Venue: TBA
Class Timings: TBA
First Meeting: TBA
Course Syllabus:
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.
Course Evaluation: Regular homework and exams will be used to assign the course grade.
Pre-requisites: Advanced calculus, some knowledge on analytical solutions of differential equations, basic coding experience.
Course outcomes: After completing this course students will develop:
- A tool kit for solving differential equations where a small or a large parameter can be taken advantage of.
- Capability to numerically solve linear and nonlinear differential equations.
- Insights on stochastic calculus and stochastic differential equations with emphasis on practical applications.
References:
- Perturbation Methods by E. J. Hinch
- Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons by Mark J. Ablowitz
- Computational Physics by Mark Newman
- Numerical Analysis by Richard L Burden and J Douglas Faires
- Spectral Methods by Claudio Canuto and Co-authors
- The Physics of Fluid Turbulence by W. D. McComb
- A First Course in Stochastic Calculus Louis-Pierre Arguin
- Teacher: Jim Thomas
Credit Score: 4