Joshua Lee Padgett

Joshua Lee Padgett
University of Arkansas | U of A · Department of Mathematical Sciences

Ph.D., Baylor University

About

34
Publications
2,921
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
124
Citations
Introduction
Joshua L. Padgett is currently an Assistant Professor in the Department of Mathematical Sciences at the University of Arkansas. Josh's research interests include Numerical Analysis, Operator Splitting, High-Dimensional Partial Differential Equations, Mathematics of Deep Learning, Geometric Numerical Integration, Stochastic Differential Equations, and Fractional Differential Equations. Most of Josh's work revolves around the rigorous analysis of novel numerical methods applied to such problems.
Additional affiliations
August 2020 - present
University of Arkansas
Position
  • Professor (Assistant)
August 2017 - August 2020
Texas Tech University
Position
  • Research Associate
Education
August 2012 - August 2017
Baylor University
Field of study
  • Mathematics
August 2008 - May 2012
Gardner-Webb University
Field of study
  • Mathematics

Publications

Publications (34)
Poster
Full-text available
We present our recent result on the modeling of the flow in porous media using Einstein's thought experiment, via jumps of the particles to model transport as a material balance equation. We model the "superfast" flow towards/from the fracture-boundary of the media. The fluid is considered to be a family of particles that are transported due to dif...
Preprint
Full-text available
Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general time horizons and Lipschitz continuous nonlinearities. However, each of the error analyses for MLP approximation...
Preprint
Full-text available
In recent years, there has been a large increase in interest in numerical algorithms which preserve various qualitative features of the original continuous problem. Herein, we propose and investigate a numerical algorithm which preserves qualitative features of so-called quenching combustion partial differential equations (PDEs). Such PDEs are ofte...
Article
This work presents an analytical investigation of anomalous diffusion and turbulence in a dusty plasma monolayer, where energy transport across scales leads to the spontaneous formation of spatially disordered patterns. Many-body simulations of 10 000-particle dusty plasma monolayers are used to demonstrate how the global dynamics depend on the sta...
Chapter
Norm estimates for strongly continuous semigroups have been successfully studied in numerous settings, but at the moment there are no corresponding studies in the case of solution operators of singular integral equations. Such equations have recently garnered a large amount of interest due to their potential to model numerous physically relevant ph...
Article
Full-text available
Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discr...
Article
Full-text available
Glycans are one of the most widely investigated biomolecules, due to their roles in numerous vital biological processes. This involvement makes it critical to understand their structure-function relationships. Few system-independent, LC-MS/MS (Liquid chromatography tandem mass spectrometry) based studies have been developed with this particular goa...
Preprint
Full-text available
This work presents a theoretical investigation of active turbulence in a dusty plasma monolayer, where energy is injected at the individual particle level and transported to larger scales, leading to the spontaneous formation of spatially disordered flow patterns. Many-body simulations of 10,000-particle dusty plasma monolayers are used to demonstr...
Preprint
Full-text available
Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discr...
Preprint
Full-text available
Norm estimates for strongly continuous semigroups have been successfully studied in numerous settings, but at the moment there are no corresponding studies in the case of solution operators of singular integral equations. Such equations have recently garnered a large amount of interest due to their potential to model numerous physically relevant ph...
Article
Full-text available
From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such processes across various scales has important application to research in materials science, finance, medicine, and...
Article
Full-text available
Glycans are one of the most widely investigated biomolecules, due to their roles in numerous vital biological processes. This involvement makes it critical to understand their structure-function relationships. Few system-independent, LC-MS/MS (Liquid chromatography tandem mass spectrometry) based studies have been developed with this particular goa...
Preprint
Full-text available
Glycans are one of the most widely investigated biomolecules, due to their roles in numerous vital biological processes. This involvement makes it critical to understand their structure-function relationships. Few system-independent, LC-MS/MS (Liquid chromatography tandem mass spectrometry) based studies have been developed with this particular goa...
Preprint
Full-text available
From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such processes across various scales has important application to research in materials science, finance, medicine, and...
Article
Full-text available
This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a one-dimensional disordered system that employs the discrete fractional Laplacian, $(-\Delta)^s,\ s\in(0,2),$ in combination with results from spectral and measure theory. It...
Article
Full-text available
The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework her...
Preprint
Full-text available
This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a one-dimensional disordered system that employs the discrete fractional Laplacian, (−∆) s , s ∈ (0, 2), in combination with results from spectral and measure theory. It is a...
Preprint
Full-text available
This work extends the applications of Anderson-type Hamiltonians to include transport characterized by anomalous diffusion. Herein, we investigate the transport properties of a one-dimensional disordered system that employs the discrete fractional Laplacian, $(-\Delta)^s,\ s\in(0,2),$ in combination with results from spectral and measure theory. It...
Preprint
Full-text available
The purpose of this work is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework, herei...
Chapter
Full-text available
In this paper, we study the convergence of a Lie-Trotter operator splitting for stochastic semilinear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the approximation for both the original and spatially discretized problems. It is known that the strong convergence of this sc...
Article
Full-text available
Self‐ and cross‐diffusion are important nonlinear spatial derivative terms that are included into biological models of predator–prey interactions. Self‐diffusion models overcrowding effects, while cross‐diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting me...
Preprint
Full-text available
Quenching solutions to a Kawarada problem with a Caputo time-fractional derivative and a fractional Laplacian are considered. The solutions exist locally in time when quenching occurs. Quenching and non-quenching solutions remain positive and are monotonically increasing in time under minor restrictions. Conditions for quenching to occur are demons...
Article
Full-text available
Quenching solutions to a Kawarada problem with a Caputo time-fractional derivative and a fractional Laplacian are considered. The solutions to such problems may only exist locally in time when quenching occurs. Quenching and non-quenching solutions are shown to remain positive and be monotonically increasing in time under minor restrictions. Condit...
Article
The numerical solution of a highly nonlinear two-dimensional degenerate stochastic Kawarada equation is investigated. A semi-discretized approximation in space is comprised on arbitrary nonuniform grids. Exponential splitting strategies are then applied to advance solutions of the semi-discretized scheme over adaptive grids in time. It is shown tha...
Preprint
Full-text available
Self-and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting met...
Preprint
Full-text available
In this paper we study the convergence of a Lie-Trotter operator splitting for stochastic semi-linear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the approximation for both the original and spatially discretized problems. It is known that the strong convergence of this sc...
Article
Full-text available
Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attem...
Article
Full-text available
The spectral approach to infinite disordered crystals is applied to an Anderson-type Hamiltonian to demonstrate the existence of extended states for nonzero disorder in 2D lattices of different geometries. The numerical simulations shown prove that extended states exist for disordered honeycomb, triangular, and square crystals. This observation sta...
Article
Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attem...
Article
Full-text available
This article studies a nonuniform finite difference method for solving the degenerate Kawarada quenching-combustion equation with a vibrant stochastic source. Arbitrary grids are introduced in both space and time via adaptive principals to accommodate the uncertainty and singularities involved. It is shown that, under proper constraints on mesh ste...
Article
This paper concerns the numerical solution of three-dimensional degenerate Kawarada equations. These partial differential equations possess highly nonlinear source terms, and exhibit strong quenching singularities which pose severe challenges to the design and analysis of highly reliable schemes. Arbitrary fixed nonuniform spatial grids, which are...
Chapter
Full-text available
This paper concerns the numerical stability of a splitting scheme for solving the three-dimensional degenerate quenching-combustion equation. The diffusion-type nonlinear equation possess highly nonlinear source terms, and is extremely important to the study of numerical combustions. Arbitrary fixed nonuniform spatial grids, which are not necessari...

Projects

Projects (5)
Project
Modeling Superfast flow towards fracture using Einstein-Brownian Model and derive singular IBVP with Fractional Laplacian, and analyze properties of the solution with iterative energy norm estimates.
Archived project
Structure-preserving methods have emerged as a central topic in computational mathematics. It has been realized that an integrator must be designed to preserve as many of the intrinsic features of the underlying problems as possible, such as conserving the mass, momentum and energy, as well as the symplecticity and multisymplecticity of physical systems. Structure-preserving algorithms can be effectively utilized for simulations of a variety of theoretical and application problems, ranging from celestial mechanics, quantum mechanics, fluid dynamics, and artificial intelligence. This special session is dedicated to recent advances in the aforementioned efforts, with a focus on high accuracy and structure-preserving algorithms when partial differential equations are targeted. We intend to accommodate a sufficiently broad spectrum of investigations, and will consider both theoretical and computational aspects of the burgeoning field. Tentative invited speakers (2-3 sessions; 6 speakers each) are: Giorgio Bornia, Texas Tech University, TX, USA Vince Ervin, Clemson University, SC, USA Tiffany N. Jones, University of Arizona, AZ, USA Julienne Kabre, Baylor University, TX, USA Abdul Khaliq, Middle Tennessee State University, TN, USA Jeonghun Lee, Baylor University, TX, USA Vu Thai Luan, Southern Methodist University, TX, USA Jorge E. Macias-Diaz, Universidad Autónoma de Aguascalientes, Mexico Romeo Martinez, Universidad Autónoma de Aguascalientes, Mexico Joshua L. Padgett, Texas Tech University, TX, USA Qin Sheng, Baylor University, TX, USA Bruce A. Wade, University of Louisiana, LA, USA Matthew Beauregard, Stephen F. Austin State University, TX, USA Jameson Graber, Baylor University, TX, USA
Project
The purpose of those project is the rigorous analysis of physically relevant FDE and FPDE, meaning ones which are derived from first principles. In addition, this project seeks to actively study numerical methods which may be effectively used for solving stiff semilinear problems. These considerations also include the study of related concepts, such as singular integral equations, Mittag-Leffler operators, etc. Motivated researchers may potentially be added after contacting Josh Padgett.