Partial Differential Equations Commons^™

Inequality In The Urban Forest: Modeling Tree Canopy Dynamics Through Demographics And Restoration Strategies, Eve Johansson

Honors Theses

Urban tree canopies play an important role in environmental quality, public health, and neighborhood livability, yet their distribution is highly uneven and often reflects historical patterns of inequality. In Brooklyn, long-term processes such as redlining, uneven development, and demographic change have contributed to persistent disparities in access to green space.

This thesis examines how urban tree canopy evolves across space and time in Brooklyn and how different restoration strategies affect long-run outcomes. The analysis uses demographic and canopy data from 1990-2020, considering race, income, employment, and educational attainment. Among these, education is the most consistent predictor of canopy coverage, with …

Globally Adaptive Exponential Integrators For Stiff Systems Of Odes, Anzhelika Vasilyeva

Honors Theses

This thesis introduces a novel method for solving systems of Ordinary Differential Equations (ODEs) resulting from the spatial discretization of Partial Differential Equations (PDEs). The proposed approach builds upon an existing technique that employs Krylov projection, which requires evaluating a matrix function at each timestep. The innovation of the new method lies in its reuse strategy, which shifts the perspective from direct matrix function evaluation to polynomial interpolation. Numerical experiments conducted on constant and variable coefficient heat equations, with both smooth and discontinuous initial data, demonstrate the computational time advantage of the new approach. The results indicate that this method …

Echolocation On Manifolds, Kerong Wang

Honors Theses

We consider the question asked by Wyman and Xi [WX23]: ``Can you hear your location on a manifold?” In other words, can you locate a unique point x on a manifold, up to symmetry, if you know the Laplacian eigenvalues and eigenfunctions of the manifold? In [WX23], Wyman and Xi showed that echolocation holds on one- and two-dimensional rectangles with Dirichlet boundary conditions using the pointwise Weyl counting function. They also showed echolocation holds on ellipsoids using Gaussian curvature.

In this thesis, we provide full details for Wyman and Xi's proof for one- and two-dimensional rectangles and we show that …

A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui

Honors Theses

In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under the symmetry group of $P$. With this measure, we prove …

The Long Time Behavior Of The Predator-Prey Model With Holling Type Iii, Regen S. Mcgee

Honors Theses

In this paper, the classical Lotka-Volterra model is expanded based on functional response of Holling type III to analyze a dynamical predator-prey relationship with hunting cooperation (a) and the Allee effect among predators. The stability of equilibrium solutions was first analyzed by deriving a Jacobian matrix from partial derivatives of our model. Newly derived eigenvalues are then used to determine the stability. The viability of the model is then demonstrated by using MATLAB. The numerical results show a clear Allee effect and a variety of possible phenomena related to stability when carrying capacity (k) is varied. Two different types of …

Efficient Denoising And Sharpening Of Color Images Through Numerical Solution Of Nonlinear Diffusion Equations, Linh T. Duong

Honors Theses

The purpose of this project is to enhance color images through denoising and sharpening, two important branches of image processing, by mathematically modeling the images. Modifications are made to two existing nonlinear diffusion image processing models to adapt them to color images. This is done by treating the red, green, and blue (RGB) channels of color images independently, contrary to the conventional idea that the channels should not be treated independently. A new numerical method is needed to solve our models for high resolution images since current methods are impractical. To produce an efficient method, the solution is represented as …