SESSIONWISE PROBLEMS ON CO-III

1. Determine the Fourier series to represent the function f(x)=(π-x)2from x= 0 to x = 2π.

2. Determine the Fourier series to represent the function f(x)=x-x2 from x= -π to x = π.

3. Identify the periodic function from the following wave form in the interval (0, 2π).

4. Determine the Fourier series to represent the function e-x from x= 0 to x = 2π.

SESSION-32

1. Express as a Fourier series of the periodic function with period 4, defined by the graph.

k

-2 -1 0 1 2

2. Obtain Fourier series of periodicity 3 of f(x) = 2x- x2 in 0<x <3. 3. Find the Fourier expansion for the function f(x) = x-x2 ; -1<x <1.

4. Obtain Fourier series of periodicity 3 of f(x) = x2-2 in -2<x<2.

SESSION-33

1. Compute the Fourier series for the function f(x) =|x| in– π x  π. 2. Determine the Fourier series for the function f(x) =│cos x│ in – π x  π 3. Express f(x) =x/2 as Fourier series in (-2, 2).

4. Compute the Fourier coefficient an for the function f(x)=x3-x in -1<x<1 and

f(x+2)=f(x). 0 π -π 2π 4π -2π -4π π

SESSION-34

1. Test even or odd for the function and determine the

Fourier series of f(x).

2. Test the function f(x)=            x x x x 0 ; -1 0 ; 1

for even and odd and hence express it

as a Fourier series.

3. Obtain the Fourier coefficients of the periodic function f(x)=         2 x 0 k; 0 x 2 k; -with f(x+4)=f(x).

4. Draw the graph of the triangular periodic signal defined by

and f(t+2)=f(t). Express the signal as series of sin

and cosine terms.

SESSION: 35 Tutorial on Fourier series

1. Construct the Periodic function for the graph in the interval (0, 2π),

(0, 0) (π,0) (2π, 0) and then express it as a Fourier Series.

2. Draw the graph for the function f(x) is defined as follows and express as Fourier series.

x2; 2x4

4. Obtain the Fourier series for f(x)=

         1 x 0 2; x 0 x 1 x;

5. Draw the graph for the function f(x) is defined as follows

f(x)=            2 x 1 1 1 x 1 -1 x 2 1

x and hence express f(x) as Fourier series.

6. Sketch the graph of the function , f(x+2 π)=f(x).

Express it as a Fourier series.

SESSION-36 MATLAB on FOURIER SERIES

1. Determine the Fourier series to represent the function f(x)=x-x2 from x= -π to x = π.

2. Find the Fourier expansion for the function f(x) = 1-x2 ; -1<x <1

3. Taking the period 4 expand the following function in the Fourier series f(x) =         4 x 2 2; x 2 x 0 x; SESSION-37

1. Expand f(x) =x in a cosine series valid for the interval 0< x< π.

2. Expand f(x) =(x-1)2 in a cosine series valid for the interval 0< x< 1`.

3. Expand f(x) =          1 2 1 if x, -1 2 1 x 0 if , 0 x

in a cosine series valid for the interval

0< x< 1.

4. Obtain the Fourier half range Fourier cosine series for

 

2

1 , 0 3

4 x

SESSION-38

1. Expand the function

            1 2 1 if 4 3 2 1 x 0 if 4 1 ) ( x x x x

f , as a series of sine terms.

2. Expand f(x) =x (π-x) in a sine series valid for the interval 0< x< π.

3. Expand f(x) =               x x x 2 if 2 x 0 if ,

in a sine series valid for the interval (0,π).

4. Construct the function by using graph; determine the half-range Fourier sine series

4.

SESSION-39 Tutorial on half range series

1. Sketch the graph for f(x) and obtain the half range Fourier sine series of

f(x)=        4 x 2 if 1, 2 x 0 if , 0 2. Expand f(x) =              x x 2 if x, -2 x 0 if ,

in a cosine series valid for the interval (0,π)

3. Expand f(x) =x sin x in a cosine series valid for the interval 0< x< π`.

4. Expand f(x)=                       x x 3 2 3 3 2 3 0 3 x 0 3

as Fourier sine series.

5. Expand f(x) =x2 in a cosine series valid for the interval 0< x< π`.

1.

2.

6. Construct the function by using graph; determine the half-range Fourier cosine series

SESSION: 40 MATLAB on Half Range Fourier Series

1. Expand f(x) =x sin x in a cosine series valid for the interval 0< x< π`. 2. Expand f(x) =x (π-x) in a sine series valid for the interval 0< x< π. 3. Expand f(x) =x2 in a cosine series valid for the interval 0< x< π`.

4. Expand f(x) =              x x 2 if x, -2 x 0 if ,

in a cosine series valid for the interval (0,π)

SESSION-41

1. Evaluate the steady state current I(t) in the RLC circuit where R=7ohms, L=1H,C=10-1F, if E(t) is given as follows with period 2

E(t)=            x 0 ; 50t 0 x ; 50t -2 2

2. Determine the general solution of the ODE x1112x35xr(t) , where r (t)=

       2 x 0 1; 0 x 2 1;

-3. Determine the transient and steady state oscillations of x5x6xr t

 

          x 0 1; 0 x 1; -) (t r

SESSION: 42 Tutorial on Sol of ODE using Fourier series

1. Determine the transient and steady state oscillations of x3x2xr t

 

         x 0 0; 0 x t;

2. Using Fourier series, Obtain the general solution of x1113x40xr(t) where

 

3. Determine the transient and steady state oscillations of x1111x30xr(t), where r(t)=         2 x 0 k; 0 x 2 k; with f(x+4)=f(x)

4. Determine the transient and steady state oscillations of x1164xr(t), where r(t)=         1 x 0 5; 0 x 1 5; with f(x+2)=f(x)

5. Obtain the general solution of x117x10xr(t)

where

 



  