### Applied Mathematical Methods - Topical (MTH 248.5)

Venue: Emmy Noether Seminar Room (ICTS) and Big Classroom (CAM)

Class Timings: 5:00 PM to 8:00 PM, Mondays. Classes will be at ICTS and CAM on alternate weeks

First Meeting: 16th January 2023 in Emmy Noether Seminar Room, ICTS

Course Description:

Part 1: Asymptotical methods: Asymptotic sequences and asymptotic expansions, solution of linear system of equations and weakly nonlinear system of equations, Regular and Singular perturbation problems, Method of multiple scales, Method of matched asymptotic expansions and boundary layer theory, WKBJ, geometric optics, and ray tracing, introduction to homogenization theory.

Part 2: Numerical methods: Time stepping methods for ordinary differential equations, error bounds of solutions, Fast Fourier Transform (FFT), spectral methods for solving partial differential equations, solutions of heat and wave equations, introduction to the pseudospectral method, dealiasing and hyperdissiaption, solution of two-dimensional Euler equation, Nonlinear Schrodinger equation, and Can-Hilliard equation.

Part 3: Stochastic methods: Basic probability, Random variables, probability distribution functions and probability density functions, Moments of random variables, sequence of random variables, Law of large numbers and Central limit theorem, Ito's theorem and Stochastic calculus, Stochastic differentiation and integration, Stochastic differential equations (SDEs), analytical and numerical solution of SDE's, applications of SDE's to practical problems.

Outcome:

After completing this course the student will develop:

1) A tool kit for solving differential equations where a small or a large parameter can be taken advantage of.
2) Skills to numerically solve linear and nonlinear differential equations.
3) Insights into stochastic calculus and stochastic differential equations with emphasis on practical applications.