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)
OMSAKTHI ADHIPARASAKTHI ENGINEERING COLLEGE, MELMARUVATHUR
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)
24) Find the Fourier transform of
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)
26) Find the Fourier transform of
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)
30) Find the Fourier transform of
,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)
OMSAKTHI ADHIPARASAKTHI ENGINEERING COLLEGE, MELMARUVATHUR
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)
OMSAKTHI ADHIPARASAKTHI ENGINEERING COLLEGE, MELMARUVATHUR
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)
OMSAKTHI ADHIPARASAKTHI ENGINEERING COLLEGE, MELMARUVATHUR
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)
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