Applied Mathematics Commons^™

Remarks On The Stability Of Some Size-Structured Population Models Iii: The Case Of Constant Inflow Of Newborns, Mohammed El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

The stability of some size-structured population dynamics models are investigated. We determine the steady states and study their stability. We also give examples that illustrate the stability results. The results in this paper generalize previous results, for example, see Calsina, et al. (2003), El- Doma (2006) and El-Doma (2008).

Remarks On The Stability Of Some Size-Structured Population Models Ii: Changes In Vital Rates Due To Size And Population Size, Mohammed El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

The stability of some size-structured population dynamics models are investigated. We determine the steady states and study their stability. We also give examples that illustrate the stability results. The results in this paper generalize previous results, for example, see Calsina, et al. (2003) and El-Doma (2006).

Oscillatory Behavior Of Second Order Neutral Differential Equations With Positive And Negative Coefficients, Jelena Manojlović, Yutaka Shoukaku, Tomoyuki Tanigawa, Norio Yoshida

Applications and Applied Mathematics: An International Journal (AAM)

Oscillation criteria are obtained for solutions of forced and unforced second order neutral differential equations with positive and negative coefficients. These criteria generalize those of Manojlović, Shoukaku, Tanigawa and Yoshida (2006).

Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner

Mathematics Faculty Publications

The surface evolution model based on geometric partial differential equation is used to numerically study the kinetics of dewetting and dynamic morphologies for the localized pinhole defect in the surface of the ultrathin solid film with the strongly anisotropic surface energy. Depending on parameters such as the initial depth and width of the pinole, the strength of the attractive substrate potential and the strength of the surface energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes phase separation into a …

Quenching For Degenerate Semilinear Parabolic Problems With Insulated Boundary Conditions, Bernard Iyawe

Theses Digitization Project

This thesis studied the existence, uniqueness, and quenching behavior of the solution to a degenerate equation subject to the initial condition and the second boundary conditions.

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Mikhail Khenner

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner

Mathematics Faculty Publications

The surface evolution model based on geometric partial differential equation is used to numerically study the kinetics of dewetting and dynamic morphologies for the localized pinhole defect in the surface of the ultrathin solid film with the strongly anisotropic surface energy. Depending on parameters such as the initial depth and width of the pinole, the strength of the attractive substrate potential and the strength of the surface energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes phase separation into a …

Stochastic Differential Equations With Jumps, Jose L. Menaldi

Mathematics Faculty Research Publications

Part I Stochastic Processes with Jumps Chapters: Probability Spaces, Semigroup Theory - Part II Stochastic Differential Equations with Jumps Chapters: Stochastic Calculus, Stochastic Differential Equations - Part III Reflected SDE with Jumps Chapters: Stochastic Differential Equations II, Stochastic Differential Equations III.

Comment: This is last version from 2014-01-07. *This Initial version 15/May/2008 was corrected and augmented to produce the others 5 volumes.

Nearly-Hamiltonian Structure For Water Waves With Constant Vorticity, Adrian Constantin, Rossen Ivanov, Emil Prodanov

Articles

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.

Optimising Design Parameters Of Enzyme-Channelling Biosensors, Dana Mackey, Tony Killard

Conference Papers

Two mathematical models for an electrochemical biosensor are proposed and compared with a view to determining the ratio of two immobilized enzymes which maximizes the amperometric signal amplitude.

Models Of Phototransduction In Rod Photoreceptors, Harihar Khanal, Vasilios Alexiades

Publications

Phototransduction is the process by which photons of light generate an electrical response in retinal rod and cone photoreceptors, thereby initiating vision. We compare the electrical response in salamander rods from increasingly more (spacialy) detailed models of phototransduction: 0-dimensional (bulk), 1-dimensional (longitudinal), 2-dimensional (axisymmetric), and 3-dimensional (with incisures). We discuss issues of finding physical parameters for simulation and validation of models, and also present some computational experiments for rods with geometry of mouse and human photoreceptors.

On An Integrable Two-Component Camassa-Holm Shallow Water System, Adrian Constantin, Rossen Ivanov

Articles

The interest in the Camassa-Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this letter we deal with such a twocomponent integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system.

Algebraic Discretization Of The Camassa-Holm And Hunter-Saxton Equations, Rossen Ivanov

Articles

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H1 and H.1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinitedimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The CH and HS equations are integrable bi-hamiltonian equations and one of their Hamiltonian structures is associated to the …

Age-Structured Population Model With Cannibalism, Mmohammed El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

An age-structured population model with cannibalism is investigated. We determine the steady states and study the local asymptotic stability as well as the global stability. The results in this paper generalize previous results.

Stability Analysis For The Gurtin-Maccamy’S Age-Structured Population Dynamics Model, Mohammed El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

The stability of the Gurtin-MacCamy’s age-structured population dynamics model is investigated. We determine the steady states and study their stability. The results in this paper generalize previous results.

The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang

Articles and Preprints

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating …

On A Class Of Backward Mckean-Vlasov Stochastic Equations In Hilbert Space: Existence And Convergence Properties, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

This investigation is devoted to the study of a class of abstract first-order backward McKean-Vlasov stochastic evolution equations in a Hilbert space. Results concerning the existence and uniqueness of solutions and the convergence of an approximating sequence of solutions (and corresponding probability measures) are established. Examples that illustrate the abstract theory are also provided.

On Backward Stochastic Evolution Equations In Hilbert Space And Optimal Control, Nazim I. Mahmudov, Mark A. Mckibben

Mathematics Faculty Publications

In this paper a new result on the existence and uniqueness of the adapted solution to a backward stochastic evolution equation in Hilbert spaces under non Lipschitz condition is established. The applicability of this result is then illustrated in a discussion of some concrete backward stochastic partial differential equation. Furthermore, stochastic maximum principle for optimal control problems of stochastic systems governed by backward stochastic evolution equations in Hilbert spaces is obtained.

Numerical Simulation Of Waves And Fronts In Inhomogeneous Solids, A. Berezovski, M. Berezovski, J. Engelbrecht, G. A. Maugin

Publications

Dynamic response of inhomogeneous materials exhibits new effects, which often do not exist in homogeneous media. It is quite natural that most of studies of wave and front propagation in inhomogeneous materials are associated with numerical simulations. To develop a numerical algorithm and to perform the numerical simulations of moving fronts we need to formulate a kinetic law of progress relating the driving force and the velocity of the discontinuity. The velocity of discontinuity is determined by means of the non-equilibrium jump relations at the front. The obtained numerical method generalizes the wave-propagation algorithm to the case of moving discontinuities …

Global Stability Results Of An Sis Age-Structured Epidemic Model With Vertical Transmission, M. El-Doma

Applications and Applied Mathematics: An International Journal (AAM)

An SIS age-structured epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility, mortality and cure rates depend on age and the force of infection of proportionate mixing assumption type. We determine the steady states and prove the global stability for the endemic equilibriums.

Parameter Estimation In Nonlinear Coupled Advection-Diffusion Equation, Robert R. Ferdinand

Applications and Applied Mathematics: An International Journal (AAM)

In this paper a coupled system of two nonlinear advection-diffusion equations is presented. Such systems of equations have been used in mathematical literature to describe the dynamics of contaminant present in groundwater flowing through cracks in a porous rock matrix and getting absorbed into it. An inverse method procedure that approximates infinite-dimensional model parameters is described and convergence results for the parameter approximants are proved. This is finally followed by a computational experiment to compare theoretical and numerical results to verify accuracy of the mathematics analysis presented.

Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne

Mathematical Sciences Technical Reports (MSTR)

Given a two-dimensional region that contains one or more circular voids, we develop mathematical methods to locate the center and radius of the voids based on thermal boundary data. These methods can be readily applied in the field of non-destructive evaluation.

Large Solutions Of Semilinear Elliptic Equations Under The Keller–Osserman Condition, Alan V. Lair

Faculty Publications

Excerpt: We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.

Mathematical Methods In Composing Melodies, Thomas Brown

Undergraduate Theses and Capstone Projects

This thesis, “Mathematical Methods in Composing Melodies,” explores the different ways in which mathematics can be used to create music. Some research has been done in this field already. Richard F. Voss and John Clarke used fractals and different frequencies of noise to create music. The Greek composer Iannis Xenakis used Markovian Stochastic trees to create some of his compositions. Explored in this thesis are seven different methods to compose melodies. After compiling the different melodies, they were categorized by different musical periods based on the musical characteristics found in the melody. This thesis differs from other research that deals …

Turing Patterns On Growing Spheres: The Exponential Case, Julijana Gjorgjieva, Jon T. Jacobsen

All HMC Faculty Publications and Research

We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations.

Conformal And Geometric Properties Of The Camassa-Holm Hierarchy, Rossen Ivanov

Articles

Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this ontribution. The squared eigenfunctions of the spectral problem, associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform (IST) for the Camassa-Holm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions …

Hamiltonian Formulation, Nonintegrability And Local Bifurcations For The Ostrovsky Equation, S. Roy Choudhury, Rossen Ivanov, Yue Liu

Articles

The Ostrovsky equation is a model for gravity waves propagating down a channel under the influence of Coriolis force. This equation is a modification of the famous Korteweg-de Vries equation and is also Hamiltonian. However the Ostrovsky equation is not integrable and in this contribution we prove its nonintegrability. We also study local bifurcations of its solitary waves.

Water Waves And Integrability, Rossen Ivanov

Articles

The Euler’s equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymtotic expansion of the Euler’s equations is taken (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modeling the motion of shallow water waves are reviewed in this contribution.