# 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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