Unit 1

Fourier series—Even and Odd functions, Half range expansions, Approximation by Trigonometric Polynomials. (Sections 11.1 to 11.4)

Unit 2

Sturm-Liouville problems, Generalized Fourier Series, Fourier Integral and Fourier transforms. (Sections 11.5 to 11.10)

Unit 3

Basic concepts of PDEs, Solution by Separating Variables, D’Alembert’s Solution of the Wave Equation. (Sections 12.1 to 12.4)

Unit 4

Heat Equation, Solution By Fourier Series, 2D Heat Equation and Dirichlet Problem, Heat equation for long bars, Solution by Fourier Integrals and Transforms, Two Dimensional Wave Equation. (Sections 12.5 to 12.8)

Unit 5

Laplacian in Polar Coordinates, Fourier Bessel Series, Laplace’s equation in Cylindrical and Spherical Coordinates, Solution of PDEs by Laplace Transforms. (Sections 12.8 to 12.12)

Course Outcomes
On successful completion of the course, students shall be able to

CO1: apply basic concepts to obtain Fourier series of simple continuous and piece-wise periodic functions, determine Fourier transforms of simple functions.
CO2: apply basic Sturm-Liouville theory to analyse different types of orthogonal functions and determine generalized Fourier series
CO3: comprehend the mathematical model of one- and two- dimensional wave and heat Equations and apply the method of separation of variables and theory of Fourier series for rectangular geometries to solve the boundary value problems.
CO4: apply the general principle in boundary value problems for PDEs to choose coordinates that make the formula for the boundary as simple as possible
CO5: apply the principles of method of separation of variables to solve the boundary value problems in polar, cylindrical and Spherical coordinates for cylindrically symmetric problems.

Textbook:

1. Erwin Kreyszig, Advanced Engineering Mathematics, 10th edition, Wiley.
2. Dennis Zill, A First Course in Differential Equations, Cengage.

References:

• W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th edition, Wiley.
• M.D. Raisinghania, Ordinary and Partial Differential equations, S Chand Publications