Scribd
Upload
102 views2 pages

Assignment 1 - Maths III

The document contains 12 problems involving Fourier transforms, Z-transforms, and difference equations. Problem 1 asks to find the Fourier transform of three functions. Problem 2 asks to find the Fourier cosine transform of three functions. Problem 3 asks to find the Fourier sine transform of three functions. Problem 4 asks to use Parseval's identity to prove an equation. Problems 5-7 involve additional Fourier transform problems and partial differential equations. Problems 8-11 involve finding and applying Z-transforms. Problem 12 asks to solve five difference equations using Z-transforms.

Uploaded by

ranjeet singh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
102 views2 pages

Assignment 1 - Maths III

The document contains 12 problems involving Fourier transforms, Z-transforms, and difference equations. Problem 1 asks to find the Fourier transform of three functions. Problem 2 asks to find the Fourier cosine transform of three functions. Problem 3 asks to find the Fourier sine transform of three functions. Problem 4 asks to use Parseval's identity to prove an equation. Problems 5-7 involve additional Fourier transform problems and partial differential equations. Problems 8-11 involve finding and applying Z-transforms. Problem 12 asks to solve five difference equations using Z-transforms.

Uploaded by

ranjeet singh
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

Assignment-1

BMA-251 (Mathematics III)

1. Find the Fourier transform of:

1 |𝑥| < 1
(i) 𝑓(𝑥) = {
0 |𝑥| > 1

1 − 𝑥2 |𝑥| < 1
(ii) 𝑓(𝑥) = {
0 |𝑥| > 1

2 2 2 ⁄2
(iii) 𝑓(𝑥) = 𝑒 −𝑎 𝑥 , 𝑎 < 0 and deduce that 𝑒 −𝑥 is self reciprocal in respect of
Fourier transform.

2. Find the Fourier cosine transform of

2
(i) 𝑓(𝑥) = 𝑒 −𝑥

𝑥 0<𝑥<1
(ii) 𝑓(𝑥) = {2 − 𝑥 1<𝑥<1
0 𝑥>2

1
(iii) 𝑓(𝑥) =
1+𝑥 2

3. Find the Fourier sine transform of

(i) 𝑓(𝑥) = 𝑒 −|𝑥|


(ii) 𝑓(𝑥) = 𝑥 𝑛−1
1
(iii)𝑓(𝑥) = 𝑥(𝑥 2+𝑎2)

∞ 𝑑𝑡 𝜋
4. Using Parseval’s identity, prove that ∫0 (𝑡 2 +𝑏2 )(𝑡 2 +𝑎2 )
= 2𝑎𝑏(𝑎+𝑏) .

2
5. Verify convolution theorem for Fourier transform for 𝑓(𝑥) = 𝑔(𝑥) = 𝑒 −𝑥 .

6. Find the finite Fourier sine and cosine transform of


(i) 𝑓(𝑥) = 2𝑥, 0 < 𝑥 < 4 .
−𝑥 𝑥<𝑐
(ii) 𝑓(𝑥) = { 𝑤ℎ𝑒𝑟𝑒 0 ≤ 𝑐 ≤ 𝜋.
𝜋−x x>𝑐
7. Solve:
𝜕𝑢 𝜕2 𝑢
(i) = 2 𝜕𝑥 2 , where u(0, t) = 0; u(x,0) = 𝑒 −𝑥 (x>0) and u(x, t) is bounded.
𝜕𝑡
𝜕𝑢 𝜕2 𝑢
(ii) = 𝜕𝑥 2 , where u(0, t) = 0; u(4,t) = 0 u(x, 0) = 2x with 0 < x < 4.
𝜕𝑡
𝜕𝑦 𝜕2 𝑦 π 𝜕𝑦
(iii) = 3 𝜕𝑥 2 , where y (2 ,t)=0 (𝜕𝑥 ) = 0 and y(x,0) =30 cos 5𝑥.
𝜕𝑡 𝑥=0
𝜕𝑢 𝜕2 𝑢 𝜕𝑢 𝑥 0<𝑥<1
(iv) = , where ( 𝜕𝑡 ) = 0 𝑎𝑛𝑑 𝑢(𝑥, 0) = {
𝜕𝑡 𝜕𝑡 2 𝑥=0 0 𝑥>1

8. Find the Z-transform of:


(i) 𝑛𝑎𝑛
(ii) 𝑛 2 𝑎𝑛
(iii) sin 𝑛𝜃
(iv) cos 𝑛𝜃
(v) 𝑎𝑛 cos n 𝜃
(vi) 𝑎𝑛 sinn 𝜃
(vii) 𝑛 2 𝑒 𝑛𝜃
−𝑧 𝑧
9. If Z(𝑢𝑛 ) = + 2 , then find Z(𝑢𝑛+2).
𝑧−1 𝑧 +1
𝑧2
10. Using convolution theorem, evaluate 𝑍 −1 ((𝑧−𝑎)(𝑧−𝑏)).
11. Find inverse Z-transforms of following functions:
2𝑧 2 +3𝑧
(i) (𝑧+2)(𝑧−4)
𝑧 3 −20𝑧
(ii) (𝑧−2)3(𝑧−4)
𝑧
(iii) (𝑧−1)2
5𝑧
(iv) (2−𝑧)(3𝑧−1)
3𝑧 2 −18𝑧+26
(v) (𝑧−2)(𝑧−3)(𝑧−4)

12. Solve following difference equations using Z-transform:


(i) 𝑢𝑛+2 +4𝑢𝑛+1 +3𝑢𝑛 = 3𝑛 , 𝑢0 = 0 𝑢1= 1
𝑛
(ii) 𝑢𝑛+2 +6𝑢𝑛+1 +9𝑢𝑛 = 2 , 𝑢0 = 𝑢1= 0
(iii) 𝑢𝑛+2-5𝑢𝑛+1 +6𝑢𝑛 = 1, 𝑢0 = 0, 𝑢1=1
𝑛
(iv) 𝑢𝑛+2+ 𝑢𝑛 = 5.2 𝑢0= 1, 𝑢1= 0
(v) 𝑦𝑛 +3𝑦𝑛−1= 𝑢𝑛 , where 𝑢𝑛 is a step function and 𝑦−1=1

You might also like