# Laplace and Fourier Transforms

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# Code(Credit) : CUTM1002(2-0-1)

## 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:•Solve differential equations with initial conditions using Laplace transform.• Evaluate the Fourier transform of a continuous function and be familiar with its basic properties.

## Course Syllabus

Module-I(T-3 h + Pj-2 h)

Laplace Transforms, Properties of Laplace transforms, Unit step function.

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.

Module-II(T-2 h + Pj-2 h)

Second shifting theorem, Laplace transforms of Derivatives and Integrals.

Project-2

Find the Laplace transform of the following functions.

Module-III(T-3 h + Pj-2 h)

Derivatives and Integrals of Transforms, Inverse Laplace transform.

Project-3

Application of Unit step function (RC- Circuit to a single square wave).

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

Solution of differential equations by using Laplace transform.

Project-4

Find the solution of differential equation by using Laplace Transform.

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

Periodic function, Fourier series, Fourier series expansion of an arbitrary period, Half range expansions.

Project-5

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

Module-VI(T-3 h + Pj-2 h)

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

Project-6

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

Module-VII(T-3 h )

Fourier Transforms, Fourier sine and cosine Transforms.

Text Book:

1. E. Kreyszig , Advanced Engineering Mathematics, Johnwilley & Sons Inc-8th Edition.Chapters:5(5.1 to 5.4(without Dirac's delta function ) ),10(10.1,10.4 and 10.7-10.9(definitions only , no proofs))
2. Highjer Engineering Mathematics by B.V.Ramana, Tata McGraw-Hill Education India, Inc-8th Edition

Reference Text Book:

1) Advanced Engineering Mathematics by P.V.O’ Neil Publisher: Thomson
2) Mathematical Methods by Potter & Goldberg ; Publisher : PHI

## Session 1

Introduction to Laplace Transform
lec-1 pdf
video-1

## Session 3

Unit Step Function
lec-3 pdf

video-3

## Session 4 & 5

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

## Session 6

Second Shifting Properties
lec-4 pdf
video-4

## Session 7

Properties of Laplace transform(Transforms of Derivatives and Integrals).
lec-5 pdf
video-5

## Session 8 & 9

Project-2

Find the Laplace transform of the following functions.

## Session 10

Derivatives and Integrals of Transforms
lec-6 pdf
video-6

## Session 11

Inverse Laplace Transform
lec-7 pdf
video-11

## Session 12

Properties of inverse Laplace Transform

## Session 13 & 14

Project-3

Application of Unit step function (RC- Circuit to a single square wave).

## Session 15

Solution of Differential Equation by using Laplace transform.
lec-9 pdf
video-15

## Session 16

Solution of initial value problem by using Laplace transform.
lec-10 pdf
video-16

## Session 17 & 18

Project-4

Find the solution of differential equation by using Laplace Transform.

## Session 20

Fourier series Expansion of a periodic functions

## Session 21

Fourier expansion of a functions of any Periods 2L
Fourier series expansion of period 2L

video-21

## Session 22

Half range Expansion of a Fourier series
Half Range Expansion of a Fourier series

video-22

## Session 23 & 24

Project-5

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

## Session 26

Fourier Integrals transform of a function f(t)
Fourier Integral
video-26

## Session 27

Different forms of Fourier Integral Theorems.
Fourier Integral

video-27

## Session 28 & 29

Project-6

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

## Session 30

Fourier sine and cosine Transforms
Fourier Sine and Transform
video-24

## Session 31

Fourier transform and Infinite Fourier Transforms

lec-26

video-26

## Session 32

First Fourier Transforms

Fourier Transform

video-27

## Case Studies

### Our Main Teachers

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Assistant Professor ,School of Applied Sciences, Centurion University of Technology and Management, Paralakhemundi, Odisha, India. He has over 11 years of teaching experiences. He has published 5 research articles in National and International refereed journals . He is continue their P.hD in the field of Fuzzy Inventory Control Models.