Volume 18

DOI: 10.1137/24S1691600

Authors

Elias Hohl (Corresponding author – ETH Zürich)

Project Advisors

Giovanni Felder (ETH Zürich)

Abstract

We find analytical solutions of the Laplace equation on a rectangular domain with Dirichlet-Neumann boundary condition type transition occurring on a side of the rectangle. The problem can be tackled either by cutting the domain in two parts and solving the obtained coupled Laplace equations using a Fourier series approach, or by creating a conformal mapping based on Christoffel-Schwarz and Mobius transformations between the original domain and another rectangular domain where the solution of the problem is less challenging. Apart from solving for the temperature field, we also compute a coefficient, dependent only on the geometry of the domain, from which the thermal resistance of a system with the given cross-section can be computed. We conduct an experiment to confirm the validity of our model. Using our results from the Christoffel-Schwarz method, we derive regularity results for the Fourier coefficients of the coupling method. We not only determine the order of decay, but also find formulas for explicit upper bounds of the absolute value of the coefficients. These results can be used to obtain maximum error bounds for a numerical computation considering only a finite number of Fourier coefficients. Furthermore, we prove that the solutions obtained via the coupling method converge to the actual solution.

DOI: 10.1137/23S1553479

Authors

Noah Dahle (Corresponding author – University of Tennessee, Knoxville), Kristina Wilson (University of Tennessee, Knoxville)

Project Advisors

Dr. Olivia Prosper (University of Tennessee, Knoxville)

Abstract

It is common for a mapping platform to relay information to the user about the traffic in the area. Traffic conditions have different degrees of traffic intensity that are represented by colors. We have studied Google Maps traffic data through a simulation to model these conditions. Our methodology explores the prospect of how different traffic condition colors will imply different travel times. This modeling was a means to develop an user interface to determine the fastest path between two points in a particular area of Knoxville, Tennessee. We accomplished this by using Dijkstra’s Algorithm. The final product is a program that accepts and corrects user input for day of week, time, and two points, and the output is the path the user should take, along with the travel time according to the Google Maps traffic conditions at that day and time. This model can be applied to different cities to study their traffic patterns or in city planning.

Supplementary Materials