Mathematics Commons^™

Qualitative Analysis Of Solutions To A General Class Of Nonlinear Difference Equations With Applications, Osama Moaaz, Mohamed F. Abouelenein, Mona Anis

Mathematical Modelling and Numerical Simulation with Applications

This work examines the qualitative behavior of a general class of difference equations. We establish criteria guaranteeing the stability, periodicity, and boundedness of the solutions of the equation under consideration. In addition, we identify its invariant intervals. The theoretical results are subsequently applied to various special cases, among them the May--Host model. Numerical simulations are presented to demonstrate the dynamics of the solutions and to validate the theoretical analysis.

Sequences That Do Frame Reconstruction, Chad Berner

Mathematics and Statistics Faculty Research & Creative Works

Frames allow all elements of a Hilbert space to be reconstructed by inner product data in a stable manner. Recently, there is interest in relaxing the definition of frames to understand the implications for stable signal recovery. In this paper, we relax the definition of a frame by allowing the operator in the frame decomposition formula to not be invertible. We provide a complete classification of sequences that allow this decomposition via a type of frame operator. In addition, we provide several examples of sequences that allow this reconstruction property that are not frames and illustrate in which ways they …

A Diffuse Interface Model And Fully Decoupled, Energy-Stable Scheme For The Two-Phase Ferrofluid Flows In Porous Media, Guo Dong Zhang, Shuai Zhou, Yunqing Huang, Xiaoming He, Xiaofeng Yang

Mathematics and Statistics Faculty Research & Creative Works

We propose a new diffuse interface model for two-phase porous media ferrofluid flows, employing the phase-field method and establishing an associated energy law. This thermodynamically consistent multi-physics model integrates the Cahn-Hilliard equations, Darcy equations, the magnetostatic equation, and magnetization equations. To efficiently solve the system by addressing its inherent nonlinearities, coupling, and saddle-point structure, we incorporate several advanced techniques, including the SAV-ZEC method to handle nonlinearities and couplings, a reformulated approach to decouple the linear coupling of the magnetic potential and magnetization, and the pressure projection method to decouple velocity and pressure. This results in a numerical scheme that is …

Existence, Uniqueness, Stability And Exact Expression Of Periodic Traveling Wave Solutions Of The Higher-Order Cubic-Quintic Nonlinear Schrödinger Equation, Nan Wang, Yuqian Zhou, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the high-order cubic-quintic nonlinear Schrödinger equation is considered. By overcoming the difficulties caused by its non-integrability, we completely explore the existence, uniqueness and stability of the limit cycle of the equation. It enables us to accurately give the explicit parameter representation of its periodic wave solution in a special form for the first time. These results provide not only a new idea of obtaining periodic solutions in complex nonlinear systems, but also the important theoretical support for understanding related wave phenomena.

(Si15-010) On Perturbations Of Gabor Frames, Jamkhongam Touthang

Applications and Applied Mathematics: An International Journal (AAM)

Stability plays a crucial role in frame theory and its applications. The present paper studies the interaction between Gabor frames and perturbations, presenting perturbation results related to small changes of the frame parameters and the window functions both in regular and irregular settings. Examples are provided for illustration. Additionally, the paper briefly discusses algorithms pertinent to Gabor frames under perturbations and highlights challenging areas in the field.

Existence And Stability Theory Of A Neurologically Inspired Parabolic Pde Model With A Nonlinear Time-Delayed Boundary Condition, Gangadhara Boregowda, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we establish the existence of a positive, bounded solution for a class of parabolic partial differential equations with nonlinear boundary conditions, where the boundary conditions depend on the solution on the boundary at a time 𝜏≥0 in the past. These equations model the production dynamics of a protein species by a single cell, where a feedback mechanism downregulates the protein’s production. Furthermore, we analyze the stability of a nontrivial steady-state solution and provide sufficient conditions on the nonlinearity parameter, boundary flux, and time-delay that ensure the occurrence of a Hopf bifurcation.

Analytical Solution And Stability Of Ψ-Prabhakar Delayed Systems: Application To Antibiotic Production, Mustafa Aydin

Turkish Journal of Mathematics

The analytical solution of the linear ψ -Prabhakar delay differential equations is explored using the method of variation of parameters, in which a ψ -general delayed exponential matrix function is introduced. The ψ -Prabhakar calculus is enhanced in terms of the semi-group property, inversion results, and several simplified calculations and relations. The system's stability is analyzed within the framework of Ulam-Hyers stability. Finally, the theoretical findings are validated through an application to antibiotic production.

Control System Properties Of Discrete-Time Linear Switched Systems With Application To Epidemic Models, Yamen Mehialdin Bawabji

Thesis/ Dissertation Defenses

This thesis studies discrete-time linear switched systems and their application to epidemic models. It focuses on three key properties: stability, controllability, and stabilizability. Stability ensures systems return to equilibrium, controllability examines reaching desired states, and stabilizability determines if unstable systems can be controlled. Using mathematical tools and examples, the research shows how switching between disease transmission modes, like quarantine or vaccination, can help control outbreaks. The work highlights the importance of discrete-time models in epidemiology, as real-world data is often collected at discrete intervals, and provides a foundation for future research on more complex epidemic models.

Isochronous And Period-Doubling Diagrams For Symplectic Maps Of The Plane, T. Zolkin, S. Nagaitsev, I. Morozov, S. Kladov, Y. -K. Kim

Physics Faculty Publications

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more …

Stability Of Optimal Spherical Codes, Károly J. Böröczky, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

For many extremal configurations of points on a sphere, the linear programming approach can be used to show their optimality. In this paper we establish the general framework for showing stability of such configurations and use this framework to prove the stability of the two spherical codes formed by minimal vectors of the lattice E8 and of the Leech lattice.

Individual Stability Of Representations Of Abelian Semigroups, Heybetkulu Mustafayev

Turkish Journal of Mathematics

Let S be a suitable subsemigroup of a locally compact abelian group and let T={T(s)}s(-S be a bounded and strongly continuous trepresentation of S on a Banach space X. In this note, we study the spectral conditions on T and the ergodic conditios on x in X which will imply that T(s)x-->0 strongly as s--> infinity through S.

Advanced Techniques In Time Series Forecasting: From Deterministic Models To Deep Learning, Xue Bai

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation discusses three instances of temporal prediction, applied to population dynamics and deep learning.

In population modeling, dynamic processes are frequently represented by systems of differential equations, allowing for the analysis of various phenomena. The first application explores modeling cloned hematopoiesis in chronic myeloid leukemia (CML) via a nonlinear system of differential equations. By tracking the evolution of different cell compartments, including cycling and quiescent stem cells, progenitor cells, differentiated cells, and terminally differentiated cells, the model captures the transition from normal hematopoiesis to the chronic and accelerated-acute phases of CML. Three distinct non-zero steady states are identified, representing …

Two-Dimensional Steady Squeezing Flow Over A Vertical Porous Channel With Free Convective Heat/Mass Transfer And Invariable Suction, Zeeshan, Waris Khan, Taoufik Saidani, Florentin Smarandache, Muhammad Shahid Khan, Hamdi Ayed, M. Modather M. Abdou

Branch Mathematics and Statistics Faculty and Staff Publications

This research reports on the combined effects of heat and mass transfer (HMT) under the influences of the Soret and Dufour in natural convection steady 2D magnetohydrodynamic flow through the boundary layer in a porous vertical tube or duct. The current study is motivated by the significant applications of HMT in engineering processes such as casting and welding. The goal of this framework is to explore the assisting and opposing movements with HMT above a vertical porous channel under the influence of invariant suction and fluid dissipation which have not been reported in the earlier studies. The governing flow equations …

An Analysis Of Plant Response To Herbivory, Lawrence Gustavo Seminario-Romero

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Herbivory is often believed to always negatively affect the growth of plants, but there are plant species that can actually benefit from being consumed by herbivores. These plants have a phenotypic trait known as “compensatory growth.” A plant that exhibits compensatory growth is able to increase its intrinsic growth rate as a response to taking damage caused by herbivory, and as a result, produce more biomass. Very few mathematical models used to describe plant-herbivore interactions have accounted for plant compensatory growth. Moreover, these models tend to assume that herbivores follow a Holling type II functional response, which may not necessarily …

The Application Of Brzdek's Fixed Point Theorem In The Stability Problem Of The Drygas Functional Equation, Mehdi Dehghanian, Yamin Sayyari

Turkish Journal of Mathematics

Using the Brzdek fixed point theorem, we establish the Hyers?Ulam stability problem of Drygas functional equations \begin{equation} \delta(x+y-z)+\delta(x-y)+\delta(-y-z)+\delta(y)=\delta(x-y-z)+\delta(y-z)+\delta(x+y)+\delta(-y)\nonumber \end{equation} for all $x,y,z\in A$.

Stability Of Cauchy's Equation On Δ+., Holden Wells

Electronic Theses and Dissertations

The most famous functional equation f(x+y)=f(x)+f(y) known as Cauchy's equation due to its appearance in the seminal analysis text Cours d'Analyse (Cauchy 1821), was used to understand fundamental aspects of the real numbers and the importance of regularity assumptions in mathematical analysis. Since then, the equation has been abstracted and examined in many contexts. One such examination, introduced by Stanislaw Ulam and furthered by Donald Hyers, was that of stability. Hyers demonstrated that Cauchy's equation exhibited stability over Banach Spaces in the following sense: functions that approximately satisfy Cauchy's equation are approximated with the same level of error by functions …

Simulation Of Mathematical Model Of Network Interference On Global System For Mobile Communication, Yidiat O. Aderinto, Yusuf Musa

Tanzania Journal of Science

Network interference is the incorporation of undesirable signals to desirable ones in an interconnected system. In this paper, the simulation of mathematical model of network interference with respect to global system of mobile communication is presented. The model was formulated and analyzed mathematically. Numerical simulation was carried out using real life data from Communication Towers Nigeria Limited, Northwest regional office Kaduna, Nigeria. The results obtained were very close to that of laboratory investigation. Keywords: Mathematical Model, Network, Interference, Congestion, Mobile communication, Stability

Modeling And Analyzing Homogeneous Tumor Growth Under Virotherapy, Chayu Yang, Jin Wang

Department of Mathematics: Faculty Publications

We present a mathematical model based on ordinary differential equations to investigate the spatially homogeneous state of tumor growth under virotherapy. The model emphasizes the interaction among the tumor cells, the oncolytic viruses, and the host immune system that generates both innate and adaptive immune responses. We conduct a rigorous equilibrium analysis and derive threshold conditions that determine the growth or decay of the tumor under various scenarios. Numerical simulation results verify our analytical predictions and provide additional insight into the tumor growth dynamics.

Qualitative Study Of A Second Order Difference Equation, Messaoud Berkal, Juan Francisco Navarro

Turkish Journal of Mathematics

In this paper, we study a second order rational difference equation. We analyze the stability of the unique positive equilibrium of the equation and prove the existence of a Neimark-Sacker bifurcation, validating our theoretical analysis via a numerical exploration of the system.

Definite Condition Of The Evolutionary (P)Over-Right-Arrow(X)-Laplacian Equation, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

For the nonlinear degenerate parabolic equations, how to find an appropriate boundary value condition to ensure the well-posedness of weak solution has been an interesting and challenging problem. In this paper, we develop the general characteristic function method to study the stability of weak solutions based on a partial boundary value condition.

Delay Dynamic Equations On Isolated Time Scales And The Relevance Of One-Periodic Coefficients, Martin Bohner, Tom Cuchta, Sabrina Streipert

Mathematics and Statistics Faculty Research & Creative Works

We are motivated by the idea that certain properties of delay differential and difference equations with constant coefficients arise as a consequence of their one-periodic nature. We apply the recently introduced definition of periodicity for arbitrary isolated time scales to linear delay dynamic equations and a class of nonlinear delay dynamic equations. Utilizing a derived identity of higher order delta derivatives and delay terms, we rewrite the considered linear and nonlinear delayed dynamic equations with one-periodic coefficients as a linear autonomous dynamic system with constant matrix. As the simplification of a constant matrix is only obtained for one-periodic coefficients, dynamic …

(R1882) Effects Of Viscosity, Oblateness, And Finite Straight Segment On The Stability Of The Equilibrium Points In The Rr3bp, Bhavneet Kaur, Sumit Kumar, Rajiv Aggarwal

Applications and Applied Mathematics: An International Journal (AAM)

Associating the influences of viscosity and oblateness in the finite straight segment model of the Robe’s problem, the linear stability of the collinear and non-collinear equilibrium points for a small solid sphere m₃ of density \rho₃ are analyzed. This small solid sphere is moving inside the first primary m₁ whose hydrostatic equilibrium figure is an oblate spheroid and it consists of an incompressible homogeneous fluid of density \rho₁. The second primary m₂ is a finite straight segment of length 2l. The existence of the equilibrium points is discussed after deriving the pertinent …

On Uniqueness And Stability For The Boltzmann-Enskog Equation, Martin Friesen, Barbara Ruediger, Padmanabhan Subdar

Faculty Publications

The time-evolution of a moderately dense gas in a vacuum is described in classical mechanics by a particle density function obtained from the Boltzmann-Enskog equation. Based on a McKean-Vlasov equation with jumps, the associated stochastic process was recently constructed by modified Picard iterations with the mean-field interactions, and more generally, by a system of interacting particles. By the introduction of a shifted distance that exactly compensates for the free transport term that accrues in the spatially inhomogeneous setting, we prove in this work an inequality on the Wasserstein distance for any two measure-valued solutions to the Boltzmann-Enskog equation. As a …

Some Convergence, Stability, And Data Dependence Results For $K^{\Ast }$ Iterative Method Of Quasi-Strictly Contractive Mappings, Ruken Çeli̇k, Neci̇p Şi̇mşek

Turkish Journal of Mathematics

In a recent paper, Yu et al. obtained convergence and stability results of the $K^{\ast }$ iterative method for quasi-strictly contractive mappings [An iteration process for a general class of contractive-like operators: Convergence, stability and polynomiography. AIMS Mathematics 2021; 6 (7): 6699-6714.]. To guarantee these convergence and stability results, the authors imposed some strong conditions on parametric control sequences which are used in the $K^{\ast }$ iterative method. The aim of the presented work is twofold: (a) to recapture the aforementioned results without any restrictions imposed on the mentioned parametric control sequences (b) to complete the work of Yu et …

Modelling The Impact Of Undetected Cases On The Transmission Dynamics Of Covid-19, Herieth Rwezaura

Tanzania Journal of Science

The spread of COVID-19 globally has caused negative impacts to the public, making understanding the dynamics of transmission a necessity. Essential factors such as undetected cases, asymptomatic cases, and several non-pharmaceutical interventions have played significant roles in the spreading mechanism of COVID-19 in the human population. It is imperative to understand the significance of these factors in order to determine whether COVID-19 will be eradicated or will continue to persist in the population. A mathematical model is formulated to investigate the impacts of vaccination and several non-pharmaceutical interventions on the dynamics of a COVID-19 accounting for asymptomatic cases, detected (identified) …

Effects Of Refuge Prey On Stability Of The Prey-Predator Model Subject To Immigrants: A Mathematical Modelling Approach, Mussa Amos Stephano, Il Hyo Jung

Tanzania Journal of Science

Prey-predator system is enormously complex and nonlinear interaction between species. Such complexity regularly requires development of new approaches which involves more factors in analysis of its population dynamics. In this paper, we formulate a modified Lotka-Volterra model that incorporates factors such as refuge prey and immigrants. We investigate the effects of refuge prey and immigrants by varying the refuge factor, with and without immigrants. The results show that with Holling’s type I functional response, the proposed model is asymptotically convergent when a refuge prey factor is introduced. Moreover, with Holling’s type II functional response, the proposed mathematical model is unstable …

Discrete Dynamics Of Dynamic Neural Fields, Eddy Kwessi

Mathematics Faculty Research

Large and small cortexes of the brain are known to contain vast amounts of neurons that interact with one another. They thus form a continuum of active neural networks whose dynamics are yet to be fully understood. One way to model these activities is to use dynamic neural fields which are mathematical models that approximately describe the behavior of these congregations of neurons. These models have been used in neuroinformatics, neuroscience, robotics, and network analysis to understand not only brain functions or brain diseases, but also learning and brain plasticity. In their theoretical forms, they are given as ordinary or …

Optimal Partial Boundary Condition For Degenerate Parabolic Equations, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

For the stability of the non-Newtonian fluid equation∂u∂t−div(a(x)|∇u|p−2∇u)−N∑i=1bi(x)Diu+c(x,t)u=f(x,t),where a(x)|x∈Ω>0, a(x)|x∈∂Ω=0 and bi(x)∈C1(¯¯¯Ω), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the …

Stochastic Delay Differential Equations With Applications In Ecology And Epidemics, Hebatallah Jamil Alsakaji

Dissertations

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the …

Efficient Time-Stepping Approaches For The Dispersive Shallow Water Equations, Linwan Feng

Dissertations

This dissertation focuses on developing efficient and stable (high order) time-stepping strategies for the dispersive shallow water equations (DSWE) with variable bathymetry. The DSWE extends the regular shallow water equations to include dispersive effects. Dispersion is physically important and can maintain the shape of a wave that would otherwise form a shock in the shallow water system.

In some cases, the DSWE may be simplified when the bathymetry length scales are small (or large) in relation to other length scales in the shallow water system. These simplified DSWE models, which are related to the full DSWEs, are also considered in …