve yn+2 + 6 y n+1 + 9yn = 2n with y0 = y1 = 0, using Z – transform.

20. Solve yn+2 - 2 y n+1 + yn = 3n + 5.