Partial Differential Equations Commons^™

Shallow Water Equations And Floor Topography Affect On Sea Surface, Chase Jones

Mathematics Senior Capstone Papers

For this research project, we have been doing research on the shallow water equations: a set of hyperbolic partial differential equations. These equations exist as a set of three primary equations [2]. However, there is another version of the shallow water equations called the Saint Venant’s equations. These equations are similar to the standard shallow water equations, but these equations have been reduced to one-dimension. The primary goal of our research has been to investigate the behavior and mathematical construction of the Saint Venant’s equations and model these equations using COMSOL. Regardless of the equation type, standard or Saint Venant’s, …

A Time Integration Method Of Approximate Particular Solutions For Nonlinear Ordinary Differential Equations, Cyril Ocloo

Master's Theses

We consider a time-dependent method which is coupled with the method of approximate particular solutions (MAPS) of Delta-shaped basis functions and the method of fundamental solutions (MFS) to solve nonlinear ordinary differential equations. Firstly, we convert a nonlinear problem into a sequence of time-dependent non-homogeneous boundary value problems through a fictitious time integration method. The superposition principle is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. Delta-shaped basis functions are used to provide an approximation of the source function at each time step. The purpose of this is to …

The Long Time Behavior Of The Predator-Prey Model With Holling Type Iii, Regen S. Mcgee

Honors Theses

In this paper, the classical Lotka-Volterra model is expanded based on functional response of Holling type III to analyze a dynamical predator-prey relationship with hunting cooperation (a) and the Allee effect among predators. The stability of equilibrium solutions was first analyzed by deriving a Jacobian matrix from partial derivatives of our model. Newly derived eigenvalues are then used to determine the stability. The viability of the model is then demonstrated by using MATLAB. The numerical results show a clear Allee effect and a variety of possible phenomena related to stability when carrying capacity (k) is varied. Two different types of …

Hadamard Well-Posedness For Two Nonlinear Structure Acoustic Models, Andrew Becklin

Department of Mathematics: Dissertations, Theses, and Student Research

This dissertation focuses on the Hadamard well-posedness of two nonlinear structure acoustic models, each consisting of a semilinear wave equation defined on a smooth bounded domain $\Omega\subset\mathbb{R}^3$ strongly coupled with a Berger plate equation acting only on a flat portion of the boundary of $\Omega$. In each case, the PDE is of the following form: \begin{align*} \begin{cases} u_{tt}-\Delta u +g_1(u_t)=f(u) &\text{ in } \Omega \times (0,T),\\[1mm] w_{tt}+\Delta^2w+g_2(w_t)+u_t|_{\Gamma}=h(w)&\text{ in }\Gamma\times(0,T),\\[1mm] u=0&\text{ on }\Gamma_0\times(0,T),\\[1mm] \partial_\nu u=w_t&\text{ on }\Gamma\times(0,T),\\[1mm] w=\partial_{\nu_\Gamma}w=0&\text{ on }\partial\Gamma\times(0,T),\\[1mm] (u(0),u_t(0))=(u_0,u_1),\hspace{5mm}(w(0),w_t(0))=(w_0,w_1), \end{cases} \end{align*} where the initial data reside in the finite energy space, i.e., $$(u_0, u_1)\in H^1_{\Gamma_0}(\Omega) \times L^2(\Omega) \, \text{ …

483— Effectiveness Of Mmr Vaccination In Orthodox Jewish Neighborhoods, Meenu Mundackal

GREAT Day Posters

Measles is a highly contagious disease, where large outbreaks arise by direct contact between susceptible (unvaccinated) and infectious individuals. Many Orthodox Jewish neighborhoods were affected by measles from 2018-2019. To quantify the vaccination effort on this susceptible population, a retrospective analysis was used to study the NYC and Rockland County populations using a differential equations model. A subsequent model, known as a realistically-structured network model, studied only the NYC population, in relation to typical household size. Vaccination strategies were applied to three cohorts: unvaccinated family members, members with 1 prior MMR dose, and members with 2 prior MMR doses. The …

The Dual Of The Compressed Shift, M. C. Câmara, William T. Ross

Department of Math & Statistics Faculty Publications

For an inner function u, we discuss the dual operator for the compressed shift P_uS∣_Ku, where K_u is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

Fluid mechanics occupies a privileged position in the sciences; it is taught in various science departments including physics, mathematics, environmental sciences and mechanical, chemical and civil engineering, with each highlighting a different aspect or interpretation of the foundation and applications of fluids. Doll’s fluid analogy [5] for this idea is especially relevant to this issue: “Emergence of creativity from complex flow of knowledge—example of Benard convection pattern as an analogy—dissipation or dispersal of knowledge (complex knowledge) results in emergent structures, i.e., creativity which in the context of education should be thought of as a unique way to arrange information so …

Existence And Stability Results Of Nonlinear Fractional Differential Equations With Nonlinear Integral Boundary Condition On Time Scales, Vipin Kumar, Muslim Malik

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we establish the existence and uniqueness of the solution to a nonlinear fractional differential equation with nonlinear integral boundary conditions on time scales.We used the fixed point theorems due to Banach, Schaefer’s, nonlinear alternative of Leray Schauder’s type and Krasnoselskii’s to establish these results. In addition, we study Ulam-Hyer’s (UH) type stability result. At the end, we present two examples to show the effectiveness of the obtained analytical results.

Numerical Analysis Of Three-Dimensional Mhd Flow Of Casson Nanofluid Past An Exponentially Stretching Sheet, Madhusudan Senapati, Kharabela Swain, Sampad Kumar Parida

Karbala International Journal of Modern Science

The convective three dimensional electrically conducting Casson nanofluid flow over an exponentially stretching sheet embedded in a saturated porous medium and subjected to thermal as well as solutal slip in the presence of externally applied transverse magnetic field (force-at-a-distance) is studied. The heat transfer phenomenon includes the viscous dissipation, Joulian dissipation, thermal radiation, contribution of nanofluidity and temperature dependent volumetric heat source. The study of mass diffusion in the presence of chemically reactive species enriches the analysis. The numerical solutions of coupled nonlinear governing equations bring some earlier reported results as particular cases providing a testimony of validation of the …

Investigating The Solution Properties Of Population Model Of Cross-Diffusion Model With Double Nonlinearity And With Variable Density, Dildora Kabilovna Muhamediyeva

Chemical Technology, Control and Management

The models of two competing populations with double nonlinear diffusion and three types of functional dependencies are considered. The first dependence corresponds to the Malthusian type, the second to the Verhühlst type (logistic population), and the third to Olli-type populations. A common element of this kind of description is the presence of a linear source. Nonlinear sinks are also present in descriptions of populations of the Verhulst and Ollie type. Suitable initial approximations for a rapidly converging iterative process are proposed. Based on a self-similar analysis and comparison of the solutions of the Cauchy problem in the domain for an …

Swirling Fluid Flow In Flexible, Expandable Elastic Tubes: Variational Approach, Reductions And Integrability, Rossen Ivanov, Vakhtang Putkaradze

Articles

Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In real-life applications like blood flow, a swirl in the fluid often plays an important role, presenting an additional complexity not described by previous theoretical models. We present a theory for the dynamics of the interaction between elastic tubes and swirling fluid flow. The equations are derived using a variational principle, with the incompressibility constraint of the fluid giving rise to a pressure-like term. In order to connect this work with the previous literature, we consider the case of inextensible and …

On The Intermediate Long Wave Propagation For Internal Waves In The Presence Of Currents, Joseph Cullen, Rossen Ivanov

Articles

A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface approximation is assumed. The fluids are incompressible and inviscid. The model equations are obtained from the Hamiltonian formulation of the dynamics in the presence of a depth-varying current. It is shown that an appropriate scaling leads to the integrable Intermediate Long Wave Equation (ILWE). Two limits of the ILWE leading to the integrable Benjamin-Ono and KdV equations are …

Camassa-Holm Cuspons, Solitons And Their Interactions Via The Dressing Method, Rossen Ivanov, Tony Lyons, Nigel Orr

Articles

A dressing method is applied to a matrix Lax pair for the Camassa–Holm equation, thereby allowing for the construction of several global solutions of the system. In particular, solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa–Holm equation are re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.

Global Existence And Asymptotic Behavior Of The Solutions To Models For Chemotaxis Systems With Chemo Attractants And Repellents, Aesha Lagha

Graduate Theses, Dissertations, and Problem Reports (ETD)

We study global existence and asymptotic behavior of the solutions to models for chemotaxis systems with chemo attractants and repellents in three dimensions. Chemo attractants and repellents may be called chemo agents. For Part I, we use the logistic model for the mass. The interactions between chemo agents and the mass are taken into account. For Part II, we consider the case when mass is conserved and we use the Lotka-Volterra type model for chemo agents. To accomplish this, we use the Fourier transform and energy method. We show the existence of global solutions by the energy method. Also, we …

Higher Accuracy Methods For Fluid Flows In Various Applications: Theory And Implementation, Dilek Erkmen

Dissertations, Master's Theses and Master's Reports

This dissertation contains research on several topics related to Defect-deferred correction (DDC) method applying to CFD problems. First, we want to improve the error due to temporal discretization for the problem of two convection dominated convection-diffusion problems, coupled across a joint interface. This serves as a step towards investigating an atmosphere-ocean coupling problem with the interface condition that allows for the exchange of energies between the domains.

The main diffuculty is to decouple the problem in an unconditionally stable way for using legacy code for subdomains. To overcome the issue, we apply the Deferred Correction (DC) method. The DC method …

Analysis On Sharp And Smooth Interface, Elizabeth V. Hawkins

College of Graduate Studies: Theses & Dissertations

In biology, minimizing a free energy functional gives an equilibrium shape that is the most stable in nature. The formulation of these functionals can vary in many ways, in particular they can have either a smooth or sharp interface. Minimizing a functional can be done through variational calculus or can be proved to exist using various analysis techniques. The functionals investigated here have a smooth and sharp interface and are analyzed using analysis and variational calculus respectively. From the former we find the condition for extremum and its second variation. The second variation is commonly used to analyze stability of …

The Analysis Of Neural Heterogeneity Through Mathematical And Statistical Methods, Kyle Wendling

Theses and Dissertations

Diversity of intrinsic neural attributes and network connections is known to exist in many areas of the brain and is thought to significantly affect neural coding. Recent theoretical and experimental work has argued that in uncoupled networks, coding is most accurate at intermediate levels of heterogeneity. I explore this phenomenon through two distinct approaches: a theoretical mathematical modeling approach and a data-driven statistical modeling approach.

Through the mathematical approach, I examine firing rate heterogeneity in a feedforward network of stochastic neural oscillators utilizing a high-dimensional model. The firing rate heterogeneity stems from two sources: intrinsic (different individual cells) and network …

Image Restoration Using Automatic Damaged Regions Detection And Machine Learning-Based Inpainting Technique, Chloe Martin-King

Computational and Data Sciences (PhD) Dissertations

In this dissertation we propose two novel image restoration schemes. The first pertains to automatic detection of damaged regions in old photographs and digital images of cracked paintings. In cases when inpainting mask generation cannot be completely automatic, our detection algorithm facilitates precise mask creation, particularly useful for images containing damage that is tedious to annotate or difficult to geometrically define. The main contribution of this dissertation is the development and utilization of a new inpainting technique, region hiding, to repair a single image by training a convolutional neural network on various transformations of that image. Region hiding is also …

Determinism Of Stochastic Processes Through The Relationship Between The Heat Equation And Random Walks, Gurmehar Singh Makker

Publications and Research

We study the deterministic characteristics of stochastic processes through investigation of random walks and the heat equation. The relationship is confirmed by discretizing the heat equation in time and space and determining the probability distribution function for random walks in dimension d = 1, 2. The existence of the relationship is presented both through theoretical analysis and numerical computation.

On A Hybrid Technique To Handle Analytical And Approximate Solutions Of Linear And Nonlinear Fractional Order Partial Differential Equations, Kamal Shah, Hammad Khalil, Ahmet Yildirim

Applications and Applied Mathematics: An International Journal (AAM)

This manuscript is devoted to consider Natural transform (NT) coupled with homotopy perturbation method (HPM) for obtaining series solutions to some linear and nonlinear fractional partial differential equations (FPDEs). By means of NT, we obtain the transformed problem which is then solved by using HPM. By means of Stehfest’s numerical algorithm and using the dual relationship of NT and Laplace transform, we calculate inverse NT for approximate solutions. The series solutions we obtain using the proposed method are in close agreement with the exact solutions. We apply the proposed method to some interesting problems to illustrate our main results.