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 expansionin
,
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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