MS, University of Cincinnati, 2024, Engineering and Applied Science: Mechanical Engineering

Within this thesis document we focus on the application of Nonlinear Model Predictive Control (NMPC) onto an epidemic compartmental model. The compartmental model is a partial differential equation (PDE) based Susceptible Latent Infected Recovered (SLIR) epidemic model. This model serves as the basis of the NMPC. In order to generate the necessary parameters for initializing and training the use of constrained optimization, a single-objective Genetic Algorithm (GA), and LSTM (Long-Short-Term-Memory) deep learning were explored. The spatial domains considered for the SLIR epidemic model includes Hamilton County, Ohio as well as the entire state of Ohio, USA. With respect to Hamilton County, Ohio three different time periods were evaluated in which varied levels of infection relating to COVID-19 were observed. At the state wide level only one time period was consider. The NMPC considers two control schemes. The first being control applied uniformly across the spatial domain of interest. While the second focuses on applying the control in a spatially targeted manner to specific geographical areas based on observed higher levels of infection. The NMPC also employs a cost function comprising the infection spread density and the associated cost of applied control measures. The latter of which in turn representing socioeconomic effects. Overall, the NMPC framework developed here is intended to aid in the evaluation of optimal Non-Pharmaceutical Interventions (NPI) towards spread mitigation of infectious diseases.

Committee: Manish Kumar Ph.D. (Committee Chair); Shelley Ehrlich M.D. (Committee Member); Subramanian Ramakrishnan Ph.D. (Committee Member); David Thompson Ph.D. (Committee Member) Subjects: Mechanical Engineering

MS, University of Cincinnati, 2024, Engineering and Applied Science: Mechanical Engineering

The COVID-19 pandemic highlighted the need for improved and precise prediction of the spatiotemporal trends of epidemic transmission. An optimized epidemic model is crucial for effectively forecasting flow of infection. By optimizing the model parameters, they can provide valuable insights into the dynamics of infection transmission and this degree of tuning helps health officials and policymakers to make data-driven decisions regarding disease control strategies, allocation of resources, and planning for healthcare. Therefore, it highlights the need of implementing reliable optimizing strategies in case of epidemic models. Similarly, the basic and effective reproductive numbers (R0, Re) are quantitative metrics widely used for estimating the rate at which the infection propagates. The limitations of existing techniques for estimating R0 and Re points the need for novel approaches to accurately estimate them using the available data. This initial part of this study presents the development of a custom GA which is capable of efficiently searching for the parameters of an epidemic model in any specified geographical region and time period. Following this, a novel computational framework for predicting the reproduction numbers from true infection data has been presented. The computational framework is derived from a reaction-diffusion based PDE epidemic model which involves fundamental mathematical derivations for obtaining their values. The PDE model is optimized using the proposed GA and the model output using the optimized parameters is found to be in correspondence with the ground truth COVID-19 data of Hamilton county, Ohio. Subsequently, the established framework for calculating the reproduction numbers was applied on the optimized model and their predictions are found to correlate with the true incidence data. In addition, these predictions are compared with a commonly used retrospective method (Wallinga-Teunis) and are found to be in harmony thereby est (open full item for complete abstract)

Committee: Manish Kumar Ph.D. (Committee Chair); Subramanian Ramakrishnan Ph.D. (Committee Member); Shelley Ehrlich M.D. (Committee Member); Derek Wolf Ph.D. (Committee Member) Subjects: Mechanical Engineering

Doctor of Philosophy (PhD), Ohio University, 2020, Physics and Astronomy (Arts and Sciences)

Actin and neurofilaments are two of the three major structural proteins that constitute the axonal cytoplasm. They are primarily synthesized in the soma of the neuron, from where they must be moved over great distances throughout the axon. Axonal transport collectively refers to the transport mechanisms, which the neuron has developed to mediate this bulk transport of proteins. The axonal transport mechanism of each protein is uniquely suited for its biophysical characteristics. The goal of this dissertation is to study and highlight the differences between the axonal transport of actin and neurofilaments. Recent experimental studies by our collaborators [1, 2] have shown that axonal actin is highly dynamic and undergoes focal assembly and disassembly on stationary endosomes, and elongates metastable fibers along the axon shaft (”actin trails”). These studies have shown that more actin trails extend away from the soma (anterogradely) than towards the soma, and a bulk, slow anterograde transport of actin along the axon. These findings led us to hypothesize that dynamic actin trails mediate slow axonal actin transport. In this study, we have simulated axonal actin trail assembly by extracting the nucleation rates of actin trails from the imaging data and by modeling the filament polymerization reactions. We have found that axonal actin trail assembly/disassembly indeed leads to a bulk anterograde transport of the actin population at rates that correspond to slow axonal transport rate found in in-vivo photoactivation experiments. Our model and associated experiments from our collaborators' laboratories suggest that actin transport is driven by the biased polymerization dynamics of actin trails. This mechanism is akin to a molecular hitchhiking process where G-actin monomers are occasionally recruited on to actin filaments. Since filaments have a preference of elongating anterogradely, this causes the entire population to move along the axon with a net anterograde flux. (open full item for complete abstract)

Committee: Peter Jung (Advisor); Charlotte Elster (Committee Member); Alexander Neiman (Committee Member); Daewoo Lee (Committee Member) Subjects: Biophysics; Neurobiology; Neurosciences; Physics

Doctor of Philosophy, Case Western Reserve University, 2019, Applied Mathematics

Conditions for bubble cavitation and behavior in non-Newtonian fluids have numerous applications in physical sciences, engineering and medicine. Non-Newtonian fluids are a rich, but relatively undeveloped area of fluid dynamics, with phenomena from di↵u- sion to bubble growth just beginning to receive attention. In the course of examining bubble cavitation, it became apparent that the random particle motion responsible for determining potential bubble formation had not been researched. As cavitation bub- bles collapse, they deform into a variety of non-spherical shapes. Due to the complex dynamics and the radial focus of current equations on bubble behavior, no accepted model has yet emerged. This work explores the behavior using numerical methods on both fluid and bubble models to examine this system from di↵erent prospectives, culminating in a time-fractional, power-law Burger's type equation showing bubble formation under these conditions.

Committee: Wojbor Woyczynski (Advisor); David Gurarie (Committee Member); Longua Zhao (Committee Member) Subjects: Applied Mathematics

Master of Science, University of Akron, 2018, Applied Mathematics

A mathematical model is developed to describe the light field in vertical line seaweed cultivation to determine the degree to which the seaweed shades itself and limits the amount of light available for photosynthesis. A probabilistic description of the spatial distribution of kelp is formulated using simplifying assumptions about frond geometry and orientation. An integro-partial differential equation called the radiative transfer equation is used to describe the light field as a function of position and angle. A finite difference solution is implemented, providing robustness and accuracy at the cost of large CPU and memory requirements, and a less computationally intensive asymptotic approximation is explored for the case of low scattering. Conditions for applicability of the asymptotic approximation are discussed, and depth-dependent light availability is compared to the predictions of simpler light models. The 3D model of this thesis is found to predict significantly lower light levels than the simpler 1D models, especially in regions of high kelp density where a precise description of self-shading is most important.

Committee: Kevin Kreider Ph.D (Advisor); Curtis Clemons Ph.D (Advisor); Gerald Young Ph.D (Advisor) Subjects: Applied Mathematics; Aquaculture; Aquatic Sciences; Ocean Engineering; Optics

Master of Mathematical Sciences, The Ohio State University, 2017, Mathematical Sciences

Partial Differential Equation simulations can produce large amounts of data that are very slow to transfer. There have been many model reduction techniques that have been proposed and utilized over the past three decades. Two popular techniques Proper Orthogonal Decomposition and Dynamic Mode Decomposition have some hindrances. Non-Uniform Dynamic Mode Decomposition (NU-DMD), which was introduced in 2015 by Gueniat et al., that overcomes some of these hindrances. In this thesis, the NU-DMD's mathematics are explained in detail, and three versions of the NU-DMD's algorithm are outlined. Furthermore, different numerical experiments were performed on the NU-DMD to ascertain its behavior with repect to errors, memory usage, and computational efficiency. It was shown that the NU-DMD could reduce an advection-diffusion simulation to 6.0075% of its original memory storage size. The NU-DMD was also applied to a computational fluid dynamics simulation of a NASA single-stage compressor rotor, which resulted in a reduced model of the simulation (using only three of the five simulation variables) that used only about 4.67% of the full simulation's storage with an overall average percent error of 8.90%. It was concluded that the NU-DMD, if used appropriately, could be used to possibly reduce a model that uses 400GB of memory to a model that uses as little as 18.67GB with less than 9% error. Further conclusions were made about how to best implement the NU-DMD.

Committee: Ching-Shan Chou (Advisor); Jen-Ping Chen (Committee Member) Subjects: Aerospace Engineering; Applied Mathematics; Computer Science; Mathematics; Mechanical Engineering

Doctor of Philosophy, University of Toledo, 2015, Mechanical Engineering

External Verification Analysis (EVA), a new approach to verifying unsteady partial differential equation codes, is presented. After a review of the relevant code verification literature, the mathematical foundation and solution method of the EVA tool is discussed in detail. The implementation of the EVA tool itself is verified through an independent Python program. A procedure for code verification with the EVA tool is described and then applied to the three-dimensional form of a high-order non-linear computational aeroacoustics code.

Committee: Ray Hixon (Advisor); Sorin Cioc (Committee Member); James DeBonis (Committee Member); Mehdi Pourazady (Committee Member); Chunhua Sheng (Committee Member) Subjects: Aerospace Engineering; Fluid Dynamics; Mechanical Engineering

Doctor of Philosophy (PhD), Ohio University, 2013, Physics and Astronomy (Arts and Sciences)

Neurons communicate with each other through dendrites and axons. Typically, dendrites are responsible for receiving signals from other neurons, while axons are the pathways to send out signals. Signal propagation through axons is closely correlated with their morphology. It is well known that the rate of signal propagation is proportional to the caliber of axons[2]. The intrinsic determinant of axonal caliber is the abundance of cytoskeletal protein, neurofilament (NF)[6]. NFs are not static but undergo "slow axonal transport", which is characterized by rapidly intermittent, asynchronous and bidirectional motion[21-23]. Many neurodegenerative diseases are related to the malfunction of neurofilament transport, either by accumulation of neurofilaments leading to swelling of the axon or by deficiency in neurofilaments resulting in axonal atrophy[9-12]. The mechanism of neurofilament transport can be explained by the "stop-and-go"; hypothesis[21, 24, 28], according to which neurofilaments spend long periods of time pausing interrupted by bouts of rapid movements. By the "stop-and-go" hypothesis, a compact and powerful mathematical model was proposed in [27], which connects the group behavior of neurofilaments as a wave to the individual neurofilament kinetics, which are observed directly from time-lapse imaging. Our main hypothesis is that axonal morphology is determined by the kinetics of NFs. According to this hypothesis, an increase in axonal caliber must go along with a decrease in speed of NFs and accordingly a modified kinetics. Two main examples, the distally increasing accumulation of NFs in the mouse optic nerve and the constrictions of myelinated axons at the nodes of Ranvier, demonstrate this hypothesis and support it with detailed experimental data. In the mouse optic nerve, sufficient data about the abundance of NFs proximal to distal as well as kinetic data are available to extract differential kinetics using our computational model. The most rema (open full item for complete abstract)

Committee: Peter Jung (Advisor); David F. J. Tees (Committee Member); Markus Böttcher (Committee Member); Ralph DiCaprio (Committee Member) Subjects: Biophysics; Physics

PhD, University of Cincinnati, 2009, Arts and Sciences : Mathematical Sciences

The Korteweg-de Vries equation models unidirectional propagation of small finite amplitude long waves in a non-dispersive medium. The well-posedness, that is the existence, uniqueness of the solution, and continuous dependence on data, has been studied on unbounded,periodic, and bounded domains.This research focuses on an initial and boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation posed on a bounded interval with general nonhomogeneous boundary conditions. Using Kato smoothing properties of an associated linear problem and the contraction mapping principle, the IBVP is shown to be locally well-posed given several conditions on the parameters for the boundary conditions, in the L²-based Sobolev space Hs(0, 1) for any s≥0.

Committee: Bingyu Zhang PhD (Committee Chair); H Dumas PhD (Committee Member); Anthony Leung PhD (Committee Member); Philip Korman PhD (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 2005, Mathematics

In the present work we study the existence and stability of multi-pulses in dynamical systems that arise as traveling-wave equations for a partial differential equation (PDE) with symmetries. The motivation comes from two different models that describe the propagation of pulses in optical fibers. In the first part of the thesis we consider reversible, ℤ2 symmetric dynamical systems with heteroclinic orbits related via symmetries. The heteroclinic orbits are assumed to undergo an orbit flip bifurcation upon changing appropriate parameters. We construct multi-bump solutions close to the heteroclinic orbits and investigate their PDE stability by using Lin's method and Lyapunov-Schmidt reduction. We apply this abstract theory to a model equation that describes the propagation of pulses in optical fibers with phase sensitive amplifiers. Our results show that stable multi-pulses exist. In the second part we consider parameter-dependent dynamical systems with reflection and SO(2) symmetry, which possess a homoclinic solution to a saddle focus. The reflection symmetry is broken by the second parameter which plays the role of the wave speed. We derive the bifurcation equations for the existence of N-pulse solutions and solve them for N=3. As a result we obtain standing and traveling 3-pulse solutions which we describe through the phase differences and the distances between consecutive bumps. We also investigate stability of these 3-pulses. We derive the stability matrix for multi-bump solutions and compute to the leading order the location of the eigenvalues for the 3-pulses.

Committee: Bjorn Sandstede (Advisor) Subjects: Mathematics

PHD, Kent State University, 2008, College of Arts and Sciences / Department of Mathematical Sciences

In this Dissertation we investigate the compact support property of the solutions of two hyperbolic stochastic partial differential equations (SPDEs) whose initial condition function is deterministic and compactly supported. First, we consider the hyperbolic semi-SPDE treated by Allouba and Goodman and others in Financial mathematics modelling. This is an SPDE in both time and space in which all the derivatives (including in the noise) are only with respect to the time parameter, and hence the name semi-SPDE. It turns out that, under appropriate conditions on the diffusion coefficient, the semi-SPDE preserves the compact support property. I.e., starting from a compactly supported initial solution u0(x), the solution u(t,x) is compactly supported in x for all times t>0. Second, we consider a rotated wave SPDE in time-space considered by Allouba. Our approach here is to use the Allouba stochastic differential-difference equations (SDDE) approach. In this approach, we start by discretizing space, leaving time continuous, thereby obtaining a simpler version of the SPDE under question. We then resolve the question for the SDDE (or SPDE on the spatial lattice) and then use a limiting argument – as the mesh size of the spatial lattice goes to zero – to transfer regularity results to the associated SPDE. We also prove a noncompact support result for the SPDE. It turns out that in the rotated wave SPDE case, the compact support property is not preserved with positive probability. The contrast between the two SPDEs' behaviors is due to the extra differentiation in space in the second SPDE which plays a crucial role in the behavior of solutions.

Committee: Prof. Hassan Allouba (Committee Chair); Prof. Brett Ellman (Committee Member); Prof. Andrew Tonge (Committee Member); Prof. Deng-Ke Yang (Committee Member); Prof. Volodymyr Andriyevskyy (Committee Member) Subjects: Mathematics

Doctor of Philosophy, Case Western Reserve University, 2012, Mathematics

We note the interesting phenomenon that the Kantorovich-Rubinstein metric is not complete on the space of signed measures. Consequently, we introduce a new metric with a useful partial completeness property. With this metric, a general result about the Hahn-Jordan decomposition of solutions of stochastic partial differential equations is shown. These general results are applied to the smoothed Stochastic Navier-Stokes equations. As an application, we derive that the vorticity of the fluid is conserved for a solution of the Stochastic Navier-Stokes equations.

Committee: Dr. Peter Kotelenez PhD (Committee Chair); Dr. Elizabeth Meckes PhD (Committee Member); Dr. Manfred Denker PhD (Committee Member); Dr. Marshall Leitman PhD (Committee Member) Subjects: Aerospace Engineering; Mathematics; Physics

Doctor of Philosophy, University of Akron, 2011, Applied Mathematics

The technology of porous bearings is well-known in industry. In classical cases, the porous medium acts as an external reservoir making their use ideal for applications where an external lubricant supply is undesirable or impractical or when fluid has to be delivered on a continual basis. The work considered here looks to extend the benefits of typical porous bearings to allow for the bearing to be sealed, containing, from the onset of operation, all necessary lubricant. The goal of this work is to demonstrate a bearing that circulates the fluid between a fluid film and an eccentric reservoir, using a porous medium as an intermediary; a system that is capable of supporting a realistic load, while simultaneously pumping the fluid back and forth between the lubricating region and the reservoir. The method used to investigate such a bearing is a mixture of analytical and numerical techniques. For the analysis, a non-dimensionalization scheme is used to analyze both the momentum and thermal governing equations at their differing orders of magnitude. Upon doing so, the governing momentum equations are reduced considerably which allows for a straight-forward numerical solution procedure. The governing thermal equations are solved using an asymptotic expansion approach, keeping the first and second order terms and equations. This is done so to more accurately model the effects the circulating fluid has on the thermal performance of the bearing. The phenomenon of cavitation is also discussed, utilizing a method that integrates cavitation into the governing equations and numerical solution procedure. Unlike other cavitation models that decouple cavitation from the governing momentum equations, this model accounts for mass flow continuity which leads to more realistic results. Practical design considerations, including how to determine the effective permeability and the effective heat transfer coefficient at the exterior wall of the bearing, are discussed. These parameters, u (open full item for complete abstract)

Committee: Gerald Young Dr. (Advisor); Minel Braun Dr. (Advisor); Kevin Kreider Dr. (Committee Member); Joseph Wilder Dr. (Committee Member); Scott Sawyer Dr. (Committee Member); Alex Povitsky Dr. (Committee Member); S.I. Hariharan Dr. (Committee Member) Subjects: Mathematics; Mechanical Engineering

Master of Science, University of Akron, 2006, Applied Mathematics

Mathematical models of physical processes often include partial differential equations (PDEs). Oftentimes solving PDEs analytically is not feasible and a numerical method is implemented to obtain an approximate solution. Too often the assumption is made that the solution should be trusted when codes are prone to implementation errors. Code verification is a field of mathematics that shows the algorithm has been implemented without mistakes and has correctly solved the problem. Currently no one method of code verification is universally accepted. The Method of Manufactured Exact Solutions (MMES) is the most commonly used, but it has a considerable disadvantage of altering the code after verification. We have developed a new method, Completely Residual Based Code Verification (CRBCV), which does not require any modification. By using several solution methods, we have shown that CRBCV is dependable when verifying the heat equation with linear and nonlinear source terms and a frontal polymerization model.

Committee: Laura Gross (Advisor) Subjects: Mathematics