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

Comments

Copyright 2024, the author. Used by permission

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.

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