# TRANSFORM CALCULUS, FOURIER SERIES AND NUMERICAL TECHNIQUES(18MAT31 )

## TRANSFORM CALCULUS, FOURIER SERIES, AND NUMERICAL TECHNIQUES

#### Course Learning Objectives:

• To have an insight into Fourier series, Fourier transforms, Laplace transforms, Difference equations
and Z-transforms.
• To develop proficiency in variational calculus and solving ODE’s arising in engineering
applications, using numerical methods.

### Module-1

Laplace Transforms Definition and Laplace transform of elementary functions. Laplace transforms of
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.

### Module-2

Fourier Series: Periodic functions, Dirichlet’s condition. Fourier series of periodic functions period 2 and
arbitrary period. Half range Fourier series. Practical harmonic analysis, examples from engineering field.

### Module-3

Fourier Transforms: Infinite Fourier transforms, Fourier sine and cosine transforms. Inverse Fourier
transforms. 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.

### Module-4

Numerical Solutions of Ordinary Differential Equations (ODE’s): Numerical solution of ODE’s of first
order 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.

### Module-5

Numerical Solution of Second Order ODE’s: Runge -Kutta method and Milne’s predictor and corrector
method.(No derivations of formulae).
Calculus of Variations: Variation of function and functional, variational problems, Euler’s equation,
Geodesics, hanging chain, problems.

### Course Outcomes:

At the end of the course the student will be able to:
• 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.

### 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.

#### Textbooks

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

#### Reference Books

1 Advanced Engineering Mathematics C. Ray Wylie, Louis C. Barrett McGraw-Hill Book Co6 th Edition, 1995
2 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

### Web links and Video Lectures:

1. http://nptel.ac.in/courses.php?disciplineID=111
2. http://www.class-central.com/subject/math(MOOCs)
4. VTU EDUSAT PROGRAMME - 20