Partial Differential Equations Commons^™

Analysis And Implementation Of Numerical Methods For Solving Ordinary Differential Equations, Muhammad Sohel Rana

Masters Theses & Specialist Projects

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It …

Evolution Of Superoscillations For Schrödinger Equation In A Uniform Magnetic Field, Fabrizio Colombo, Jonathan Gantner, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, in optics, and in signal processing. An important issue is the study of the evolution of superoscillations using the Schrödinger equation when the initial datum is a weak value. Some superoscillatory functions are not square integrable, but they are real analytic functions that can be extended to entire holomorphic functions. This fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of …

A Regression Model To Predict Stock Market Mega Movements And/Or Volatility Using Both Macroeconomic Indicators & Fed Bank Variables, Timothy A. Smith, Alcuin Rajan

Publications

In finance, regression models or time series moving averages can be used to determine the value of an asset based on its underlying traits. In prior work we built a regression model to predict the value of the S&P 500 based on macroeconomic indicators such as gross domestic product, money supply, produce price and consumer price indices. In this present work this model is updated both with more data and an adjustment in the input variables to improve the coefficient of determination. A scheme is also laid out to alternately define volatility rather than using common tools such as the …

Interplay Of Quantum Size Effect, Anisotropy And Surface Stress Shapes The Instability Of Thin Metal Films, Mikhail Khenner

Mathematics Faculty Publications

Morphological instability of a planar surface ([111], [011], or [001]) of an ultra-thin metal film is studied in a parameter space formed by three major effects (the quantum size effect, the surface energy anisotropy and the surface stress) that influence a film dewetting. The analysis is based on the extended Mullins equation, where the effects are cast as functions of the film thickness. The formulation of the quantum size effect (Z. Zhang et al., PRL 80, 5381 (1998)) includes the oscillation of the surface energy with thickness caused by electrons confinement. By systematically comparing the effects, their contributions into the …

Thermodynamics Of Coherent Structures Near Phase Transitions, Julia M. Meyer, Ivan Christov

The Summer Undergraduate Research Fellowship (SURF) Symposium

Phase transitions within large-scale systems may be modeled by nonlinear stochastic partial differential equations in which system dynamics are captured by appropriate potentials. Coherent structures in these systems evolve randomly through time; thus, statistical behavior of these fields is of greater interest than particular system realizations. The ability to simulate and predict phase transition behavior has many applications, from material behaviors (e.g., crystallographic phase transformations and coherent movement of granular materials) to traffic congestion. Past research focused on deriving solutions to the system probability density function (PDF), which is the ground-state wave function squared. Until recently, the extent to which …

Regularized Solutions For Terminal Problems Of Parabolic Equations., Sujeewa Indika Hapuarachchi

Electronic Theses and Dissertations

The heat equation with a terminal condition problem is not well-posed in the sense of Hadamard so regularization is needed. In general, partial differential equations (PDE) with terminal conditions are those in which the solution depends uniquely but not continuously on the given condition. In this dissertation, we explore how to find an approximation problem for a nonlinear heat equation which is well-posed. By using a small parameter, we construct an approximation problem and use a modified quasi-boundary value method to regularize a time dependent thermal conductivity heat equation and a quasi-boundary value method to regularize a space dependent thermal …

Some Problems Arising From Mathematical Model Of Ductal Carcinoma In Situ., Heng Li

Electronic Theses and Dissertations

Ductal carcinoma in situ (DCIS) is the earliest form of breast cancer. Three mathematical models in the one dimensional case arising from DCIS are proposed. The first two models are in the form of parabolic equation with initial and known moving boundaries. Direct and inverse problems are considered in model 1, existence and uniqueness are proved by using tool from heat potential theory and Volterra integral equations. Also, we discuss the direct problem and nonlocal problem of model 2, existence and uniqueness are proved. And approximation solution of these problems are implemented by Ritz-Galerkin method, which is the first attempt …

Eignefunctions For Partial Differential Equations On Two-Dimensional Domains With Piecewise Constant Coefficients, Abdullah Muheel Momit Aurko

Master's Theses

In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.

Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to …

Numerical Solution Of Partial Differential Equations Using Polynomial Particular Solutions, Thir R. Dangal

Dissertations

Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their …

Solution Of Pdes For First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods, Somayyeh Sheikholeslami

Dissertations

We solve the first order reaction-diffusion equations which describe binding-diffusion kinetics using a photobleaching scanning profile of a confocal laser scanning microscope approximated by a Gaussian laser profile. We show how to solve these equations with prebleach steady-state initial conditions using a time-domain method known as a Krylov Subspace Spectral (KSS) method. KSS methods are explicit methods for solving time- dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of their stability and their superior scalability. These advantages are obtained by applying Gaussian quadrature rules in the spectral domain developed by Golub and Meurant. …

Numerically Solving A System Of Pdes Modeling Chronic Wounds Treated With Oxygen Therapy, Stefan Stryker

Mahurin Honors College Capstone Experience/Thesis Projects

Chronic wounds such as diabetic foot ulcers are the leading cause of non-traumatic amputations in developed countries. For researchers to better understand the physiology of these wounds, a mathematical model describing oxygen levels at the wound site can be used to help predict healing responses. The model utilizes equations that are modified from work by Guffey (2015) that consists of four variables – oxygen, bacteria, neutrophils, and chemoattractant within a system of partial differential equations. Our research focuses on numerically solving these partial differential equations using a finite volume approach. This numerical solver will be important for future research in …

Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of …

Finite Difference Schemes For Variable Order Time-Fractional First Initial Boundary Value Problems, Gunvant A. Birajdar, M. M. Rashidi

Applications and Applied Mathematics: An International Journal (AAM)

The aim of the study is to obtain the numerical solution of first initial boundary value problem (IBVP) for semi-linear variable order fractional diffusion equation by using different finite difference schemes. We developed the three finite difference schemes namely explicit difference scheme, implicit difference scheme and Crank-Nicolson difference scheme, respectively for variable order type semi-linear diffusion equation. For this scheme the stability as well as convergence are studied via Fourier method. At the end, solution of some numerical examples are discussed and represented graphically using Matlab.

Application Of Kudryashov Method For The Ito Equations, Mozhgan Akbari

Applications and Applied Mathematics: An International Journal (AAM)

In this present work, the Kudryashov method is used to construct exact solutions of the (1+1)- dimensional and the (1+2)-dimensional form of the generalized Ito integro-differential equation. The Kudryashov method is a powerful method for obtaining exact solutions of nonlinear evolution equations. This method can be applied to non-integrable equations as well as integrable ones.

Asymptotic Behavior Of Waves In A Nonuniform Medium, Nezam Iraniparast, Lan Nguyen, Mikhail Khenner

Applications and Applied Mathematics: An International Journal (AAM)

An incoming wave on an infinite string, that has uniform density except for one or two jump discontinuities, splits into transmitted and reflected waves. These waves can explicitly be described in terms of the incoming wave with changes in the amplitude and speed. But when a string or membrane has continuous inhomogeneity in a finite region the waves can only be approximated or described asymptotically. Here, we study the cases of monochromatic waves along a nonuniform density string and plane waves along a membrane with nonuniform density. In both cases the speed of the physical system is assumed to tend …

Inverse Problem For A Parabolic System, Reza Pourgholi, Amin Esfahani, Hassan D. Mazraeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper a numerical approach combining the least squares method and a genetic algorithm is proposed for the determination of the source term in an inverse parabolic system (IPS). A numerical experiment confirm the utility of this algorithm as the results are in good agreement with the exact data. Results show that a reasonable estimation can be obtained by the genetic algorithm within a CPU with clock speed 2.7 GHz.

Effect Of Nonlinear Thermal Radiation On Mhd Chemically Reacting Maxwell Fluid Flow Past A Linearly Stretching Sheet, A. M. Ramireddy, J. V. Ramana Reddy, N. Sandeep, V. Sugunamma

Applications and Applied Mathematics: An International Journal (AAM)

This communication addresses the influence of nonlinear thermal radiation on magneto hydrodynamic Maxwell fluid flow past a linearly stretching surface with heat and mass transfer. The effects of heat generation/absorption and chemical reaction are taken into account. At first, we converted the governing partial differential equations into nonlinear ordinary differential equations with the help of suitable similarity transformations and solved by using Runge-Kutta based shooting technique. Further, the effects of various physical parameters on velocity, temperature and concentration fields were discussed thoroughly with the help of graphs obtained by using bvp5c MATLAB package. In view of many engineering applications we …

New Structure For Exact Solutions Of Nonlinear Time Fractional Sharma-Tasso-Olver Equation Via Conformable Fractional Derivative, Hadi Rezazadeh, Farid S. Khodadad, Jalil Manafian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper new fractional derivative and direct algebraic method are used to construct exact solutions of the nonlinear time fractional Sharma-Tasso-Olver equation. As a result, three families of exact analytical solutions are obtained. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.

High Performance Computation Of Cardiac Models In Real-Time Using Webgl, Abouzar Kaboudian, Flavio H. Fenton

Biology and Medicine Through Mathematics Conference

No abstract provided.

A Large Reaction-Diffusion Model For Cell Polarization In Yeast, Marissa Renardy

Biology and Medicine Through Mathematics Conference

No abstract provided.