Mathematics - Student Works
Permanent link for this collection
Browse
Browsing Mathematics - Student Works by Author "Bica, Ion"
Now showing 1 - 5 of 5
Results Per Page
Sort Options
Item Hydrostatic balance in meteorology(2020) Mucalica, Ana; Bica, IonMeteorology is a branch of geophysics that studies the properties of the atmosphere that “cushions” the Earth and all the phenomena that happen within it. The hydrostatic balance occurs when the pressure at any point in the fluid equals the weight of an air column of the unit section from above the point, and in these ideal conditions, we have the hydrostatic equation for the fluid in hydrostatic balance. The hydrostatic equation is viewed as the hydrostatic equilibrium condition, which provides an accurate approximation for the vertical dependence of the pressure field in the real atmosphere. In this talk, we manipulate a mathematical model describing the variation of pressure to altitude in the atmosphere using partial differential equations, and we will show how realistic charts based on real data are obtained, and compare them to charts obtained by using a climate change scenario when the temperature at sea-level is warmer by 2℃ than the one used in current charts. Work done in collaboration with Anneliese Ansorger, Cory Efird, Cassandra Lisitza, Ghristopher Macyk, Tarig Mergani, Adam Reeves, Rebecca Walton, Yaying Zhong, and Jett Ziehe, MacEwan University.Item Lorenz’s system - analysis of a sensitive system(2021) Lisitza, Cassandra; Bica, IonMeteorology is a branch of geophysics concerned with atmospheric processes and phenomena and atmospheric effects on our weather. Edward Lorenz was a devoted meteorologist who made several significant contributions to this field. We first describe the fascinating history of Lorenz’s discoveries and his revolutionary additions to the area of meteorology. In particular, he noted the extremely sensitive dependence on the initial conditions of a Chaotic system in the atmosphere, which is commonly referred to as the Butterfly Effect and pertains to Lorenz’s system of three Ordinary Differential Equations (ODEs) that models the atmospheric convection in the atmosphere. We conducted a novel, in-depth mathematical analysis of the Theorem of Existence and Uniqueness for a system of ODEs in general and addressed how it applies to Lorenz’s system. Further, we exhibited how Lorenz’s system is ill-posed using an application where we varied the initial parameters by minimal variations and noted relatively quick and drastic differences in the trajectories of the system.Item The two physics governing the one-dimensional cubic nonlinear Schrödinger equation(2021) Mucalica, Ana; Bica, IonIn 1926, in his quest to explain the quantum probabilistic nature of particles, Erwin Schrödinger proposed a nonrelativistic wave equation that required only one initial condition, i.e., the initial displacement of an electron. His equation describes the wave-particle duality discovered by Louis de Broglie in 1924. Furthermore, Schrödinger's wave equation is dimensionless, allowing the equation to be a mathematical model describing different physical phenomena. Introducing nonlinearity into the Schrödinger equation, we worked with the so-called self-focusing nonlinear Schrödinger equation. We showed that when the nonlinearity is perfectly balanced with the dispersion, the self-focusing nonlinear Schrödinger model describes the propagation of a soliton. In 1968 Peter Lax introduced the "Lax Pair," a pair of time-dependent matrices/operators describing the nature of a nonlinear evolution partial differential equation, to discuss solitons in continuous media. This procedure is what we call the scattering method for describing mathematically nonlinear processes in physics. We used the scattering method to find the Lax Pair for the nonlinear Schrodinger model, and we showed that the equation is a compatibility condition for the AKNS system. In 1974, Ablowitz, Kaup, Newell, and Segur (AKNS) introduced the inverse scattering transform to solve evolution nonlinear partial differential equations arising from compatibility conditions for the AKNS system. Rather than using the inverse scattering transform, we showed an intuitive approach in revealing the formation and propagation of a soliton for the self-focusing nonlinear Schrodinger equation, using a novel approach via cnoidal waves. The work will also include a novel theorem describing the steepening of the wavefront due to nonlinearity.Item Van der Pol oscillator – analysis of a non-conservative system(2020) Reeves, Adam; Bica, IonThe Van der Pol oscillator was introduced by Balthasar van der Pol, who was ”a famous scholar, a famous scientist, a famous administrator at the international level, he was equally well known for the clarity of his lectures (in several languages), his knowledge of the classics, his warm personality and his talents for friendship, and his love for music.” [2] The oscillator describes the nonlinear oscillations for systems like a triode circuit, which produce selfsustained oscillations known as relaxation oscillations. Extensive studies have been done on the oscillator, for understanding it and for using it as an applied model for the heartbeat, for example. In this thesis, we will explain the nature of the oscillator from an original point of view, in the low-friction regime. First, we will give an intuitive physical explanation of the first order averaging method, a perturbation theory method, applied onto the oscillator. We will follow with an analytical approach of the first order averaging method, and we will show the mathematical complexity of it. We will conclude with the application of the first order averaging method to the Van der Pol oscillator, confirming the findings from the intuitive approach.Item Van der Pol oscillator – analysis of a non-conservative system(2020) Reeves, Adam; Bica, IonThe Van der Pol oscillator was introduced by Balthasar van der Pol, who was ”a famous scholar, a famous scientist, a famous administrator at the international level, he was equally well known for the clarity of his lectures (in several languages), his knowledge of the classics, his warm personality and his talents for friendship, and his love for music.” The oscillator describes the nonlinear oscillations for systems like a triode circuit, which produce self-sustained oscillations known as relaxation oscillations. Extensive studies have been done on the oscillator, for understanding it and for using it as an applied model for the heartbeat, for example. In this thesis, we will explain the nature of the oscillator from an original point of view, in the low-friction regime. First, we will give an intuitive physical explanation of the first order averaging method, a perturbation theory method, applied onto the oscillator. We will follow with an analytical approach of the first order averaging method, and we will show the mathematical complexity of it. We will conclude with the application of the first order averaging method to the Van der Pol oscillator,confirming the findings from the intuitive approach.