MTH-213.4 Advanced Mathematical 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.

Introduction to coding with Python, 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 pseudo-spectral method, dealiasing and hyperdissiaption, solution of two-dimensional Euler equation, Nonlinear Schrodinger equation, and Can-Hilliard equation.  

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.