tpde 8 marks

OMSAKTHI ADHIPARASAKTHI ENGINEERING COLLEGE, MELMARUVATHUR

DEPARTMENT OF MATHEMATICS

UNIT I (8 Marks)

FOURIER SERIES

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

20 x . (May/June 2007)

2) Find the Half range cosine series for the function xxxf in x0

Deduce that

90

.......3

1

2

1

1

1 4

444

. (May/June 2007) (May/June 2009)

3) Find the complex form of Fourier series for the function xexf in 11 x .

(May/June 2007)

4) Determine the Fourier series for the function

x

x

x

xxf

0

0

,1

,1 Hence

deduce that 4

........5

1

3

11

. (May/June 2007)

5) By finding the Fourier cosine series for xxf in x0 , Show that

14

4

12

1

96 n n

(Nov/Dec 2005)

6) Find the complex form of the Fourier series of the function xexf when

x and xfxf 2 (N/D07)

7) Find the Half range cosine series of 2xxf in the interval ,0 .Hence find

the sum of the series .......3

1

2

1

1

1444

.(Nov/Dec 2006)

8) Find the Fourier series as the second harmonic to represent the function given un

the following data:

X : 0 1 2 3 4 5

Y : 9 18 24 28 26 20. (N/D 06)

10) Find the Fourier series expansion of period and period for the

function

ll

l

in

in

xl

xxf

,2

2,0

Hence deduce the sum of the series

14)12(

1

n n

(Nov/Dec 2006)

11) Obtain the Fourier series of xf of period 2l and defined as follows

lxl

lxxlxf

2

0

,0

, .Hence deduce that

4........

5

1

3

11

and

8

.......5

1

3

1

1

1 2

222

(Nov/Dec 2007),(Dec 2008)

12) Determine the Fourier expansion of xxf in the interval x .

(Apr/May 2004)

13) Find the Half range cosine series for xxsin in ,0 .(A/M 2004)

14) Obtain the Fourier series for the function

21

10

,2

,

x

x

x

xxf

.

(Apr/May 2004)

15) Find the Fourier series of period 2 for the function

2,

,0

2

1

in

inxf

and

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

1

3

1

1

1222

. (Apr/May 2004),

(Apr/May 2005)

16) Obtain the Fourier expansion for xcos1 in x . (March 1996)

17) Find the Fourier series for the function

21

10

,1

,

x

x

in

in

x

xxf .Deduce that

8.......

5

1

3

1

1

1 2

222

. (Nov/Dec 2005)

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

(Nov/Dec 2005)

19) Find the Fourier series for 2xxf in , .Hence find

.......3

1

2

1

1

1444

. (Nov/Dec 2005) (Dec 2008)

20) Expand in Fourier series of periodicity 2 of 2

2x

e

.

21) Find the Fourier series expansion of the periodic function xf of period 2l defined

by

lx

xl

xl

xlxf

0

0

,

, Deduce that

812

1 2

12

n n

(Oct/Nov 2002)

22) Obtain the half range cosine series for 22 xxf in the interval (0,2).

(Dec 2008)

23) Expand 2xxxf in , as a full range Fourier series and hence deduce the

sum of the series

12

1

n n (Dec 2008)

24) Expand 20,2 2 xxxxf as a series of cosines. (Dec 2008)

25) Give the sine series of 1xf in ,0 and prove that 8

1 2

...3,12

n (Dec 2008)

26) Find the Fourier series up to second Harmonic for the data

x : 0 60 120 180 240 300 360

f(x): 1 1.4 1.9 1.7 1.5 1.2 1 (Dec 2008)

27) Find the cosine series of 2xxf in ,0 (Dec 2008)

28) Find Fourier series of

lxl

lxxlxf

2

0

,0

, (Dec08) (N/D07)

29) Find the Fourier series of 2xxf in 2,0 and periodic with period 2 . Hence

deduce that

1

2

2 6

1

n n

(May/June 2009)

30) If a is not an integer , find the complex Fourier series of axxf cos in , .

(May/June 2009)

31) Compute the first two harmonics of the Fourier series of xf given in the

following table:

x : 0 3/ 3/2 3/4 3/5 2

xf : 1.0 1.4 1.9 1.7 1.5 1.2 1.0

(May/June 2009)


DEPARTMENT OF MATHEMATICS UNIT-II (8 Marks)

FOURIER TRANSFORM

1) Find the Fourier transform of f(x) =

0

1

otherwise

xfor 1 .Hence prove that

0 0

2

2

sinsin dx

x

xdx

x

x. (Nov 2002),(Nov/Dec 2003)

2) Find the Fourier transform of

0

sin xxf

x

x

0. (Nov 2002)

3) Find the Fourier cosine transform of 2,0)2(4)1(3)( nnynyny .Deduce

that

0

8

2 816

2cosedx

x

x ,

0

8

2 816

2sinedx

x

xx .

4) State and prove the Convolution theorem for Fourier transforms.

(Nov 2002)

5) Find the Fourier transform of 0,

aexa

. Deduce that (i)

0

3222 4

1

adx

ax

,

(ii) 222

22

sa

asixeF

xa

. (April 2003)

6) Find the Fourier Sine transform of 2

2x

xe

. (April 2003)

7) Find the Fourier cosine transform of 22xae .Hence evaluate the Fourier Sine

transform of 22xaxe

. (Nov/Dec 2006)

8) Find the Fourier transform of 22xae . Hence prove that 2

2x

e

is self-reciprocal.

(May 2006), (May 2007)

9) Find the Fourier cosine transform of

0

1 2xxf

otherwise

x 10 . Hence prove that

0

3.

16

3

2cos

cossin dx

x

x

xxx (April 2003)

10) Derive the Parseval’s identity for Fourier transforms. (April 2003)

11) Find the Fourier Sine and cosine transform of xe . Hence find the Fourier Sine

transform of 21 x

x

and Fourier cosine transform of

21

1

x.

(Nov/Dec 2003)

12) Show that Fourier transform of

0

22 xaxf

:

:

ax

ax

is

3

cossin22

aaa. Hence deduce that

0

3.

4

cossin dt

t

ttt

(Apr/May 2004)

13) Find the Fourier Sine and cosine transform of

0

2 x

x

xf

:

:

:

2

21

10

x

x

x

(Apr/May 2004)

14) If f is the Fourier transform of f(x) , find the Fourier transform of f(x-a) and

f(ax). (Apr/May 2004)

15) Verify Parseval’s theorem of Fourier transform for the function

xe

xf0

:

:

0

0

x

x

(Apr/May 2004)

16) Find the Fourier transform of f(x),

0

1 2xxf

:

:

1

1

x

x. Hence evaluate

(i)

0

3 2cos

cossindx

x

x

xxx (ii)

0

2

3 15

cossin ds

s

sss.

(April 2005) (Dec 2008)

17) Find the Fourier Sine transform of 0

ax

e ax

. (Nov/Dec 2006)

18) Find the Fourier Sine and cosine transform of xe 2 . Hence find the value of the

following integrals (i)

0

22 4x

dx (ii)

0

22

2

4dx

x

x. (A.U.Model Qu)

19) Evaluate (i)

0

2222 bxax

dx (ii)

0

22 41 xx

dxusing Fourier transform.

(Nov/Dec 2008)

20) Find the Fourier Sine and cosine transform of 1nx . (May 2006)

21) Using Parseval’s identity for Fourier cosine transform of ax

e

evaluate

0

222 xa

dx .

(Nov/Dec 2007)

22) Find the Fourier Sine transform of 0, ae ax . Hence find ax

S xeF . Hence deduce

the inversion formula. (May/June 2007)

23) Find the Fourier Sine transform of f(x) defined as

0

sin xxf

where

where

ax

ax

0. (Dec 2008)


0

1 xxf

otherwise

for

1x. Hence find the values

of (i)

0

4sin

dtt

t and (ii)

0

2sin

dxx

x (Dec 2008)

25) Find the finite sine and cosine transform of 2

1

xxf in the interval ,0 .

(Dec 2008)


0

xaxf

,

,

ax

ax

. (Dec 2008)

27) Evaluate

0

2222

2

xbxa

dxx using Parseval’s identity. (Dec 2008)

28) Find the Fourier transform of xf if

,0

,1xf

if

if

0ax

ax

. Hence deduce that

0

2

2

sin dt

t

t . (May/June 2009)

29) Find the Fourier Cosine transform of 22 xae for any a>0 and hence prove that 2/2xe

is self-reciprocal under Fourier Cosine transform. (May/June 2009)


,0

,22 xaxf

if

if

0ax

ax

. Hence deduce that

0

3 4

cossin dt

t

ttt. (May/June 2009)

31) Find ax

C eF ,

21

1

xFC and

21 x

xFC . (Hence CF stands for Fourier Cosine

transform) (May/June 2009)


DEPARTMENT OF MATHEMATICS UNIT-III (8 Marks)

PARTIAL DIFFERENTIAL EQUATIONS

1) Form the PDE by eliminating the arbitrary constants ‘a’ and ‘b’ from the expression

2222czbyax . (May/June 2007)

2) Solve yxZDDDD 2sin5252 22 . (May/June 2007)

3) Solve 2222 yxzqzxypzyx . (May/June 2007)

4) Solve p(1+q)=qz. (May/June 2007)

5) Solve zxxyqpzyx 22)( 222 . (Nov/Dec 2007) (May/June 2009)

6) Solve yxeyxZDDDD 4322 322 . (Nov/Dec 2007)

7) Solve 22222 yxqpz . (Nov/Dec 2007)

8) Solve yxzDDDD sin)43( 22 . (Nov/Dec 2007)

9) Find the singular integral of 22 qpqpqypxz . (Nov/Dec 2006)

10) Solve xyZDDDD sin65 22 . (Nov/Dec 2006)

11) Solve xyqzxpyz 322443 .(N/D 2006), (N/D 2003)

12) Solve yxeyxzDDDD 222 )2( . (Nov/Dec 2006)

13) Find the singular integral of PDE 22 qpqypxz .(N/D 2003)

14) Solve yxezDDDD yx 2sin3)54( 222 .(N/D 2003)

15) Find the general solution of 222222 xyzqzxypyzx .

16) Solve 22322 )2()2332( yx eezDDDDDD . (N/D 2003)

17) Solve yxzqxzypzyx . (Apr/May 2004)

18) Solve yxeyxzDDDD 2323 2sin)67( . (A/M 2004)

19) Solve 22 yxzqyxzpyx . (Nov/Dec 2005)

20) Solve yxezDDDDDD 222 )1222( . (N/D 2005)

21) Solve 221 qpqypxz . (May/June 2009) (Nov/Dec 2005),(Apr/May 2004)

22) Solve xyy

z

yx

z

x

zcos6

2

22

2

2

. (Nov/Dec 2005)

23) Solve yxzqxzyxpyzyx 2222 .(N/D 2005)

24) Solve yxezDDDD yx 4sin)20( 522 .(Nov/Dec 2005)

25) Solve 22 qpz . (Nov/Dec 2005)

26) Solve yxeyxzDDDD 3222 )6( . (Nov/Dec 2005)

27) Form the PDE by eliminating the arbitrary functions f and g in

yxgyxfz 22 33 . (Oct/Nov 2002)

28) Solve yxyxqxyzpxzy . (Oct/Nov 2002)

29) Solve yexyzDDDD x 622 )30( .

30) Solve 222 1 qpz . (April 1996) (Dec 2008)

31) Solve zxqyxpzy 22 . (Apr/May 2003)

32) Form the PDE by eliminating the arbitrary functions f and g in xgyyfxz 22

33) Form the p.d.e by eliminating the function f and g from yxxgyxfz 22

34) Solve zxqyp (Dec 2008)

35) Solve 22222 zyqxp (Dec 2008)

36) Solve yxyxzDDD 223 32sin2 (Dec 2008)

37) Form the p.d.e by eliminating the arbitrary function f and g from

yxxgyxfz 3232 (Dec 2008)

38) Solve 022 xzxyqpzy (Dec 2008)

39) Solve xytsr cos6 (Dec 2008)

40) Obtain complete solution of the equation pqqypxz 2 (Dec 2008)

41) Solve yxzDDDD 2cos6 22 . (Dec 2008)

42) Solve xyyzqxzp (Dec 2008)

43) Solve yxezDDDD 222 252 (Dec 2008)

44) Find the complete solution of 2zpqxy (May/June 2009)

45) Solve the equation yxeZDD yx 2sin22 . (May/June 2009)


DEPARTMENT OF MATHEMATICS UNIT –IV (8 Marks)

APPLICATIONS OF PDE

1) A tightly stretched string of length ‘ l ’ has its ends fastened at x=0 & x=l . The midpoint

of the string is then taken to a height ‘ h ’ and then released from rest in that position.

Obtain an expression for the displacement of the string at any subsequent time. (N 2002)

2) A tightly stretched flexible string has its ends fixed at x=0 and x=l. At time t=0 , the

string is given a shape defined by ),()( 2 xlkxxf where k is a constant , and then

released from rest. Find the displacement of any point x of the string at any time t > 0.

A2003)

3) A tightly stretched string with fixed end points x=0 and x=l is initially in a position given

by

l

xyxy

3

0 sin)0,( . It is released from rest from this position. Find the

displacement at anytime ‘ t ’. (Nov 2004)

4) A tightly stretched string of length ‘ 2l ’ has its ends fastened at x=0 , x=2l. The midpoint

of the string is then taken to height ‘ b ’ and then released from rest in that position. Find

the lateral displacement of a point of the string at time ‘ t ’ from yhe instant of release.

(May 2005)

5) A string of length ‘ l ’ has its ends x=0 , x=l fixed. The point where 3

lx is drawn

6) side a small distance ‘ h ’,the displacement ),( txy satisfies .2

22

2

2

x

ya

t

y

Find ),( txy at

any time ‘ t ’.

7) An elastic string of length ‘ 2l ’ fixed at both ends is disturbed from its equilibrium

position by imparting to each point an initial velocity of magnitude ).2( 2xlxk Find the

displacement function ),( txy . (May ‘06)

8) A uniform string is stretched and fastened to two points ‘ l ’ apart. Motion is started by

displacing the string into the form of the curve ),( xlkxy and then releasing it from

his position at time t=0. Find the displacement of the point of the string at a distance ‘ x ’

from one end at time ‘ t ’. (A.U.Tri. Nov/Dec 2008) (Dec 2008) (May/June 2009)

9) If a string of length ‘ l ’ is initially at rest in its equilibrium position and each of its points

is given a velocity ‘ v ’ such that

)( xlc

cxv

for

for

lxl

lx

2

20

show that the

displacement at any time‘ t ’ is given by

...

3sin

3sin

3

1sinsin

4),(

33

2

l

at

l

x

l

at

l

x

a

cltxy

. (Nov/Dec2008)

10) A string is stretched between two fixed points at a distance 2l apart and the points of the

string are given initial velocities ‘ v ’ where

)2( xll

cl

cx

v in

in

lxl

lx

2

0

‘ x ’

being the distance from one end point .Find the displacement of the string at any

subsequent time. (April/May 2004)

11) The ends A and B of a rod ‘ l ’ cm long have the temperatures C

40 and C

90 until

steady state prevails. The temperature at A is suddenly raised to C

90 and at the same

time that at B is lowered to C

40 . Find the temperature distribution in the rod at time ‘ t

’ . Also show that the temperature at the midpoint of the rod remains unaltered for all

time , regardless of the material of the rod. (April 2003)

12) A metal bar 10 cm long with insulated sides , has its ends A and B kept at C

20 and

C

40 until steady state conditions prevail. The temperature at A is then suddenly raised

to C

50 and at the same instant that at B is lowered to C

10 . Find the subsequent

temperature at any point of the bar at any time .

(Nov/Dec 2005)

13) The ends A and B of a rod ‘ l ’cm long have their temperatures kept at C

30 and

C

80 , until steady state conditions prevail. The temperature at the end B is suddenly

reduced to C

60 and that of A is increased to C

40 . Find the temperature distribution

in the rod after time ‘ t ’. (M/J’ 07)

14) The boundary value problem governing the steady state temperature distribution in a flat,

thin , square plate is given by

,02

2

2

2

y

u

x

u ax 0 , ay 0 , 0)0,( xu ,

a

xaxu

3sin4),( , ax 0

0),0( yu , 0),( yau , ay 0 . Find the steady-state temperature distribution

in the plate. (Nov 2002)

15) A rectangular plate with insulated surface is 10 cm wide so long compared to its width

that it may be considered infinite length. If the temperature along short edge y=0 is

given by

10sin8)0,(

xxu

when 100 x , while the two long edges x=0 and x=10

as well as the other short edge are kept at C

0 , find the steady state temperature

function ),( yxu . (Nov 2003)

16) An infinitely long rectangular plate with insulated surface is 10 cm 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

)10(20

20

y

yu

for

for

105

50

y

y . Find the steady

state temperature in the plate. (Nov/Dec 2005), (Nov 2004) (Dec 2008)

17) A rectangular plate with insulated surface is 10 cm wide and so long compared to its

width that it may be considered infinite in length without introducing appreciable error.

The temperature at short edge y=0 is given by

)10(20

20

x

xu

for

for

105

50

x

x and all

the other three edges are kept at C0 . Find the steady state temperature at any point in

the plate. (May 2005)

18) Find the steady state temperature distribution in a rectangular plate of sides a and b

insulated at the lateral surface and satisfying the boundary conditions

0),(),0( yauyu for by 0 , 0),( bxu and )()0,( xaxxu for ax 0 .

(Nov/Dec 2005)

19) An infinitely long plate in the form of an area is enclosed between the lines

yy ,0 for positive values of x. The temperature is zero along the edges

yy ,0 and the edge at infinity. If the edge x=0 is kept at temperature ‘ Ky(l-y)’ ’

find the steady state temperature distribution in the plate.

(May 2006)

20) An infinitely long uniform plate is bounded by two parallel edges and an end at right

angle to them. The breadth of this edge x=0 is , this end is maintained at temperature

as )( 2yyKu at all points while the other edges are at zero temperature . Find the

temperature ),( yxu at any point of the plate in the steady state.

21) A rod of length ‘‘l ’ has its ends ‘A’ and ‘B’ kept at C0 and C120

respectively

until steady state conditions prevail. If the temperature at ‘B’ is reduced to C0 and

kept so while that of ‘A’ is maintained, find the temperature distribution in the rod.

(Dec 2008)

22) Find the steady state temperature in a circular plate of radius ‘a’ cm, which has one half

of its circumference at C0 and the other half at C100

.

(Dec 2008)

23) Find the steady state temperature distribution in a square plate bounded by the lines

20,20,0,0 yxyx . Its surfaces are insulated, satisfying the boundary conditions

xxxuxuyuyu 2020,&00,,20,0 . (Dec 2008)

24) A rectangular plate with insulated surface is 10 cm wide and so long compared to its

width that it may be considered infinite in length without introducing appreciable error. If

the temperature of the short edge y=0 is given by xu for 50 x and x10 for

105 x and the two long edges x=0,x=10 as well as the other short edges are kept at

C0 . Find the temperature yxu , at any point yx, of the plate in the steady state.

(May/June 2009)


DEPARTMENT OF MATHEMATICS UNIT-V (8 Marks)

Z-TRANSFORM

1) Find

)2()1( 2

31

zz

zZ using partial fraction. (N/D2005)(Dec2008)

2) Solve the difference equation 0)(4)1(4)2( kykyky where .0)1(,1)0( yy

(Nov/Dec 2005)

3) Prove that

1log

1

1

z

zz

nZ . (Nov/Dec 2005)

4) State and prove second shifting theorem in Z-transform. (Nov/Dec 2005)

5) Using convolution theorem evaluate inverse Z-transform of

)3)(1(

2

zz

zZ .

(Dec 2008) (May/June 2006)

6) Using Z-transform solve 2,0)2(4)1(3)( nnynyny given that

.2)1(,3)0( yy (May/June 2006)

7) Find

2

21

)1)(1(

)2(

zz

zzzZ by using method of partial fraction.(N/D 2006)

8) Find Z-transform of )2)(1(

1

nn. (Nov/Dec 2006)

9) Using convolution theorem evaluate

)2)(1(

21

zz

zZ .(Nov/Dec 2006)

10) Find Z-transform of na and nan cos . (May/June 2007)

11) Using the Z-transform method solve 22 nn yy given that 010 yy .

(May/June 2007)

12) State and prove final value theorem in Z-transform.(May/Jun 2007)

13) Find the inverse Z-transform of 3)1(

)1(

z

zz. (May/June 2007)

14) State and prove first shifting theorem on Z-transform. Also find teZ at .

15) Use Z-transform to solve n

nnn yyy 2127 12 given 010 yy .

(Dec 2008)

16) Find iateZ and hence deduce the values of atZ cos and atZ sin .

17) Find

)1()1( 2

1

zz

zZ .

18) Prove that 1 pp nZdz

dznZ where p is any positive integer. Deduce that

2)1(

z

znZ and

3

22

)1(

2

z

znZ .

19) Find the inverse Z-transform of )4)(2(

32 2

zz

zz.

20) Find the inverse Z-transform of 21 nn . (Dec 2008)

21) Using Z-transforms, solve ,1,04132 nnynyny given that 30 y and

21 y . (Dec 2008)

22) Find the Z-transform of the sequence 1

1

nfn . (Dec 2008)

23) Find the inverse Z-transform of 2

2

2

42

z

zzZF using residue theorem. (Dec 2008)

24) By using convolution theorem, prove the inverse of bzaz

z

2

is 111

nn

n

abab

.

(May/June 2009).

25) By the method of Z – transform solve nnynyny 29162 given that

00 y and 01 y . (May/June 2009)

26) Find the Z – transform of ncos and hence find nnZ cos .

(May/June 2009)

27) Solve the equation (using Z – transform) 366152 nynyny given that

.010 yy (May/June 2009)

tpde 8 marks

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