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• 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.

- 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.

**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

Properties of Laplace Transform

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.

Project-3

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

Project-4

Find the solution of differential equation by using Laplace Transform.

Periodic Function

https://www.slideshare.net/UmangGupta5/fourier-series-31466525

video-19

Fourier series Expansion of a periodic functions

Project-5

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

Project-6

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

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.

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