NUMERICAL SOLUTIONS OF ELLIPTIC PDE WITH POINT SOURCES The branch of mathematics known as partial differential equations (PDE) has numerous applications to varying fields of science and engineering. Ranging from theoretical physics to weather prediction, partial differential equations plays an integral role in the mathematical modeling of various physical phenomena. Our particular interest is on the numerical computation of a 2nd order nonlinear elliptic PDE with delta point sources defined over a doubly periodic domain. Our encounter, with such a PDE, stems from. the study of the two-Higgs electroweak model of Bimonte and Lozano governing two electroweak Higgs doublets and possessing vortex-like solitons representing a system. of non-interacting particles. The existence and uniqueness, up to an additive constant,. of solutions over a doubly periodic domain and over more general surfaces has been established by Aubin. However, analytic solutions, even in the one-dimensional case are non-trivial. We explore integral representations, Green’s functions, and Fourier series approaches to the numerical solutions of the PDE over a periodic domain.
KYE DRAGESET BS Mathematics 2017 Mid-Pacific Institute. Honolulu, Hawaii Faculty Luciano Medina NYU College of Arts & Sciences
DYNAMICAL MODELS OF DNA: EXISTENCE OF DNA BREATHERS DNA molecules form the familiar double helix that has been known for decades. While studying the dynamics of the DNA molecule, an interesting problem arises, where the appearance of localized finite-amplitude oscillations of the strands suggest the existence of DNA breathers. Developing a rigorous mathematical theory for the existence of DNA breathers would add to the understanding of the DNA and important processes such. as denaturation, which is when the double-strand of the DNA molecule unzips into two single strands.
MAXIMILIAN KANTOR BS Applied Physics and Mathematics 2016
Our focus is on the pioneering work of Englander et al, and Peyrard and Bishop modeling DNA dynamics. The proposed Hamiltonian encapsulates the interactions due to hydrogen bonding between the nucleotides or bases within a base pair and the stacking force between the base pairs. The resulting Hamiltonian equations, governing the DNA dynamics, are a coupled system of nonlinear differential equations. Using modern techniques in the calculus of variation and global functional analysis, we propose to find necessary conditions for the existence of localized solutions. Additionally, we develop numerical solutions to the Hamiltonian equations and simulate the DNA dynamics using. a MATLAB environment.
Xavier High School. New York, New York Faculty Luciano Medina NYU School of Engineering
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