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# Anna University Coimbatore 2009 B.E Transforms and Partial Differential Equations( 2 ks with Answers for Unit-1) - Question Paper

Wednesday, 16 January 2013 06:35Web

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

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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