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Nonlocal models with a finite range of nonlocal interactions

Xiaochuan Tian

Title:
Nonlocal models with a finite range of nonlocal interactions
Author(s):
Tian, Xiaochuan
Thesis Advisor(s):
Du, Qiang
Date:
Type:
Theses
Degree:
Ph.D., Columbia University
Department(s):
Applied Physics and Applied Mathematics
Persistent URL:
Abstract:
Nonlocal phenomena are ubiquitous in nature. The nonlocal models investigated in this thesis use integration in replace of differentiation and provide alternatives to the classical partial differential equations. The nonlocal interaction kernels in the models are assumed to be as general as possible and usually involve finite range of nonlocal interactions. Such settings on one hand allow us to connect nonlocal models with the existing classical models through various asymptotic limits of the modeling parameter, and on the other hand enjoy practical significance especially for multiscale modeling and simulations. To make connections with classical models at the discrete level, the central theme of the numerical analysis for nonlocal models in this thesis concerns with numerical schemes that are robust under the changes of modeling parameters, with mathematical analysis provided as theoretical foundations. Together with extensive discussions of linear nonlocal diffusion and nonlocal mechanics models, we also touch upon other topics such as high order nonlocal models, nonlinear nonlocal fracture models and coupling of models characterized by different scales.
Subject(s):
Mathematical models
Multiscale modeling
Mathematics
Calculus, Integral
Item views
115
Metadata:
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Suggested Citation:
Xiaochuan Tian, , Nonlocal models with a finite range of nonlocal interactions, Columbia University Academic Commons, .

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