Honors Program


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Montemayor, Layla. Nonlocal Frameworks for Differential Equations--Analysis and Numerical Investigations . Undergraduate Honors Thesis, University of Nebraska-Lincoln, 2024


Copyright Layla Montemayor 2024.


Systems of differential equations have been employed successfully in nearly every scientific field, including natural phenomena, finance, A.I. and machine learning. In a classical set up, a differentiable equation will be associated with a number of initial conditions that will help determine a unique solution to the system. The setting of classical differential equations requires assumptions on the regularity or smoothness of the input functions. Thus, we cannot produce with classical differential equations solutions which are irregular or even have discontinuities.

To overcome some of these challenges brought up by investigations of irregular or singular behavior, different formulations of nonlocal models have been proposed. Many fields have benefited from the introduction of nonlocal frameworks, such as dy- namical fracture [Sil00], nonlocal diffusion [IR07], image processing [GO09], sand pile formation [And10], swarming and collective behavior, phase separation. In this fast growing field, an increasing body of literature has studied applied/modeling aspects, numerical implement, and theoretical analysis of solutions to these systems. Part of these studies include well-posedness of solutions (existence, uniqueness, dependence on data), properties of solutions, estimates and asymptotic behavior.

In this thesis I studied aspects of the nonlocal gradient, nonlocal equations with a focus on a nonlocal derivative. A particular interest was the behavior of a non- local gradient with different selections of kernel functions. An initial study of finite difference methods for nonlocal equations has produced some conjectures towards the formulation of nonlocal versions of initial value problems. Finally, this work has identified a growing list of open problems which could be the focus of future investigations.