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- Applied Mathematics Research - |
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| An Overview | ||||||
| As an applied mathematician I am
always looking for a good balance between mathematics and close applications to scientific problems. In particular, I am
interested in problems which have a solid interplay between
mathematical modeling, nonlinear analysis, and numerical
methods.
Mathematical modeling entails exactly what one would expect: take something physical and write a dynamic set of equations governing the system. One can catch trivial glimpses of this with word problems as early as your first or second year in grade school, but of course the complexity is greatly enhanced at this part in the game! The examples are endless but simple models could be the position of a baseball being thrown from a pitcher (preferably a Yankee) or something more complex of modelling the atmosphere. Nonlinear analysis entails taking a mathematical model, which is typically in the form of partial differential equations, and examine the behavior around a point of interest. Typically one starts with a linear analysis, examining solely the linear parts of the model. Only through a more detailed nonlinear analysis, which introduces the nonlinear terms via perturbation methods, issues of stability or instability can be better understood. Numerical methods is taking the governing system of equations, the model, and simulating the equations on a computer. This entails recasting the governing equations into a discrete form, which becomes readily available to write in code on a computer...boy it sounds almost simple. |
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| Nonlinear Dynamics of Elastic Filaments | ||||||
| In my current research I am studying the dynamical instabilities of fluid flow down flexible tubes. This has numerous applications in biology: umbilical cords, flow through our veins, etc. The tubes or rods are given such that the maximum radius of the cross-section is small compared with its overall length. Think of a cable wire for example. Through a perturbation technique, developed here, on the Kirchhoff equations, which model the dynamics of an elastic rod, a nonlinear analysis about certain configurations of the rod can be further examined. Another goal of my research is to provide a full time dependent, numerical study of the Kirchhoff equations. | ||||||
| Nonlinear Optics | ||||||
| My undergraduate thesis examined the nonlinear dynamics of an actively mode-locked laser. The brunt of my thesis entailed developing finite element methods to simulate the couple nonlinear equations governing the laser. Ultimately the most suitable method was a spectral element method, using Hermite polynomials (not to be confused with Hermite interpolating polynomials). After this spatial discretization an implicit Runge-Kutta method was used to iterate in time. If you are more interested here is the link to a talk I gave to present my results: | ||||||
| The Slinky | ||||||
| A fun little topic I love to talk about. | ||||||