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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:

• 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.

**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)

Properties of Laplace Transform

**Practice-1**

Laplace Transform and Inverse Laplace Transform Using Matlab.

https://www.youtube.com/watch?v=Epq128CiGrQ&feature=youtu.be

**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.

**Practice-2**

To solve integral Equation by using Matlab

https://www.youtube.com/watch?v=sZIzlRmyG00

**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.

**Practice-3**

Solve the differential equations by Laplace Transform Using Matlab.

https://www.youtube.com/watch?v=igbwPMFC13w

**Project-7**

Solve the differential equation by using Laplace transform.

**Project-8**

Solve the integral equation by using Laplace transform

**Practice-4**

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

https://www.youtube.com/watch?v=r6sGWTCMz2k

**Project-9(Flip Class)**

Find the Fourier series expansion of the following functions.

**Project-10(Flip Class)**

Find the half range expansion of Fourier sine and cosine series

**Practice-2**

Find the Fourier Series Coefficients by using Matlab.

https://www.youtube.com/watch?v=o7O7vC0Bmms

**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.

**Practice-6**

First Fourier transform by using Matlab.

**Project-12**

A report of an Application of Convolution With FFTs .

**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.

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.