Anna University Coimbatore 2009 B.E Transforms and Partial Differential Equations( 2 ks with Answers for Unit-1) - Question Paper
TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS
UNIT-I
FOURIER SERIES
TWO MARKS
1)
Determine 
in
the Fourier series expansion of 
in
with
period 
.
Solution:
2)
Define root
mean square value of 
in
.
(OR) Define root mean square value of a function 
over
the range (a,b) .
Solution:
Let 
be
the function defined in the interval 
then
is
called the Root Mean Square value and is denoted by 
.
The RMS value is otherwise known as Effective value 
.
3)
If 
and
for
all x,
find the sum of the Fourier series of 
at
.
Solution:
Sum of the Fourier Series
4)
Find the
value of 
in
the cosine series expansion of 
in
the interval (0,10).
Solution:
5)
Find the
root mean square value of the function 
in
the interval ( 0, l ).
Solution:
6) State Dirichlets conditions for a given function to expand in Fourier series.
(OR) State the Dirichlets condition for the convergence of the Fourier series of
in
with
period 
.
Solution:
Any function 
can
be developed as a Fourier series 
where
are
constants provided
is
a periodic, single valued, finite.
is
continuous with finite number of discontinuities in any one period.
has
atmost finite number of maxima and minima.
7)
If the
Fourier series of the function 
in
the interval
is
,
then find the value of the infinite series 
Solution:
Here 
is
a point of discontinuity. Sum of Fourier series
8)
Find the
Fourier sine series of the function 
,
.
Solution:
9)
If the
Fourier series of the function 
is
deduce
that 
.
Solution:
Here 
is
a point of continuity. Sum of Fourier series
10)
Does 
possess
a Fourier expansion ?
Solution:
Here 
is
a Periodic function with period 
,
not 
and
it does not satisfy the Dirichlets condition.
11) State Parsevals Theorem on Fourier series. (OR) State Parsevals identity of Fourier series.
Solution:
If 
is
expressed as a Fourier series of periodicity 
in
then
.
12)
Find in
the expansion of 
as
a Fourier series in 
.
Solution:
Here 
So,
is
an Even function. Hence 
.
13)
If 
is
an odd function defined in ( -l , l ) , what are the values of
and
?
Solution:
Here 
is
an Odd function in 
.
So 
and
.
14)
Find the
constant term in the Fourier series corresponding to 
expressed
in the interval 
.
Solution:
It
is an Even function. So 
15)
To which
value the Half range sine series corresponding to 
expressed
in the interval (0,2) converges to 
?
Solution:
Here 
is
a point of discontinuity. Sum of Fourier series
i.e., Sum of Fourier series
16)
If 
is
expressed as a Fourier series in the interval ( - 2, 2 ) to which value this
series converges at 
?
Solution:
Here 
is
a point of discontinuity.
Sum
of Fourier series
17)
If the
Fourier series corresponding to 
in
the interval 
is
,without
finding the values of 
Find
the value of 
Solution:
.
Hence
.
18)
Find the
Half range sine series for 
in
.
Solution:
19)
If the
cosine series for 
for
is
given by 
Prove
that 
.
Solution:
Here 
is
a point of continuity.
Sum of Fourier series 
20) What do you mean by Harmonic analysis? (OR) Define Harmonic analysis.
Solution:
The process of finding the Fourier series for a function given by
Numerical value is known as Harmonic Analysis,
2[Mean
value of 
in
2[Mean
value of 
in
2[Mean
value of 
in
.
21)
In the
Fourier expansion
in
,
find the value of ,the
coefficient of 
Solution:
Here 
is
an Even function. So .
22)
Find 
in
expanding 
as
Fourier series in
.
Solution:
23)
If 
in
,
find the sum of the series 
Solution:
24)
The Fourier
series of 
in
(0,2) and that of 
in
(-2,0) are identical or not. Give reason.
Solution:
For 
,
For 
Since 
for
and
for
are
not equal, hence their Fourier series are not identical.
25)
Define the
value of the Fourier series of 
at
a point of discontinuity.
Solution:
is
defined in 
.
Sum of Fourier series
If point of discontinuity is at 
.
Sum of Fourier series
.
26)
If 
is
defined in 
,
write the values of Fourier coefficients 
and
.
Solution:
.
So, it is an Odd function
Hence, the values of 
and
.
27)
If 
in
,
prove that 
.
Solution:
By Parsevals identity, 
28)
The
functions 
,
cannot
be expanded as a
Fourier series. Why ?
Solution:
It does not satisfy the Dirichlets condition. So, it is not possible to
expand the function as Fourier series.
29)
Expand the
function 
,
as
a series of sines.
Solution:
30)
Find the
Fourier Cosine series of 
,
.
Solution:
31)
,
which
one of the following is correct a) an even function b) an odd function
c) neither even nor odd
Solution:
c) neither even nor odd
(
is
Even function only in the interval 
.But
here the interval is 
).
32)
Let 
be
defined in 
by
and
.
Find the value of 
.
Solution:
Sum of Fourier series
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