Partial Differential Equations

This paper will discuss methods for solving many different partial differential equations, as well as real world applications in physics. We are interested in finding solutions to the wave and heat equations in one dimension, the wave equation in two dimensions, as well as a solution to Schrodinger’s equation. In order to do this, we will study different methods including Fourier series, Bessel functions, and Hermite polynomials. I will use these methods to derive solutions for the mentioned problems, as well as to produce visualizations for many of them

The Relationship between Tobacco Retailer Density and Neighborhood Demographics in Ohio

Introduction: Studies from various parts of the country suggest that tobacco-related health disparities are exacerbated by disparitiesin the distribution of tobacco retailers (convenience stores, tobacco shops, etc.). The purpose of the present study was to use advancedspatial modeling techniques for count data to estimate current disparities in tobacco retailer density in Ohio.Methods: We identified and geocoded 11,392 tobacco retailers in Ohio. Next, we obtained census tract-level information on race/ethnicity, poverty, and age and obtained county-level information on whether an area was Urban, Suburban, or Rural. Finally, we used negative binomial generalized linear models, adapted for residual spatial dependence, to determine the association between per capita tobacco retailer density and demographic characteristics—summarized by adjusted rate ratios.Results: There were more (from 1.4-1.9 times as many) retailers per capita in high-poverty vs. low-poverty tracts. Poverty also interacted with age: the association between high poverty and high retailer density was stronger for tracts with a low youth population. Density was also greater in tracts with a high (vs. low) prevalence of African Americans (1.1 times as many) and Hispanics (1.2 times as many). Finally, density was generally greater in rural (vs. suburban or urban) tracts, although the effect was modified by a three-way interaction: density was particularly high for rural tracts that also had both a high prevalence of poverty and a low youth population.Discussion: Overall, our findings indicate that Ohio’s vulnerable populations are exposed to a greater per capita density of tobaccoretailers. There is a need for state and local-level tobacco control policies that will improve equity and reduce health disparities