Transform Calculus, Fourier Series and Numerical Techniques.pdf

Model Question Paper-II with effect from 2022
USN

Third Semester B.E Degree Examination
Transform Calculus, Fourier Series and Numerical Techniques
(21MAT31)
TIME: 03 Hours Max. Marks: 100

Note: Answer any FIVE full questions, choosing at least ONE question from each module.

Module -1 Marks
Find the Laplace transform of
1−𝑒 𝑡
Q.01 a (𝑖) 𝑒 −3𝑡 𝑠𝑖𝑛5𝑡 𝑐𝑜𝑠3𝑡 (𝑖𝑖) 06
𝑡

Find the Laplace transform of the square–wave function of period a given by

b 1, 0 < 𝑡 < 𝑎/2 07
𝑓(𝑡) = {
−1, 𝑎/2 < 𝑡 < 2

1
Using the convolution theorem find the inverse Laplace transform of (𝑠2 +1)(𝑠2 +9)
c 07

OR
Using the unit step function, find the Laplace transform of
cos 𝑡 , 0 ≤ 𝑡 ≤ 𝜋
Q.02 a 𝑓(𝑡) = { cos 2𝑡 , 𝜋 ≤ 𝑡 ≤ 2𝜋 06
cos 3𝑡 , 𝑡 ≥ 2𝜋

2𝑠2 −6𝑠+5
b Find the inverse Laplace transform of 𝑠3 −6𝑠2 +11𝑠−6 07

Solve by using Laplace transform techniques
𝑑2 𝑥 𝑑𝑥
c
𝑑𝑡 2
−2 + 𝑥 = 𝑒 𝑡 with 𝑥(0) = 2 𝑎𝑛𝑑 𝑥 ′ (0) = −1 07
𝑑𝑡

Module-2
Find a Fourier series to represent 𝑓(𝑥) = 𝑥 2 𝑖𝑛 −𝜋 ≤𝑥 ≤𝜋
Q. 03 a 06
Obtain the half-range cosine series for 𝑓(𝑥) = 𝑥 𝑠𝑖𝑛𝑥 in (0, 𝜋) and hence show that
𝜋−2 1 1 1
b = − + −⋯ ∞ 07
4 1.3 3.5 5.7

The following table gives the variation of periodic current over a period.
t sec 0 T/6 T/3 T/2 2T/3 5T/6 T
A amp 1.98 1.30 1.05 1.30 -0.88 -0.25 1.98
c 07
Show that there is a direct current part of 0.75 amp in the variable current and obtain the
amplitude of the first harmonic.

, OR
Find the Fourier series expansion of 𝑓(𝑥) = 2𝑥 − 𝑥 2 , 𝑖𝑛 ( 0, 3)
Q.04 a 6

Obtain half-range sine series for 07
𝑙
b 𝑘𝑥 , 0≤𝑥≤ 2
𝑓(𝑥) = { 𝑙
𝑘(𝑙 − 𝑥) , 2
≤𝑥≤𝑙
Expand y as a Fourier series up to the first harmonic if the values of y are given by
c x 0° 30° 60° 90° 120° 150° 180° 210° 240 270 300 330 07
y 1.80 1.10 0.30 0.16 1.50 1.30 2.16 1.25 1.30 1.52 1.76 2.00

Module-3
1, |𝑥| ≤ 1
Find the Fourier transform of 𝑓(𝑥) = {
0, |𝑥| > 1
Q. 05 a ∞ 𝑠𝑖𝑛𝑥 06
Hence evaluate ∫0
𝑥
𝑑𝑥

Find the Fourier cosine and sine transforms of 𝑒 −𝑎𝑥
b 07
Find the Z-transforms of (𝑖) (𝑛 + 1)2 and (𝑖𝑖) sin(3𝑛 + 5)
c 07
OR
−𝑎2 𝑥 2
Find the Fourier transform of 𝑒 , 𝑎 > 0. Hence deduce that it is self-reciprocal in
Q. 06 a respect of Fourier series 06

2𝑧 2 +3𝑧
Find the inverse z –transform of (𝑧+2)(𝑧−4)
b 07

Using z-transformation, solve the difference equation 𝑢𝑛+2 + 4𝑢𝑛+1 + 3𝑢𝑛 = 3𝑛 ,
c 𝑢0 = 0 , 𝑢1 = 1 07

Module-4
Classify the following partial differential equations
(i) 𝑢𝑥𝑥 + 4𝑢𝑥𝑦 + 4𝑢𝑦𝑦 − 𝑢𝑥 + 2𝑢𝑦 = 0
Q. 07 a (ii) 𝑥 2 𝑢𝑥𝑥 + (1 − 𝑦 2 )𝑢𝑦𝑦 = 0 ,−1 < 𝑦 < 1 10
(iii) (1 + 𝑥 2 )𝑢𝑥𝑥 + (5 + 2𝑥 2 )𝑢𝑥𝑡 + (4 + 𝑥 2 )𝑢𝑡𝑡 = 0
(iv) 𝑦 2 𝑢𝑥𝑥 − 2𝑦𝑢𝑥𝑦 + 𝑢𝑦𝑦 − 𝑢𝑦 = 8𝑦
Find the values of 𝑢(𝑥, 𝑡) satisfying the parabolic equation 𝑢𝑡 = 4𝑢𝑥𝑥 and the boundary
𝑥2
conditions 𝑢(0, 𝑡) = 0 = 𝑢(8, 0) and 𝑢(𝑥, 0) = 4𝑥 − 2
at the points
b 𝑗
10
𝑥 = 𝑖 ∶ 𝑖 = 0,1,2, … ,8 and 𝑡 = ∶ 𝑗 = 0,1,2,3,4.
8

OR
𝜕𝑢 𝜕2 𝑢
Solve the equation 𝜕𝑡 = 𝜕𝑥 2 subject to the conditions 𝑢(𝑥, 0) = sin 𝜋𝑥 , 0 ≤ 𝑥 ≤ 1
Q. 08 a 𝑢(0, 𝑡) = 𝑢(1, 𝑡) = 0, Carry out computations for two levels, taking 10
1 1
ℎ = 34 𝑎𝑛𝑑 𝑘 = 36

, The transverse displacement u of a point at a distance x from one end and at any time t of a
vibrating string satisfies the equation 𝑢𝑡𝑡 = 25 𝑢𝑥𝑥, with the boundary conditions
20𝑥 , 0 ≤ 𝑥 ≤ 1
b 𝑢(𝑥, 𝑡) = 𝑢(5, 𝑡) = 0 and the initial conditions 𝑢(𝑥, 0) = { 10
5(5 − 𝑥), 1 ≤ 𝑥 ≤ 5
and 𝑢𝑡 (𝑥, 0) = 0. Solve this equation numerically up to 𝑡 = 5 taking ℎ = 1, 𝑘 = 0.2.

Module-5
𝑑2 𝑦 𝑑𝑦
Q. 09 a
Using Runge –Kutta method of order four, solve 𝑑𝑥 2 = 𝑦 + 𝑥 𝑑𝑥 for x = 0.2 , Given 06
that , 𝑦(0) = 1 , 𝑦 ′ (0) = 0
𝑥
Find the extremals of the functional ∫𝑥 2[𝑦 2 + (𝑦 ′ )2 + 2𝑦𝑒 𝑥 ]𝑑𝑥
b 1 07

Find the path on which a particle in the absence of friction, will slide from one point
c to another in the shortest time under the action of gravity 07

OR
𝑑2 𝑦 𝑑𝑦
Apply Milne’s method to solve =1+ at x = 0.4. given that
𝑑𝑥 2 𝑑𝑥
Q. 10 a 𝑦(0) = 1 , 𝑦(0.1) = 1.1103, 𝑦(0.2) = 1.2427 , 𝑦(0.3) = 1.399 06
𝑦 ′ (0) = 1 , 𝑦 ′ (0.1) = 1.2103, 𝑦 ′ (0.2) = 1.4427, 𝑦 ′ (0.3) = 1.699
2
2 𝑥 (𝑦 ′ )
b Find the extremals of the functional ∫𝑥1 𝑑𝑥
𝑥3 07

𝜋/2
Find the curve on which the functional ∫0 [(𝑦 ′ )2 + 12𝑥𝑦] 𝑑𝑥
c with 𝑦(0) = 0 𝑎𝑛𝑑 𝑦(𝜋/2) = 0 can be extremised 07

Table showing the Bloom’s Taxonomy Level, Course Outcome and Program Outcome

Question Bloom’s Taxonomy Course Program Outcome
Level attached Outcome
(a) L1 CO 01 PO 01
Q.1 (b) L2 CO 01 PO 02
(c) L2 CO 01 PO 02
(a) L2 CO 01 PO 02
Q.2 (b) L2 CO 01 PO 02
(c) L2 CO 01 PO 02
(a) L2 CO 02 PO 02
Q.3 (b) L2 CO 02 PO 02
(c) L3 CO 02 PO 02
(a) L2 CO 02 PO 02
Q.4 (b) L2 CO 02 PO 02
(c) L2 CO 02 PO 02
(a) L2 CO 03 PO 02
Q.5 (b) L2 CO 03 PO 02
(c) L2 CO 03 PO 02
Q.6 (a) L2 CO 03 PO 02

, (b) L2 CO 03 PO 02
(c) L3 CO 03 PO 02
(a) L1 CO 04 PO 01
Q.7
(b) L2 CO 04 PO 02
(a) L2 CO 04 PO 02
Q.8
(b) L3 CO 04 PO 02
(a) L2 CO 05 PO 01
Q.9 (b) L2 CO 05 PO 02
(c) L3 CO 05 PO 02
(a) L2 CO 05 PO 01
Q.10 (b) L2 CO 05 PO 02
(c) L2 CO 05 PO 02


Lower order thinking skills
Bloom’s Remembering Understanding Applying
Taxonomy (knowledge): L1 (Comprehension): L2 (Application): L3
Levels Higher-order thinking skills
Analyzing Valuating Creating
(Analysis): L4 (Evaluation): L5 (Synthesis): L6