Course

Unit 1

Fourier analysis:

Periodic Functions, Trigonometric Series, Fourier Series, Functions of any Period p = 2L, Even and Odd Functions, Half Range Expansions (theorem statement only), Complex Fourier Series, Applications of Parseval’s Identity.

Unit 2

Fourier Integrals, Sine and Cosine Integrals, Fourier Transforms – Sine and Cosine Transforms, Properties, Convolution Theorem, diffraction theory- Fourier method.

Unit 3

Laplace Transforms:
Laplace Transforms, Inverse Transforms, Properties, Transforms of Derivatives and Integrals, Second Shifting Theorem, Unit Step Function and Dirac-Delta Function,

Unit 4

Differentiation and Integration of Transforms, Convolution, Initial and Final Value Theorems, Periodic Functions, Solving Linear Ordinary Differential Equations with Constant Coefficients, System of Differential Equations and Integral Equations.

UNIT 1: Partial Differential Equations
Basic Concepts, Modelling; Vibrating String, Wave Equation, Separation of Variables, Use of Fourier Series, D’Alembert’s Solution of the Wave Equation, Heat Equation; Solution by Fourier Series.

Unit 1

Partial Differential Equations

Basic Concepts, Modelling; Vibrating String, Wave Equation, Separation of Variables, Use of Fourier Series, D’Alembert’s Solution of the Wave Equation, Heat Equation; Solution by Fourier Series.

Course Objectives:
The objective of this course is to introduce the student to the two important transforms – the Fourier and Laplace transforms. Properties of the transforms, series, complex forms are introduced. The uses of these transforms in the solution of partial differential equations are also taught to the students. This course is intended to lay a mathematical foundation to other theoretical courses such as quantum mechanics and act as a primer to a student who opts to take up a higher course in physics.

Pre-requisites:
Since this is an undergraduate level course, the student is expected to be familiar with basic differential and integral calculus only.

1. E Kreyszig, Advanced Engineering Mathematics, 10th Ed., John Wiley and Sons, 2015.
2. P. P. G. Dyke,An Introduction to Laplace Transforms and Fourier series, 2nd Ed.,Springer, 2014.
3. Larry C. Andrews and Bhimson, K. Shivamoggi, The Integral Transforms for Engineers,Prentice Hall India Learning Private Limited, 2003.