 # Integral Transformation Teacher Category

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# Code(Credit) :CUTM1521(3-1-2)

## Course Objectives

• To describe the ideas of Fourier and Laplace Transforms and indicate their applications in the fields such as application of PDE, Digital Signal Processing, Image Processing, Theory of wave equations, Differential Equations and many others.
• To use Fourier series for solving boundary value problems appearing in scientific & engineering problems.

## Learning Outcomes

• Upon successful completion of this course, students will be able to:
• Learn application of MATLAB programming.

•Solve differential & integral equations with initial conditions using Laplace transform.

• Evaluate the Fourier transform of a continuous function and be familiar with its basic properties.

• Solution of integral equation and their application.

## Course Syllabus

Module-I(T-4 h + Pr-2 h+ Pj-4 h)

Introduction to Integral Transforms, Laplace Transforms,Properties of Laplace transforms,unit step function,second shifting theorem.

Practice-1

Laplace Transform and Inverse Laplace Transform Using Matlab.

Project-1

Make a short draft of properties of Laplace transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications.

Project-2

Find the Laplace transform of the following functions

Module-II(T-4 h + Pj-4 h)

Unit impulse function,Laplace transforms of Derivatives and Integrals, Derivatives and Integrals of Transforms.

Project-3

Find the inverse Laplace transform of Derivative and integral properties.
Project-4
Find the Laplace transform of unit impulse function and unit step function.

Module-III(T-4 h + Pr-2 h+ Pj-4 h)

Evaluation of integrals,convolution theorem,inverse Laplace transform

Practice-2

To solve integral equations by using Matlab.

Project-5

Evaluation of integrals by using Laplace transform.

Project-6

Find the inverse Laplace transform by using convolution theorem.

Module-IV(T-4 h + Pr-2 h+ Pj-4 h)

Solution of differential equations,integral equations.

Practice-3

Find the solution of differential equations by Laplace Transform Using Matlab.

Project-7

Solve the differential equation by using Laplace transform.

Project-8

Solve the integral equation by using Laplace transform.

Module-V(T-4 h + Pr-2 h+ Pj-4 h)

Periodic function,Fourier series,Fourier series expansion of an arbitrary period,even and odd functions,half Range Expansions of a Fourier series.

Practice-4

Find the Fourier series expansion of a periodic function by using Matlab.

Project-9

Find the Fourier series expansion of a 2-pi periodic function.

Project-10

Find the Half range expansion of Fourier cosine and sine series.

Module-VI(T-4 h + Pr-2 h+ Pj-2 h)

Complex form of Fourier series, Fourier Integrals, Different forms of Fourier Integral Theorem.

Practice-5

Find the Fourier Series Coefficients by using Matlab.

Project-11

Find the Fourier sine and cosine integral of the following functions.

Module-VII(T-4 h + Pr-2 h+ Pj-2 h)

Fourier Transforms, Infinite Fourier Transforms,Finite Fourier transform, Properties Fourier Transform.

Practice-6

First Fourier transform by using Matlab.

Project-12

Make a short draft of properties of Fourier transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications.

Text Book:

M.D. Raisinghania, H.C. Saxena and H.K. Dass: Integral Transforms, S. Chand & Company LTD.Chapters: 1(1.1-1.5,1.7 to 1.10,1.12) 2(2.2,2.4,2.5,2.6), 3(3.1 to 3.21), 4(4.1, 4.4, 4.5 to 4.13, 4.15 to 4.17), 5(5.1, 5.16), 6(6.1 to 6.3, 6.6 to 6.8, 6.11), 7(7.1 to 7.3).

Reference Text Book:

1.Higher Engineering Mathematics by B.V.Ramana
2.An introduction to non harmonic Fourier series,Author(s):Robert M. Young

3.Advanced Engineering Mathematics by E.Kreyszig (10th edition)

## Session 1

Introduction to Laplace Transform
PDF-10
video-10

## Session 3

Shifting Properties
PDF-14
video-14

## Session 4

Unit Step Function
PDF-15

## Session 5 & 6

Practice-1

Laplace Transform and Inverse Laplace Transform Using Matlab.

## Session 12

Properties of Laplace transform(Transforms of Derivatives and Integrals)
PDF-11
video-11

## Session 13

Shifting Properties
PDF-14
video-14

## Session 14

Derivatives and Integrals of Transforms
PDF-13
video-13

## Session 15 & 16

Project-3

Find the inverse Laplace transform of Derivative and integral properties.

## Session 17 & 18

Project-4
Find the Laplace transform of unit impulse function and unit step function.

## Session 19

Convolution Theorem
PDF-18
video-18

## Session 20

Evaluation of Integrals
PDF-16
video-16

## Session 21

Inverse Laplace Transform
PDF-17
video-17

## Session 22 & 23

Practice-2
To solve integral Equation by using Matlab

## Session 24 & 25

Project-5
Make a short draft of application of Laplace transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications.

PROJECT

## Session 26 & 27

Project-6
Application of Unit step function by using Matlab

PROJECT

## Session 28

Solution to Differential Equation by using Laplace transform
PDF-19
video-19

## Session 29 & 30

Practice-3

Solve the differential equations by Laplace Transform Using Matlab.

## Session 31

Solution to Differential Equation by using dirac delta function
PDF-20
video-20

## Session 32 & 33

Project-7
Solve the differential equation by using Laplace transform.

## Session 34

Integral Equation
PDF-21
video-21

## Session 35 & 36

Project-8
Solve the integral equation by using Laplace transform

## Session 38

Fourier series Expansion of a periodic functions

## Session 39

Fourier expansion of a functions of any Periods 2L
PDF-4

video-4

## Session 40 & 41

Practice-4

Find the Fourier series expansion of a periodic function by using Matlab.

## Session 42& 43

Project-9(Flip Class)

Find the Fourier series expansion of the following functions.

## Session 44 & 45

Project-10(Flip Class)

Find the half range expansion of Fourier sine and cosine series

## Session 46

Even and Odd Periodic extension of a Functions.

video-5

## Session 47

Half range Expansion of a Fourier series
PDF-6

video-6

## Session-48 &49

Practice-2

Find the Fourier Series Coefficients by using Matlab.

## Session 50

Complex form of Fourier series

## Session 51 & 52

Project-11(Flip Class)

Find the steady-state current I(t) m the RLC-circuit ,where R = 100 ohm, L = 10 H, C = 1/100 F and E(t) V as follows and periodic with period 2pi. Sketch or graph the first four partial sums.

## Session 53

Fourier Integrals transform of a function f(t)
PDF-8
video-8

## Session 54

Different forms of Fourier Integral Theorems.
PDF-9

video-9

## Session 55

Fourier transform and Infinite Fourier Transforms
PDF-24
video-24

## Session 56

Fourier sine and cosine Transforms
PDF-25
video-24

## Session 57

Fourier cosine Transforms and inverse fourier transform
PDF-26
video-24

## Session 58 & 59

Practice-6
First Fourier transform by using Matlab.

## Session 60 & 61

Project-12
A report of an Application of Convolution With FFTs .

## Session 62

First Fourier Transforms
PDF-27
video-23

## Session 63

Properties of Fourier Transforms
PDF-28
video-24

## Session 64

Properties of Fourier transform
PDF-29
video-25

## Session 65 & 66

Project-12

Make a short draft of properties of Fourier transform from memory. Then compare your notes with the text and write a report of 2-3 pages on these operations and their significance in applications. 