Applied Mathematics Commons^™

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.

Application Of Reduced Differential Transform Method For Solving Two-Dimensional Volterra Integral Equations Of The Second Kind, Seyyedeh R. Moosavi Noori, Nasir Taghizadeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we propose new theorems of the reduced differential transform method (RDTM) for solving a class of two-dimensional linear and nonlinear Volterra integral equations (VIEs) of the second kind. The advantage of this method is its simplicity in using. It solves the equations straightforward and directly without using perturbation, Adomian’s polynomial, linearization or any other transformation and gives the solution as convergent power series with simply determinable components. Also, six examples and numerical results are provided so as to validate the reliability and efficiency of the method.

A New Method To Solve Fractional Differential Equations: Inverse Fractional Shehu Transform Method, Ali Khalouta, Abdelouahab Kadem

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we propose a new method called the inverse fractional Shehu transform method to solve homogenous and non-homogenous linear fractional differential equations. Fractional derivatives are described in the sense of Riemann-Liouville and Caputo. Illustrative examples are given to demonstrate the validity, efficiency and applicability of the presented method. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature.

Numerical Solution Of Fractional Partial Differential Equations With Normalized Bernstein Wavelet Method, Mahsa Entezari, Saeid Abbasbandy, Esmail Babolian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, normalized Bernstein wavelets are presented. Next, the fractional order integration and Bernstein wavelets operational matrices of integration are derived and finally are used for solving fractional partial differential equations. The operational matrices merged with the collocation method are used in order to convert fractional problems to a number of algebraic equations. In the suggested method the boundary conditions are automatically taken into consideration. An assessment of the error of function approximation based on the normalized Bernstein wavelet is also presented. Some numerical instances are given to manifest the versatility and applicability of the suggested method. Founded numerical …

Multi-Point Flux Approximations Via The O-Method, Christen Leggett

Master's Theses

When an oil refining company is drilling for oil, much of the oil gets left behind after the first drilling. Enhanced oil recovery techniques can be used to recover more of that oil, but these methods are quite expensive. When a company is deciding if it is worth their time and money to use enhanced oil recovery methods, simulations can be used to model oil flow, showing the behavior and location of the oil. While methods do exist to model this flow, these methods are often very slow and inaccurate due to a large domain and wide variance in coefficients. …

Blow Up Of Solutions For A Coupled Kirchhoff-Type Equations With Degenerate Damping Terms, Erhan Piskin, Fatma Ekinci

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we investigate a system of coupled Kirchhoff-type equations with degenerate damping terms. We prove a nonexistence of global solutions with positive initial energy. Later, we give some estimates for lower bound of the blow up time.

Local Non-Similar Solution Of Powell-Eyring Fluid Flow Over A Vertical Flat Plate, Hemangini Shukla, Hema C. Surati, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

Our objective is to obtain the non-similarity solution of non-Newtonian fluid for Powell-Eyring model by a local non-similarity method. Here, free stream velocity is considered in power-law form (𝑈=𝑥^m). The governing equations are transformed using non-similar transformations and derived equations are treated as ordinary differential equations. Non-similar solutions are obtained for different values of power-law index 𝑚 and stream-wise location 𝜉. Influence of various parameters on velocity and temperature field are presented graphically using MATLAB bvp4c solver.

Analyzing Student Loan Debt Using Seir Compartmental Model Of Epidemiology, Kavya Ravishankar, Dr. Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Sperm Motility In Groups, Julie Simons

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Multidisciplinary Education And Research In Biomathematics For Solving Global Challenges, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Parallel Multipole Expansion Algorithms And Their Biology Applications, Jiahui Chen

Mathematics Theses and Dissertations

N-body pairwise interactions are ubiquitous in scientific areas such as astrophysics, fluids mechanics, electrical engineering, molecular biology, etc. Computing these interactions using direct sum of an O(N) cost is expensive, whereas multipole expansion methods, such as the fast multipole method (FMM) or treecode, can reduce the cost to O(N) or O(N log N). This thesis focuses on developing numerical algorithms of Cartesian FMM and treecode, as well as using these algorithms to directly or implicitly solve biological problems involving pairwise interactions. This thesis consists of the following topics. 1) A cyclic parallel scheme is developed to handle the load balancing …

Smoothness Of Defining Functions And The Diederich-Fornæss Index, Felita Nadia Humes

Graduate Theses and Dissertations

Let Ω ⊂ Cn be a smooth, bounded, pseudoconvex domain, and let M ⊂ ∂Ω be a complex submanifold with rectifiable boundary. In 2017, Harrington studied the equation dM A = α ̃ on M, where α ̃ is D’Angelo’s 1-form and A is real. In this thesis, we will study a non-pseudoconvex example in which M has a non-rectifiable boundary. In spite of the lack of topological obstructions on the boundary, there are no continuous solutions to dM A = α ̃.

Krylov Subspace Spectral Methods With Non-Homogenous Boundary Conditions, Abbie Hendley

Master's Theses

For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent …

Radial Basis Function Finite Difference Approximations Of The Laplace-Beltrami Operator, Sage Byron Shaw

Boise State University Theses and Dissertations

Partial differential equations (PDEs) are used throughout science and engineering for modeling various phenomena. Solutions to PDEs cannot generally be represented analytically, and therefore must be approximated using numerical techniques; this is especially true for geometrically complex domains. Radial basis function generated finite differences (RBF-FD) is a recently developed mesh-free method for numerically solving PDEs that is robust, accurate, computationally efficient, and geometrically flexible. In the past seven years, RBF-FD methods have been developed for solving PDEs on surfaces, which have applications in biology, chemistry, geophysics, and computer graphics. These methods are advantageous, as they are mesh-free, operate on arbitrary …

Design Of Metamaterials For Optics, Abiti Adili

LSU Doctoral Dissertations

First part of this dissertation studies the problem of designing metamaterial crystals with double negative effective properties for applications in optics by investigating the conditions necessary for generating novel dispersion properties in a metamaterial crystal with subwavelength microstructure. This provides novel optical properties created through local resonances tied to the geometry of the media in subwavelength regime.

In the second part, this dissertation studies the representation formula used to describe band structures in photonic crystals with plasmonic inclusions. By using layer potential techniques, a magnetic dipole operator describing the tangential component of the electrical field generated by magnetic distribution is …

Two Dimensional Mathematical Study Of Flow Dynamics Through Emphysemic Lung, Jyoti Kori, Pratibha _

Applications and Applied Mathematics: An International Journal (AAM)

There are various lung diseases, such as chronic obstructive pulmonary disease, asthma, fibrosis, emphysema etc., occurred due to deposition of different shape and size particles. Among them we focused on flow dynamics of viscous air through an emphysemic lung. We considered lung as a porous medium and porosity is a function of tidal volume. Two dimensional generalized equation of momentum is used to study the flow of air and equation of motion is used to study the flow of nanoparticles of elongated shape. Darcy term for flow in porous media and shape factor for nonspherical nanoparticles are used in mathematical …

Solitary And Periodic Exact Solutions Of The Viscosity-Capillarity Van Der Waals Gas Equations, Emad A. Az-Zo’Bi

Applications and Applied Mathematics: An International Journal (AAM)

Periodic and soliton solutions are derived for the (1+1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. The system is handled via the e-^φ(ξ) -expansion method. The obtained solutions have been articulated by the hyperbolic, trigonometric, exponential and rational functions with arbitrary constants. Mathematical analysis and numerical graphs are provided for some solitons, periodic and kink solitary wave solutions to visualize the dynamics of equations. Obtained results reveal that the method is very influential and effective tool for solving nonlinear partial differential equations in applied mathematics.

Laplace Adomian Decomposition Method To Study Chemical Ion Transport Through Soil, Poonam Soni, Arun Kumar, A. Rani

Applications and Applied Mathematics: An International Journal (AAM)

The paper deals with a theoretical study of chemical ion transport in soil under a uniform external force in the transverse direction, where the soil is taken as porous medium. The problem is formulated in terms of boundary value problem that consists of a set of partial differential equations, which is subsequently converted to a system of ordinary differential equations by applying similarity transformation along with boundary layer approximation. The equations hence obtained are solved by utilizing Laplace Adomian Decomposition Method (LADM). The merit of this method lies in the fact that much of simplifying assumptions need not be made …

Transient Thermal Stresses Due To Axisymmetric Heat Supply In A Semi-Infinite Thick Circular Plate, S. D. Warbhe, K. C. Deshmukh

Applications and Applied Mathematics: An International Journal (AAM)

The present paper deals with the determination of thermal stresses in a semi-infinite thick circular plate of a finite length and infinite extent subjected to an axisymmetric heat supply. A thick circular plate is considered having constant initial temperature and arbitrary heat flux is applied on the upper and lower face. The governing heat conduction equation has been solved by using integral transform technique. The results are obtained in terms of Bessel’s function. The thermoelastic behavior has been computed numerically and illustrated graphically for a steel plate.

Qualitative Analysis Of A Modified Leslie-Gower Predator-Prey Model With Weak Allee Effect Ii, Manoj K. Singh, B. S. Bhadauria

Applications and Applied Mathematics: An International Journal (AAM)

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for …