Undergrad Research | Aslihan Demirkaya
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Undergrad Research

Aslihan Demirkaya, working with Robert Decker, helped supervise the following research conference projects.

2016—2017

The Hudson River Undergraduate Mathematics Conference (HRUMC)

April 8, 2017, at Westfield State University in Westfield, MA
  • Iliana Albion-Poles, Mitchell Sugar, and Yonatan Shavit studied the scattering of periodic standing and traveling wave solutions (kink and anti-kink) of a nonlinear beam equation. Waves traveling in opposite directions collide, and the number of collisions depends on the initial velocity. After the collisions, the waves escape from each other with a constant velocity, called escape velocity. The objective of the research has been to find a relation between the initial and escape velocities and the number of collisions.

  • Robert Galvez and Matthew Bernocco studied the interaction of two wave pulses of a model in neuroscience: The model has recently been developed and attempts to explain how signals are conducted within neurons. They have considered two pulses traveling at different speeds in the same direction. They did observe the interaction and look for possible bounces.

2015—2016

The Hudson River Undergraduate Mathematics Conference (HRUMC)

April 2, 2016, at St Michael’s College, Colchester, VT
  • Salem Moges and Reid Bassette studied the scattering of traveling wave solutions (kink and anti-kink) of a nonlinear Klein-Gordon equation. Waves traveling in opposite directions collide, and the number of collisions depends on the initial velocity. After the collisions, the waves escape from each other with a constant velocity, called escape velocity. They studied the relation between the initial and escape velocities and the number of collisions as the diffusion constant varies.

  • Damaris Zachos and Gianmarco Molino studied the traveling wave solutions of the Korteweg-de Vries Equation and Regularized Long-Wave Equation. Even though the equations differ from each other, both have been used to describe shallow-water waves because their traveling wave solutions are similar to each other for specific parameters. The objective of their research was to compare numerical properties of the two, in terms of speed and stability. The numerical results were supported by a stability analysis.

2014—2015

10th Annual Spuyten Duyvil Undergraduate Conference

April 10, 2015, at Manhattan College, Riverdale, NY
  • Steven Kingston, Jarrett Lagler, and Gianmarco Molino worked on the soliton model in neuroscience which is a recently developed model that attempts to explain how signals are conducted within neurons.

2013—2014

The Joint Hudson River and Spuyten Duyvil Undergraduate Mathematics Conference

April 26, 2014, at Marist College, Poughkeepsie, NY
  • Ethan Bourdeau, John Cunsole, Virginia Demske and Scott Rubin worked on the numerical existence, stability, and simulation of traveling wave solutions (solitons) to the Sine-Gordon equation. The existence of localized traveling wave solutions to various nonlinear partial differential equations has been intensely studied in the last fifty years. When such a wave solution retains its shape and speed after a collision with another such wave, the traveling wave is called a soliton. Solitons have found application in many areas of science, including particle physics, fiber optics, and biology.

  • Hala Al-Khalil and Jamie Nagode worked on the numerical existence, stability, and simulation of traveling wave solutions to a distributed Spruce-Budworm PDE. The ODE model (undistributed) has been used extensively both in practice as a model for outbreaks of the spruce budworm in forests of fir trees and as a useful model for studying the concepts of fixed points and bifurcations in introductory differential equation courses.

2012—2013

The Hudson River and Spuyten Duyvil Undergraduate Mathematics Conference

April 6, 2013, at Williams College, Williamstown, MA
  • Gino Cordone and Jeffrey Knecht worked on a forced Klein-Gordon equation. The objective of the research was to explore the nonlinear forced Klein-Gordon equation numerically to see if there were chaos and period-doubling. It was known that there existed period doubling and chaotic behavior for a certain range of parameters for a forced duffing equation. Influenced by this fact, and using spectral methods we looked for the parameters that resulted in a change of the behavior of the system.

  • Catherine Brennan and James Pellissier worked on the numerical existence and the stability of the steady state solutions of distributed Spruce Budworm equation. (In 1978 D. Ludwig, D.D. Jones, and C.S. Holling used the logistic equation modified by a predation rate to model the relationship between the prolific spruce budworm and predatory birds. The Spruce Budworm Model has since been a topic of interest in population dynamics and differential equations.) The results, later on, were combined with the results obtained by Hala Al-Khalil and Jamie Nagode and accepted by Involve [SM-B7].

  • Jessica Andersen and Cole Murphy focused on the effects of a forcing term on the limit cycle of the Van der Pol Equation. The initial analysis examined the ODE version of the equation. A pattern of interest was discovered and examined. Further study was devoted to converting the Van der Pol equation into a PDE and analyzing as before.

2011—2012

The same project was presented at two different conferences.

The Hudson River Undergraduate Mathematics Conference (HRUMC)

April 21, 2012, at Western New England University, Springfield, MA

The 7th Annual Spuyten Duyvil Undergraduate Mathematics Conference

April 14, 2012, at Ramapo College, Mahwah, NJ

Miles Aron, Peter Bowers, Nicole Byer and Jun Hwan Ryu numerically studied the existence and stability of the steady-state solutions of the Reaction-Diffusion equation and the Klein-Gordon equation with the Dirichlet boundary conditions on a specific interval. The numerical methods were based on spectral methods developed by Trefethen.