Syllabus

Fourier Series and Transform: Expansion of a function in Fourier series for a given range and its convergence, Even and odd functions, Half range sine and cosine expansions, Fourier integrals, Complex Fourier series, Fourier transform, Inverse Fourier transform, Properties of Fourier transform, Convolution theorem, Discrete Fourier transform. Second order partial differential equations, Normal Form, Solutions of wave equation, Heat equation and Laplace’s equation and their use in problems of vibrating string, One dimensional unsteady heat flow and two dimensional steady state heat flow.

Integral Equations: Classification of Integral equations, Neumann’s iterative method for Fredholm’s equation of second kind, Volterra type integral equation, Integral equations of first kind, Convolution type Integral Equations.

Calculus of Variations: Functionals, Variation of functionals, Example of variation problems, Euler's equation, sufficient conditions for the extremum of a functional, conditional extremum, Rayleigh-Ritz method.

To introduce Fourier series, Fourier transforms and their applications.

A thorough introduction of partial differential equations, their classifications and applications.

To introduce integral equations, their classifications and applications in engineering.

To introduce concepts of calculus of variations and it applications.

After completing the course students will learn applications of Fourier series and Fourier transformations to solve differential equations. They will become familiar with partial differential equations and their applications. They will also learn different ways to solve integral equations. Students will become familiar with Euler equation to find extremum of a functional and its real-life applications.

Erwin Kreyszig, Advanced Engineering Mathematics, Wiley , Available in Amazon

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications

R. P. Kanwal, Linear Integral Equations, Birkhäuser Boston

R. V. Churchill, Operational Mathematics, McGraw Hill