Course

UNIT 1:Periodic Functions, Trigonometric Series, Fourier Series, Functions of any Period p = 2L, Even and Odd Functions, Half Range Expansions.UNIT 2:Fourier Integrals, Sine and Cosine Integrals, Fourier Transforms – Sine and Cosine Transforms and inverse transforms, Properties, Convolution Theorem, Differentiation and Integration of Transforms.UNIT 3:Laplace Transforms, Inverse Transforms, Properties, Transforms of Derivatives and Integrals, Second Shifting Theorem, Unit Step Function and Dirac-Delta Function; Differentiation and Integration of Transforms.UNIT 4:Solving Linear Ordinary Differential Equations with constant coefficients – use of the integral transforms; solving linear circuits integro differential equations with Laplace transforms.UNIT 5:Types of PDEs, Separation of Variables method, Wave equation – Use of Fourier Series to solve PDE, Heat Equation; Solution by Fourier Series

Course Objectives:The objective of this course is to introduce the Fourier and Laplace transforms, and the uses of these transforms in the solution of partial differential equations. 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.Course Outcomes:At the end of the course, the students will be able to:CO1: Understand and Apply Fourier Analysis to various types of periodic functions CO2: Understand and Analyze Fourier Integrals and TransformsCO3: Understand the significance of Laplace Transforms and apply it to various problems CO4: Understand the methods of solving PDEs using Series and Transforms.

Text Books: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.