Graduate Studies
First Advisor
Petronela Radu
Degree Name
Doctor of Philosophy (Ph.D.)
Department
Mathematics
Date of this Version
8-9-2024
Document Type
Dissertation
Citation
A dissertation presented to the faculty of the Graduate College of the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy
Major:
Under the supervision of Professor
Lincoln, Nebraska, August 2024
Abstract
Nonlocal models are have recently seen an explosive interest and development in the context of fracture mechanics, diffusion, image processing, population dynamics due to their ability to approximate differential-like operators with integral operators for inherently discontinuous solutions. Much of the work in the field focuses on how concepts from partial differential equations (PDEs) can be extended to the nonlocal domain. Boundary conditions for PDEs are crucial components for applications to physical problems, prescribing data on the domain boundary to capture the behavior of physical phenomena accurately with the underlying model. In this thesis we specifically examine a Neumann-type boundary condition for a weakly singular interaction kernel with finite horizon. We establish a pointwise convergence result in 1D for the operator with position dependent scaling as the horizon approaches zero. Furthermore, we establish well-posedness and uniqueness of solutions for the mixed (Robin) nonlocal boundary value problem, and we produce numerical simulations using the nonlocal FEM software PyNucleus developed by Christian Glusa at Sandia National Laboratories.
Recommended Citation
Hootman-Ng, Scott Alex, "On Neumann Boundary Conditions for Nonlocal Models with Finite Horizon" (2024). Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–. 197.
https://digitalcommons.unl.edu/dissunl/197
Comments
Copyright 2024, the author. Used by permission