# 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.
•  COs Course Outcomes Mapping with COs with POs (High-H, Low-L, Medium-M) CO1 Have deep knowledge of Laplace Transformation and its real life application. PO1, PO2, PO5 CO2 Solve initial value problem and boundary value problem using Laplace Transform. PO1, PO2, PO4, PO5 CO3 Understand Fourier transform and its properties and will be able to solve the examples based on it. PO1, PO4, PO5 CO4 Have deep understanding to handle various types of problems using different kind of integral Transforms. PO1, PO2, PO5

## 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(sinat,cos at,sinh at,cosh at,e^at,t^n,etc).

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 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

## Session 7

Properties of Laplace transform (Transform of Derivative and Integrals)

Lecture Notes:(Transform of Derivative and Integrals).

## Session 8 & 9

Project-2

Find the Laplace transform of the following functions.

## Session 13 & 14

Project-3

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

## 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 23 & 24

Project-5

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

## Session 28 & 29

Project-6

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

Assignment-1

Assignment-2&3

Question Bank