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Full-Text Articles in Physical Sciences and Mathematics
A Doubly Nonlocal Laplace Operator And Its Connection To The Classical Laplacian, Petronela Radu, Kelseys Wells
A Doubly Nonlocal Laplace Operator And Its Connection To The Classical Laplacian, Petronela Radu, Kelseys Wells
Department of Mathematics: Faculty Publications
In this paper, motivated by the state-based peridynamic frame- work, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces addi- tional degrees of exibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connec- tions with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero.
Fractional Difference Operators And Related Boundary Value Problems, Scott C. Gensler
Fractional Difference Operators And Related Boundary Value Problems, Scott C. Gensler
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation we develop a fractional difference calculus for functions on a discrete domain. We start by showing that the Taylor monomials, which play a role analagous to that of the power functions in ordinary differential calculus, can be expressed in terms of a family of polynomials which I will refer to as the Pochhammer polynomials. These important functions, the Taylor monomials, were previously described by other scholars primarily in terms of the gamma function. With only this description it is challenging to understand their properties. Describing the Taylor monomials in terms of the Pochhammer polynomials has made it …