Exercises for Module-III (Transform Calculus) - Nptel

Exercises for Module-III (Transform Calculus)

1) Discuss the piecewise continuity of the following functions:

a)

( ) 1 , 2

f t 2 t

=t ≠

− b)

2

2 , 1

1 , 1

( )

t t

t t

f t

≤

+ >

= 



c)

1 , 0

0, 0

( )

e t t t

t

f t

− − ≠

=

= 

 d)

sin 1, 0 0, 0

( ) ^{t t t}

t

f t

  

  ≠

=

= 



2) Show that the function ^{f t( )=t et2sin}

( )

^et2 possesses a Laplace transform.

3) Find Laplace transform of the following functions:

a)

cos , 2 0 e⁻t t t >

b)

0, 0 t<2 , 2

( )

eat t

f t

≤

≥

= 



c)

( ) sin , 0

f t = t t > d) f t( )=t H t( −a), t>0

4) Find the inverse Laplace transform of the following functions:

a) ^{F s( )=}

(

^s2+s6+s3+13

)

^{2 b) F s( )=}

⁽

^s−14

^{) (}

^s2+s5+2

⁾

c)

²

( ) 2

e s

F s s

−π

= − d)

( ) ( )

2 2 2 2

2 3

( )

1 1

F s s

s s

= +

+ +

5) Solve the following initial value problems for 𝑡𝑡> 0 using Laplace transform method:

a) d y²₂ 1; (0) '(0) 0

y y y

dt + = = =

b) d y²2 dy

(

1 ( 1) ; (0)

)

1, '(0) 1

H t y y

dt + dt = − − = = −

c) ²2

cos , 0 0,

( ); (0) '(0) 0; where ( )

t t

t

d y y f t y y f t

dt

π π

≤ ≤

>

+ = = = = 



6) Solve the following boundary value problems using Laplace transform methods:

a) ²2 sin , (0) 1, ,

2

d y y t y y

dt

π π

+ = =    =

b) ²₂ 9 , (0) 1, ' 1

3

d y t y y

dt

λ  π

+ = =  = −

 

7) Solve the following differential equations using Laplace transform method:

a) d y²₂ dy 1; (0) 0

t y

dt − dt = − =

b) d y²₂ ( 1)dy 2 ^t; (0) 0

t t y e y

dt dt

+ + + = − =

8) Solve the following integral equations using Laplace transform method:

a)

0

( ) sin 2 ( ) cos( ) d

t

y t = t+

∫

y u t−u u

b)

( )

0

( ) 3 ( ) 2 ( ) d ; 0 1 dy t t

y t y u u t y

dt + +

∫

= =

9) With the application of Laplace transform, solve the Heat equation ²u₂ u; 0, 0

x t

x t

∂ =∂ > >

∂ ∂ subject to the conditions ( , 0 ) 0, 0

u x + = x>

(0, ) ( ), 0 u t = f t t>

lim ( , ) 0

x u x t

→∞ =

10) Using Laplace transform solve the wave equation ²₂y ²y₂ ; 0 1, 0

x t

t x

∂ = ∂ < < >

∂ ∂

with the initial and boundary conditions ( , 0 ) sin , 0 1

y x + = πx < <x , y x_t( , 0 )+ =0, 0< <x 1 (0, ) 0, 0,

y t = t> y(1, )t =0, t>0

11) Expand f x( )=e^−x, 0< <x 2π in a Fourier series.

12) Find the half range sine series of the function f x( )=x², 0< <x π 13) Determine the Fourier integral representation of the function

sin ,

0, and

( ) .

x x

x x

f x

π π

π π

− < <

<− >

= 



14) Let the function f x( ) is given as

1 , 0 1

0, 1

( ) .

x x

x

f x

− < <

< <∞

= 



Determine the Fourier sine integral of f x( ) and find the value of

( )

2

0

( sin ) sin

2 d .

α α α

α α

∞ −

∫

15) Find the Fourier sine transform of f t( )=te^−t, t≥0.

16) Determine the Fourier cosine transform for f t( )=e^−tcos , t t≥0and show that

2 4 0

2 ( 2) cos

cos d , 0

4

t t

e t α α α t

π α

− ∞ +

= ≥

∫

+ ^.

17) Find Fourier transform of 𝑓𝑓(𝑡𝑡) =𝑡𝑡exp(−𝑎𝑎|𝑡𝑡|).

18) Find 𝑓𝑓(𝑡𝑡) if ² ₂

( ) 1 .

Fs α π

α α +

 

=  

19) Solve the heat equation u ²u₂, 0

k x

t x

∂ ∂

= < < ∞

∂ ∂ subject to the conditions ( , 0) ^x;

u x =e⁻ u_x(0, )t =0, t>0, uand u_xboth tend to zero asx→ ∞. 20) Solve the following Laplace equation ²u₂ ²u₂ 0; 0

x y y

∂ +∂ = >

∂ ∂ subject to the

conditions uxand u →0as x²+y² → ∞ and

1, 1 0, 1

( , 0)

x x

u x

≤

>

= 

 .

Answers/Hints:

1)

a) Function is not piecewise continuous since

lim2 ( )

t f t

→ ± do not exists.

b) Function is continuous everywhere.

c) Function has a jump discontinuity at t=0 and hence the function is piecewise continuous.

d) Function is continuous everywhere.

2)

Definition of Laplace transform and integration by parts give

{ }

²

( )

²

{ ( )

²

}

0

( ) sin d 1 cos(1) cos

2

st t t t

L f t e te e t sL e

∞ −  

=

∫

=  − 

Note that ^L

{

^cos

( )

^et2

}

exists because ^cos

( )

^et2 is continuous and is of exponential order.

Hence ^{L f t}

{ }

^{( ) exists.}

3)

a)

{ } ^{( )}

( ) ( { ) } ^{( ) ( )}

2 2

2

2 2

1 2 2 3

cos 1 1 4 1 2 5

t s s s

L e t

s s s

s s

− = + + = + +

+ + +

+ + +

b) ^{f t( ) e H tat}

(

²

)

^{ea t( )2e H t2a}

(

²

)

^{e 2(s a) 1}

s a

− − −

= − = − =

− c)

{

^{( )}

}

^{1 1 0 sin d}

(

^{1 1}

)(

^{2 1}

)

s st

s s

L f t e t t e

e e s

π π

π π

− −

− −

= = +

−

∫

− +

d)

{ }

2 2

[ ]

( ) 1 1

as as

as e e

L f t e a as

s s s

− −

= − + = +

4)

a) ^L−1

(

^{s2 s6s313}

)

^{2 L−1}

(

^{s s3}

)

^{23 4 2 e−3tL−1}

(

^{s2 s4}

)

^{2 14te−3tsin 2t}

 

 +  +  

 =  =  =

     

 

+ + +

   + +   

      

b) ^{1 3 1 2}

3 3

t t t

e + te − e⁻

c) ^{1 ( ) sinh}

{

²

( ) }

2 H t−π t−π

d) 3 5 3 1 1

cos sin sin

2 4 2 4 4

t t

e⁻ + te⁻ − t+ t− t t

5)

a) ^{y t( )= −1 cost}

b) ^{y= − +t 1 2e−t−H t}

(

⁻¹

) (

^{t− +2 e− +t 1}

)

c) ^{1 sin}

( )( ) (

^sin

)

y= 2t t+H t−π t−π t−π 

6)

a) 1

cos cos sin

y= − 2t t+ t+π t

 

b) ^{cos 3 4sin 3}

9 9

y= +t t+ t

7)

a) y= +t ct² b) y=te^−t

8)

a) y t( )=te^t

b) ^{( ) 1 2 5 2}

2 2

t t

y t = − e⁻ + e⁻

9)

2 3

0

( , ) exp ( ) d

4( )

2 ( )

t x x

u x t f u u

t u t u

π

 

=

∫

− − − 

10)

( , ) sin cos y x t = πx πt

11)

1 2 1 1 1 1 2

( ) cos cos 2 ... sin sin 2 ...

2 2 5 2 5

f x e x x x x

π

π

− −  + + +    + + + 



12)

4 2 4

( ) 2 sin sin 2 sin 3 sin 4 ...

3 9 2

f x π x π x π x π x

π π

 −  − +  −  − −

   

   

 ]

13)

2 0

2 sin sin 1 d

απ αx α

π α

∞

∫

− 14)

2 0

2 sin

( ) sin d

f x α α αx α

π α

∞ −

∫

 and the value is

4 π .

15)

2 2

2 2

(1 )

α π +α 16)

2 4

2 2

(4 )

α

π α

+ + 17)

2 2 2

2 2

( )

ia a

α

π +α

18)

f(t) = exp (−t) 19)

2

2

2 1

( , ) cos d

1

k t

u x t e ^α αx α

π α

∞ −

−∞

=

∫

+

20)

¹ 1 ¹ 1

( , ) y tan x tan x

u x y

x y y

− −

  −   + 

=   +  

   

 