Engineering Mathematics 2015 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1 SUBJECT NAME : Transforms and Partial Diff. Eqn. SUBJECT CODE : MA2211 MATERIAL NAME : University Questions REGULATION : R2008 WEBSITE : www.hariganesh.com UPDATED ON : May-June 2015 TEXT BOOK FOR REFFERENCE : Hariganesh Publications (Author: C. Ganesan) To buy the book visit www.hariganesh.com/textbook Name of the Student: Branch: Unit – I (Fourier Series) Fourier Series in the interval (0,2ℓ) 1. Expand () 2 fx x x as Fourier series in 0, 2 and hence deduce that the sum of 2 2 2 2 1 1 1 1 ... 1 2 3 4 . (A/M 2011) Text Book Page No.: 2.6 2. Find the Fourier series of 2 ( ) fx x in 0, 2 of periodicity 2 . (M/J 2012) Text Book Page No.: 2.3 3. Find the Fourier series expansion of for 0 ( ) 2 for 2 x x fx x x . Also, deduce that 2 2 2 2 1 1 1 ... 1 3 5 8 . (N/D 2010) Text Book Page No.: 2.12
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Engineering Mathematics 2015
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1
SUBJECT NAME : Transforms and Partial Diff. Eqn.
SUBJECT CODE : MA2211
MATERIAL NAME : University Questions
REGULATION : R2008
WEBSITE : www.hariganesh.com
UPDATED ON : May-June 2015
TEXT BOOK FOR REFFERENCE : Hariganesh Publications (Author: C. Ganesan) To buy the book visit www.hariganesh.com/textbook
Name of the Student: Branch:
Unit – I (Fourier Series)
Fourier Series in the interval (0,2ℓ)
1. Expand ( ) 2f x x x as Fourier series in 0,2 and hence deduce that the
sum of 2 2 2 2
1 1 1 1...
1 2 3 4 . (A/M 2011)
Text Book Page No.: 2.6
2. Find the Fourier series of 2
( )f x x in 0,2 of periodicity 2 .
(M/J 2012)
Text Book Page No.: 2.3
3. Find the Fourier series expansion of for 0
( )2 for 2
x xf x
x x
. Also, deduce
that 2
2 2 2
1 1 1...
1 3 5 8
. (N/D 2010)
Text Book Page No.: 2.12
Engineering Mathematics 2015
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 2
4. Find the Fourier Series Expansion of 1 for 0
( )2 for 2
xf x
x
. (N/D 2013)
Text Book Page No.: 2.10
5. Determine the Fourier series for the function ( ) sinf x x x in 0 2x .
(N/D 2014)
Text Book Page No.: 2.21
6. Find the Fourier series expansion of , 0 1
( )2 , 1 2
x xf x
x x
. Also, deduce
2
2 2 2
1 1 1...
1 3 5 8
. (N/D 2012)
Text Book Page No.: 2.38
7. Find the Fourier series for 2( ) 2f x x x in the interval 0 2x . (A/M 2010)
Text Book Page No.: 2.38
8. Obtain the Fourier series of periodicity 3 for 2( ) 2f x x x in 0 3x .
Text Book Page No.: 2.33 (N/D 2011),(N/D 2014)
Fourier Series in the interval (-ℓ,ℓ)
1. Find the Fourier series of 2x in , and hence deduce that
4
4 4 4
1 1 1...
1 2 3 90
. (M/J 2013)
Text Book Page No.: 2.42
2. Obtain the Fourier series of ( ) sinf x x x in , . (N/D 2011)
Text Book Page No.: 2.47
3. Obtain the Fourier series to represent the function ( )f x x , x and
deduce
2
21
1
82 1n n
. (M/J 2012)
Text Book Page No.: 2.45
Engineering Mathematics 2015
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 3
4. Obtain the Fourier series of the periodic function defined by
0( )
0
xf x
x x
. Deduce that
2
2 2 2
1 1 1...
1 3 5 8
. (N/D 2009)
Text Book Page No.: 2.66
5. Expand
21 , - 0
( )2
1 , 0
xx
f xx
x
as a full range Fourier series in the interval
, . Hence deduce that 2
2 2 2
1 1 1...
1 3 5 8
. (M/J 2014)
Text Book Page No.: 2.58
6. Obtain the Fourier series for the function ( )f x given by1 , 0
( )1 , 0
x xf x
x x
.
Hence deduce that2
2 2 2
1 1 1...
1 3 5 8
. (A/M 2011)
Text Book Page No.: 2.61
7. Find the Fourier series expansion of 1, 0
( )1, 0
x xf x
x x
. (N/D 2013)
Text Book Page No.: 2.61
8. Find the Fourier series of the function0, 0
( )sin , 0
xf x
x x
and hence evaluate
1 1 1...
1.3 3.5 5.7 . (N/D 2011)(AUT)
Text Book Page No.: 2.78
9. Expand 2( )f x x x as a Fourier series in L x L and using this series find the
root mean square value of ( )f x in the interval. (N/D 2009)
Text Book Page No.: 2.79
10. Find the Fourier series expansion of 2( )f x x x in , . (N/D 2012)
Text Book Page No.: 2.62
Engineering Mathematics 2015
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 4
11. Find the Fourier series expansion of 2( ) 1f x x in the interval 1,1 .
Text Book Page No.: 2.72 (N/D 2010)
Half Range Fourier Series
1. Obtain the half range cosine series for ( )f x x in 0, and hence show that
4
4 4
1 11 ...
3 5 96
. (N/D 2010),(N/D 2012),(N/D 2014)
Text Book Page No.: 2.81
2. Find the half range cosine series of the function ( ) ( )f x x x in the interval
0 x . Hence deduce that4
4 4 4
1 1 1...
1 2 3 90
. (A/M 2010)
Text Book Page No.: 2.108
3. Find the half-range Fourier cosine series of 2
( )f x x in the interval (0, ) .
Hence find the sum of the series 4 4 4
1 1 1...
1 2 3 . (M/J 2012)
Text Book Page No.: 2.95
4. Obtain the Fourier cosine series of 2
1 , 0 1x x and hence show that
2
2 2 2
1 1 1...
1 2 3 6
. (M/J 2013)
Text Book Page No.: 2.88
5. Obtain the half range cosine series for ( )f x x in 0, . (N/D 2010),(N/D 2012)
Text Book Page No.: 2.81
6. Obtain the Fourier cosine series expansion of sinx x in 0, and hence find the value
of 2 2 2 2
1 ...1.3 3.5 5.7 7.9
. (N/D 2011)
Text Book Page No.: 2.83
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Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 5
7. Find the half-range sine series of 2( ) 4f x x x in the interval 0,4 . Hence deduce
the value of the series 3 3 3 3
1 1 1 1...
1 3 5 7 . (M/J 2014)
Text Book Page No.: 2.109
8. Find the half range sine series of 2( )f x x x in 0, . (N/D 2013)
Text Book Page No.: 2.86
9. Obtain the sine series for
in 02
( )
in 2
x x
f x
x x
. (A/M 2011)
Text Book Page No.: 2.109
10. Obtain the Fourier cosine series for
in 02
( )
in 2
kx x
f x
k x x
. (M/J 2013)
Text Book Page No.: 2.91
Complex Form of Fourier Series
1. Find the complex form of the Fourier series of ( )ax
f x e , x .(A/M 2010)
Text Book Page No.: 2.113
2. Show that the complex form of Fourier series for the function ( )x
f x e when
x and ( ) ( 2 )f x f x is 2
sinh 1( ) 1
1
n inx
n
inf x e
n
.
Text Book Page No.: 2.113 (N/D 2014)
3. Find the complex form of the Fourier series of ( )x
f x e in 1 1x .(N/D 2009)
Text Book Page No.: 2.117
4. Find the complex form of Fourier series of cosax in , , where " "a is not an
integer. (M/J 2013)
Engineering Mathematics 2015
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 6
Text Book Page No.: 2.115
5. Expand ( ) sinf x x as a complex form Fourier series in , . (M/J 2014)
Text Book Page No.: 2.124
Harmonic Analysis
1. Compute upto first harmonics of the Fourier series of ( )f x given by the following table
x 0 T/6 T/3 T/2 2T/3 5T/6 T
( )f x 1.98 1.30 1.05 1.30 – 0.88 – 0.25 1.98
(N/D 2009),(N/D 2011)
Text Book Page No.: 2.131
2. Find the Fourier series as far as the second harmonic to represent the function ( )f x
with the period 6, given in the following table.
(N/D 2009),(N/D 2010),(M/J 2012),(N/D 2012)
x 0 1 2 3 4 5
( )f x 9 18 24 28 26 20
Text Book Page No.: 2.129
3. Find the Fourier series up to second harmonic for ( )y f x from the following values.
x: 0 π/3 2 π/3 π 4π/3 5 π/3 2 π
y: 1.0 1.4 1.9 1.7 1.5 1.2 1.0
(A/M 2011),(N/D 2013),(M/J 2014)
Text Book Page No.: 2.127
4. Calculate the first 3 harmonics of the Fourier of ( )f x from the following data (N/D 2011)