(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, 2024 NMIMS Deemed to be University

#### (R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

*Applications and Applied Mathematics: An International Journal (AAM)*

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

Advances In Computational And Statistical Inverse Problems, 2024 Dartmouth College

#### Advances In Computational And Statistical Inverse Problems, Dylan Green

*Dartmouth College Ph.D Dissertations*

Inverse problems are prevalent in many fields of science and engineering, such as signal processing and medical imaging. In such problems, indirect data are used to recover information regarding some unknown parameters of interest. When these problems fail to be well-posed, the original problems must be modified to include additional constraints or optimization terms, giving rise to so-called regularization techniques. Classical methods for solving inverse problems are often deterministic and focus on finding point estimates for the unknowns. Some newer methods approach the solving of inverse problems by instead casting them in a statistical framework, allowing for novel point estimate …

Identifiability For Pde Models Of Fluorescence Microscopy Experiments, 2024 Duke University

#### Identifiability For Pde Models Of Fluorescence Microscopy Experiments, Veronica Ciocanel

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Multiscale Modeling Of Microtubule Polarity Mechanisms Following Neuronal Axotomy, 2024 Virginia Commonwealth University

#### Multiscale Modeling Of Microtubule Polarity Mechanisms Following Neuronal Axotomy, Hannah Scanlon

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Effect Of Recommending Users And Opinions On The Network Connectivity And Idea Generation Process, 2024 Binghamton University, SUNY

#### Effect Of Recommending Users And Opinions On The Network Connectivity And Idea Generation Process, Sriniwas Pandey, Hiroki Sayama

*Northeast Journal of Complex Systems (NEJCS)*

The growing reliance on online services underscores the crucial role of recommendation systems, especially on social media platforms seeking increased user engagement. This study investigates how recommendation systems influence the impact of personal behavioral traits on social network dynamics. It explores the interplay between homophily, users’ openness to novel ideas, and recommendation-driven exposure to new opinions. Additionally, the research examines the impact of recommendation systems on the diversity of newly generated ideas, shedding light on the challenges and opportunities in designing effective systems that balance the exploration of new ideas with the risk of reinforcing biases or filtering valuable, unconventional …

Analytical And Numerical Analysis Of The Sirs Model, 2024 University of Mary Washington

#### Analytical And Numerical Analysis Of The Sirs Model, Catherine Nguyen

*Student Research Submissions*

Mathematical models in epidemiology describe how diseases affect and spread within a population. By understanding the trends of a disease, more effective public health policies can be made. In this paper, the Susceptible-Infected-Recovered-Susceptible (SIRS) Model was examined analytically and numerically to compare with the data for Coronavirus Disease 2019 (COVID-19). Since the SIRS model is a complex model, analytical techniques were used to solve simplified versions of the SIRS model in order to understand general trends that occur. Then by Euler's Method, the Runge-Kutta Method, and the Predictor-Corrector Method, computational approximations were obtained to solve and plot the SIRS model. …

Proof-Of-Concept For Converging Beam Small Animal Irradiator, 2024 The Texas Medical Center Library

#### Proof-Of-Concept For Converging Beam Small Animal Irradiator, Benjamin Insley

*Dissertations & Theses (Open Access)*

The Monte Carlo particle simulator TOPAS, the multiphysics solver COMSOL., and

several analytical radiation transport methods were employed to perform an in-depth proof-ofconcept

for a high dose rate, high precision converging beam small animal irradiation platform.

In the first aim of this work, a novel carbon nanotube-based compact X-ray tube optimized for

high output and high directionality was designed and characterized. In the second aim, an

optimization algorithm was developed to customize a collimator geometry for this unique Xray

source to simultaneously maximize the irradiator’s intensity and precision. Then, a full

converging beam irradiator apparatus was fit with a multitude …

Mathematical Modeling And Examination Into Existing And Emerging Parkinson’S Disease Treatments: Levodopa And Ketamine, 2024 William & Mary

#### Mathematical Modeling And Examination Into Existing And Emerging Parkinson’S Disease Treatments: Levodopa And Ketamine, Gabrielle Riddlemoser

*Undergraduate Honors Theses*

Parkinson’s disease (PD) is the second most common neurodegenerative disease across the world, affecting over 6 million people worldwide. This disorder is characterized by the progressive loss of dopaminergic neurons within the substantia nigra pars compacta (SNpc) due to the aggregation of α-synuclein within the brain. Patients with PD develop motor symptoms such as tremors, bradykinesia, and postural instability, as well as a host of non-motor symptoms such as behavioral changes, sleep difficulties, and fatigue. The reduction of dopamine within the brain is the primary cause of these symptoms. The main form of treatment for PD is levodopa, a precursor …

Convergence Estimate Of Minimal Residual Methods And Random Sketching Of Krylov Subspace Methods, 2024 Clemson University

#### Convergence Estimate Of Minimal Residual Methods And Random Sketching Of Krylov Subspace Methods, Peter Westerbaan

*All Dissertations*

This study concerns two main issues in numerical linear algebra: convergence estimate of minimal residual methods based on explicit construction of approximate min-max polynomials for in- definite matrices, and development and analysis of Krylov subspace methods using non-orthonormal basis vectors based on random sketching. For a matrix A with spectrum Λ(A), it is well known that the min-max polynomial problem min max |pk (z)| pk ∈Pk, pk (0)=1, z∈Λ(A) is used to bound the relative error of Krylov subspace minimum residual methods or similar methods. For a symmetric positive definite matrix A, the min-max polynomial for the Conjugate Gradient (CG) …

Domain Decomposition Methods For Fluid-Structure Interaction Problems Involving Elastic, Porous, Or Poroelastic Structures, 2024 Clemson University

#### Domain Decomposition Methods For Fluid-Structure Interaction Problems Involving Elastic, Porous, Or Poroelastic Structures, Hemanta Kunwar

*All Dissertations*

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and Schwarz waveform relaxation (SWR) method using Robin transmission conditions (or the Robin method), for solving fluid-structure interaction systems involving elastic, porous, or poroelastic structure. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent fluid and the structure subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent multiphysics phenomena. For the fluid-porous …

Analysis Of Nonsmooth Neural Mass Models, 2024 University of Maine - Main

#### Analysis Of Nonsmooth Neural Mass Models, Cadi Howell

*Honors College*

Neural activity in the brain involves a series of action potentials that represent “all or nothing” impulses. This implies the action potential will only “fire” if the mem- brane potential is at or above a specific threshold. The Wilson-Cowan neural mass model [6, 28] is a popular mathematical model in neuroscience that groups excita- tory and inhibitory neural populations and models their communication. Within the model, the on/off behavior of the firing rate is typically modeled by a smooth sigmoid curve. However, a piecewise-linear (PWL) firing rate function has been considered in the Wilson-Cowan model in the literature (e.g., see …

Modeling An Infection Outbreak With Quarantine: The Sibkr Model, 2024 The Ohio State University

#### Modeling An Infection Outbreak With Quarantine: The Sibkr Model, Mikenna Dew, Amanda Langosch, Theadora Baker-Wallerstein

*Rose-Hulman Undergraduate Mathematics Journal*

Influenza is a respiratory infection that places a substantial burden in the world population each year. In this project, we study and interpret a data set from a flu outbreak in a British boarding school in 1978 with mathematical modeling. First, we propose a generalization of the SIR model based on the quarantine measure in place and establish the long-time behavior of the model. By analyzing the model mathematically, we determine the analytic formulas of the basic reproduction number, the long-time limit of solutions, and the maximum number of infection population. Moreover, we estimate the parameters of the model based …

Applications Of Survival Estimation Under Stochastic Order To Cancer: The Three Sample Problem, 2024 St. Mary's University

#### Applications Of Survival Estimation Under Stochastic Order To Cancer: The Three Sample Problem, Sage Vantine

*Honors Program Theses and Research Projects*

Stochastic ordering of probability distributions holds various practical applications. However, in real-world scenarios, the empirical survival functions extracted from actual data often fail to meet the requirements of stochastic ordering. Consequently, we must devise methods to estimate these distribution curves in order to satisfy the constraint. In practical applications, such as the investigation of the time of death or the progression of diseases like cancer, we frequently observe that patients with one condition are expected to exhibit a higher likelihood of survival at all time points compared to those with a different condition. Nevertheless, when we attempt to fit a …

Modeling And Numerical Analysis Of The Cholesteric Landau-De Gennes Model, 2024 Louisiana State University and Agricultural and Mechanical College

#### Modeling And Numerical Analysis Of The Cholesteric Landau-De Gennes Model, Andrew L. Hicks

*LSU Doctoral Dissertations*

This thesis gives an analysis of modeling and numerical issues in the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs) with cholesteric effects. We derive various time-step restrictions for a (weighted) $L^2$ gradient flow scheme to be energy decreasing. Furthermore, we prove a mesh size restriction, for finite element discretizations, that is critical to avoid spurious numerical artifacts in discrete minimizers that is not well-known in the LC literature, particularly when simulating cholesteric LCs that exhibit ``twist''. Furthermore, we perform a computational exploration of the model and present several numerical simulations in 3-D, on both slab geometries and spherical …

Tools For Biomolecular Modeling And Simulation, 2024 Southern Methodist University

#### Tools For Biomolecular Modeling And Simulation, Xin Yang

*Mathematics Theses and Dissertations*

Electrostatic interactions play a pivotal role in understanding biomolecular systems, influencing their structural stability and functional dynamics. The Poisson-Boltzmann (PB) equation, a prevalent implicit solvent model that treats the solvent as a continuum while describes the mobile ions using the Boltzmann distribution, has become a standard tool for detailed investigations into biomolecular electrostatics. There are two primary methodologies: grid-based finite difference or finite element methods and body-fitted boundary element methods. This dissertation focuses on developing fast and accurate PB solvers, leveraging both methodologies, to meet diverse scientific needs and overcome various obstacles in the field.

Homotopy Perturbation Laplace Method For Boundary Value Problems, 2024 Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore, Pakistan

#### Homotopy Perturbation Laplace Method For Boundary Value Problems, Mubashir Qayyum, Khadim Hussain

*International Journal of Emerging Multidisciplinaries: Mathematics*

Most of the real situations are typically modeled as differential equations (DEs). Accurate solutions of such equations is one of the objective of researchers for the analysis and predictions in the physical systems. Typically, pure numerical approaches are utilized for the solution of such problems. These methods are usually consistent, but due to discretization and round-off errors, accuracy can be compromised. Also, pure numerical schemes may be computationally expensive and have large memory requirement. Due to this reason, current manuscript proposed a hybrid methodology by combining homotopy perturbation method (HPM) with Laplace transformation. This scheme provides excellent accuracy in less …

A Novel Fuzzy Time Series Forecasting Method Based On Probabilistic Fuzzy Set And Cpbd Approach, 2024 Shri Ram Murti Smarak College of Engineering and Technology

#### A Novel Fuzzy Time Series Forecasting Method Based On Probabilistic Fuzzy Set And Cpbd Approach, Krishna Kumar Gupta, Suneet Saxena

*Applications and Applied Mathematics: An International Journal (AAM)*

Probabilistic fuzzy set is used to model the non-probabilistic and probabilistic uncertainties simultaneously in the system. This study proposes a cumulative probability-based discretization and probabilistic fuzzy set based novel fuzzy time series forecasting method. It also proposes a novel discretization approach based on cumulative probability to tackle the probabilistic uncertainty in partitioning of datasets. Gaussian probability distribution function has been used to construct probabilistic fuzzy set. The advantage of the proposed work is that it addresses the uncertainties due to randomness and fuzziness simultaneously and also improves accuracy rate in time series forecasting. A proposed forecasting method is applied on …

New Algorithmic Support For The Fundamental Theorem Of Algebra, 2024 The Graduate Center, City University of New York

#### New Algorithmic Support For The Fundamental Theorem Of Algebra, Vitaly Zaderman

*Dissertations, Theses, and Capstone Projects*

Univariate polynomial root-finding is a venerated subjects of Mathematics and Computational Mathematics studied for four millenia. In 1924 Herman Weyl published a seminal root-finder and called it an algorithmic proof of the Fundamental Theorem of Algebra. Steve Smale in 1981 and Arnold Schonhage in 1982 proposed to classify such algorithmic proofs in terms of their computational complexity. This prompted extensive research in 1980s and 1990s, culminated in a divide-and-conquer polynomial root-finder by Victor Pan at ACM STOC 1995, which used a near optimal number of bit-operations. The algorithm approximates all roots of a polynomial p almost as fast as one …

Year-2 Progress Report On Numerical Methods For Bgk-Type Kinetic Equations, 2024 University of Tennessee, Knoxville

#### Year-2 Progress Report On Numerical Methods For Bgk-Type Kinetic Equations, Steven M. Wise, Evan Habbershaw

*Faculty Publications and Other Works -- Mathematics*

In this second progress report we expand upon our previous report and preliminary work. Specifically, we review some work on the numerical solution of single- and multi-species BGK-type kinetic equations of particle transport. Such equations model the motion of fluid particles via a density field when the kinetic theory of rarefied gases must be used in place of the continuum limit Navier-Stokes and Euler equations. The BGK-type equations describe the fluid in terms of phase space variables, and, in three space dimensions, require 6 independent phase-space variables (3 for space and 3 for velocity) for each species for accurate simulation. …

Utilization Of Adomain Decomposition Method And Laplace Transform To Study Fractional Kdv And Fractional Benjamin Models Via Caputo Fractional Operator, 2024 Institute of Mathematics, Khwaja Fareed University of Engineering & Information Technology, Rahim Yar Khan 64200, Pakistan

#### Utilization Of Adomain Decomposition Method And Laplace Transform To Study Fractional Kdv And Fractional Benjamin Models Via Caputo Fractional Operator, Muhammad Sohail, Hina Younis

*International Journal of Emerging Multidisciplinaries: Mathematics*

In the present study, we implement Adomian decomposition method (ADM) to solve fractional potential Korteweg-de Vries (p-KdV) and Benjamin models. The investigated approach is a hybrid of the Adomian decomposition method and the Laplace transform, and the fractional operator developed by Caputo has been utilized in the present research. In a vast accessible domain, the proposed solution tackle impacts and regulates the gained conclusions. Additionally, it provides a simple technique for determining the point of convergence region of the derived result. To ensure that the LADM is realistic and dependable, mathematical simulations for each equation were run, and the results …