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__TRANSFORM CALCULUS, FOURIER SERIES, AND NUMERICAL TECHNIQUES__

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Course Code:18MAT31

CIE Marks:40

SEE Marks:60

Teaching Hours/Week (L:T:P):(2:2:0)

Credits 03 Exam Hours:03

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__Course Learning Objectives:__

• To have an insight into Fourier series, Fourier transforms, Laplace transforms, Difference equationsand Z-transforms.

• To develop proficiency in variational calculus and solving ODE’s arising in engineering

applications, using numerical methods.

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Module-1

Laplace Transforms Definition and Laplace transform of elementary functions. Laplace transforms ofModule-1

Periodic functions and unit-step function – problems.

Inverse Laplace Transforms: Inverse Laplace transform - problems, Convolution theorem to find the inverse

Laplace transform (without proof) and problems, solution of linear differential equations using Laplace

transform.

### Click here to download Module-1

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Module-2

Fourier Series: Periodic functions, Dirichlet’s condition. Fourier series of periodic functions period 2 andModule-2

arbitrary period. Half range Fourier series. Practical harmonic analysis, examples from engineering field.

### Click here to download Module-2

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__Module-3__

Fourier Transforms: Infinite Fourier transforms, Fourier sine and cosine transforms. Inverse Fouriertransforms. Simple problems.

Difference Equations and Z-Transforms: Difference equations, basic definition, z-transform-definition,

Standard z-transforms, Damping and shifting rules, initial value and final value theorems (without proof) and

problems, Inverse z-transform. Simple problems.

### Click here to download Module-3

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__Module-4__

Numerical Solutions of Ordinary Differential Equations (ODE’s): Numerical solution of ODE’s of firstorder and first degree- Taylor’s series method, Modified Euler’s method. Range - Kutta method of fourth

order, Milne’s and Adam-Bashforth predictor and corrector method (No derivations of formulae), Problems.

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__Module-5__

Numerical Solution of Second Order ODE’s: Runge -Kutta method and Milne’s predictor and correctormethod.(No derivations of formulae).

Calculus of Variations: Variation of function and functional, variational problems, Euler’s equation,

Geodesics, hanging chain, problems.

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Course Outcomes:

At the end of the course the student will be able to:Course Outcomes:

• CO1: Use Laplace transform and inverse Laplace transform in solving differential/ integral equation

arising in network analysis, control systems and other fields of engineering.

• CO2: Demonstrate Fourier series to study the behaviour of periodic functions and their applications in

system communications, digital signal processing and field theory.

• CO3: Make use of Fourier transform and Z-transform to illustrate discrete/continuous function arising

in wave and heat propagation, signals and systems.

• CO4: Solve first and second order ordinary differential equations arising in engineering problems

using single step and multistep numerical methods.

• CO5:Determine the extremals of functionals using calculus of variations and solve problems

arising in dynamics of rigid bodies and vibrational analysis.

Important Links:

Important Links:

### 1. click here to Download the Previous year's Question paper with solutions

2. click here to download Model Question Paper with solutions

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__Question paper pattern:__

• The question paper will have ten full questions carrying equal marks.• Each full question will be for 20 marks.

• There will be two full questions (with a maximum of four sub- questions) from each module.

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Textbooks

1 Advanced Engineering mathematics. Kreyszig John Wiley & Sons 10th Edition, 2016Textbooks

2 Higher Engineering Mathematics B. S. Grewal Khanna Publishers 44th Edition, 2017

3 Engineering Mathematics Srimanta Pal et al Oxford University Press 3rd Edition, 2016

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__Reference Books__

1 Advanced Engineering Mathematics C. Ray Wylie, Louis C. Barrett McGraw-Hill Book Co6 th Edition, 19952 Introductory Methods of Numerical Analysis S. S. Sastry Prentice Hall of India 4 th Edition 2010

3 Higher Engineering Mathematics B.V. Ramana McGraw-Hill 11th Edition,2010

4 A Text Book of Engineering Mathematics N. P. Bali and Manish Goyal Laxmi Publications 2014

5 Advanced Engineering Mathematics Chandrika Prasad and Reena Garg Khanna Publishing, 2018

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__Web links and Video Lectures:__

1. http://nptel.ac.in/courses.php?disciplineID=1112. http://www.class-central.com/subject/math(MOOCs)

3. http://academicearth.org/

4. VTU EDUSAT PROGRAMME - 20