Applied Mathematics Commons^™

New Exact Solutions Of The Perturbed Nonlinear Fractional Schr¨Odinger Equation Using Two Reliable Methods, Nasir Taghizadeh, Mona N. Foumani, Vahid S. Mohammadi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the fractional derivatives in the sense of the modified Riemann-Liouville derivative and the first integral method and the Bernoulli sub-ODE method are employed for constructing the exact complex solutions of the perturbed nonlinear fractional Schr ¨odinger equation and comparing the solutions.

A Hybrid Variational Iteration Method For Blasius Equation, M. Sajid, N. Ali, T. Javed

Applications and Applied Mathematics: An International Journal (AAM)

The objective of this paper is to present the hybrid variational iteration method. The proposed algorithm is based on the combination of variational iteration and shooting methods. In the proposed algorithm the entire domain is divided into subintervals to establish the accuracy and convergence of the approximate solution. It is found that in each subinterval a three term approximate solution using variational iteration method is sufficient. The proposed hybrid variational iteration method offers not only numerical values, but also closed form analytic solutions in each subinterval. The method is implemented using an example of the Blasius equation. The results show …

Implicit-Explicit Higher-Order Time Integration Schemes For Computations Of Structural Dynamics With Fluid-Structure Interaction, José C. Pedro, Mapundi K. Banda, Precious Sibanda

Applications and Applied Mathematics: An International Journal (AAM)

In this paper higher order implicit Runge-Kutta schemes are applied to fluid-structure interaction (FSI) simulations. A staggered approach with a structural predictor is applied to an FSI problem. The equations governing the dynamics of the structure are integrated in time by the Explicit Single Diagonal Implicit Runge-Kutta (ESDIRK) schemes and the arbitrary high order finite volume scheme is taken as the fluid solver. The performance of the ESDIRK scheme of order of convergence three to five is tested. Comparative studies with other time integration schemes which have been successfully applied to FSI problems are undertaken. Comparisons to test the performance …

Hydrodynamic Analogues Of Hamiltonian Systems, Francisco J. Jauffred

Graduate Masters Theses

A one-dimensional Hamiltonian system can be modeled and understood as a two-dimensional incompressible fluid in phase space. In this sense, the chaotic behavior of one-dimensional time dependent Hamiltonians corresponds to the mixing of two-dimensional fluids. Amey (2012) studied the characteristic values of one such system and found a scaling law governing them. We explain this scaling law as a diffusion process occurring in an elliptical region with very low eccentricity. We prove that for such a scaling law to occur, it is necessary for a vorticity field to be present. Furthermore, we show that a conformal mapping of an incompressible …

Unsteady Boundary Layer Flow Of Thermophoretic Mhd Nanofluid Past A Stretching Sheet With Space And Time Dependent Internal Heat Source/Sink, N. Sandeep, C. Sulochana, C. S. K. Raju, M. J. Babu, V. Sugunamma

Applications and Applied Mathematics: An International Journal (AAM)

In this study we analyze the boundary layer flow of a thermophoretic magnetohydrodynamic dissipative nanofluid over an unsteady stretching sheet in a porous medium with space and time dependent internal heat source/sink. The governing equations are transformed to ordinary differential equations by using similarity transformation. Numerical solutions of these equations are obtained by using the Shooting Technique. The effects of non-dimensional governing parameters on the velocity, temperature, concentration profiles, friction factor, Nusselt and Sherwood numbers are discussed and presented through graphs and tables. Accuracy of the results compared with the existing ones. Excellent agreement is found with earlier studies.

Transition Orbits Of Walking Droplets, Joshua Parker

Physics

It was recently discovered that millimeter-sized droplets bouncing on the surface of an oscillating bath of the same fluid can couple with the surface waves it produces and begin walking across the fluid bath. These walkers have been shown to behave similarly to quantum particles; a few examples include single-particle diffraction, tunneling, and quantized orbits. Such behavior occurs because the drop and surface waves depend on each other to exist, making this the first and only known macroscopic pilot-wave system. In this paper, the quantized orbits between two identical drops are explored. By sending a perturbation to a pair of …

Efficient General Computational Method For Estimation Of Standard Atmosphere Parameters, Nihad E. Daidzic Ph.D., Sc.D.

International Journal of Aviation, Aeronautics, and Aerospace

Knowledge of standard air temperature, pressure, density, speed of sound, and viscosity as a function of altitude is essential information in aircraft design, performance testing, pressure altimeter calibration, and several other aeronautical engineering and aviation science applications. A new efficient computational method for rapid calculations of standard atmospheric parameters up to 86 orthometric km is presented. Additionally, mass and weight of each standard atmospheric layer were calculated using a numerical integration method. The sum of all fractional masses and weights represents the total mass and weight of Earth’s atmosphere. The results obtained here agree well with measurements and models of …

A Contribution Toward Better Understanding Of Overbanking Tendency In Fixed-Wing Aircraft, Nihad E. Daidzic

Journal of Aviation Technology and Engineering

The phenomenon of overbanking tendency for a rigid-body, fixed-wing aircraft is investigated. Overbanking tendency is defined as a spontaneous, unbalanced rolling moment that keeps increasing an airplane’s bank angle in steep turns and must be arrested by opposite aileron action. As stated by the Federal Aviation Administration, the overbanking tendency may lead to a loss of control, especially in instrument meteorological conditions. It was found in this study that the speed differential over wing halves in horizontal turns indeed creates a rolling moment that achieves maximum values for bank angles between 45 and 55 degrees. However, this induced rolling moment …

Four Tails Problems For Dynamical Collapse Theories, Kelvin J. Mcqueen

Philosophy Faculty Articles and Research

The primary quantum mechanical equation of motion entails that measurements typically do not have determinate outcomes, but result in superpositions of all possible outcomes. Dynamical collapse theories (e.g. GRW) supplement this equation with a stochastic Gaussian collapse function, intended to collapse the superposition of outcomes into one outcome. But the Gaussian collapses are imperfect in a way that leaves the superpositions intact. This is the tails problem. There are several ways of making this problem more precise. But many authors dismiss the problem without considering the more severe formulations. Here I distinguish four distinct tails problems. The first (bare tails …

Spin Glass Reflection Of The Decoding Transition For Quantum Error Correcting Codes, Alexey Kovalev, Leonid P. Pryadko

Department of Physics and Astronomy: Faculty Publications

We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette Z₂ gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of "post-topological" extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with …

Conditions For Deterministic Limits Of Markov Jump Processes : The Kurtz Theorem In Chemistry, Ada Anna Sedova

Legacy Theses & Dissertations (2009 - 2024)

A theorem by Kurtz on convergence of Markov jump processes is presented as it relates

Periodic State Revivals In Commensurate Waveguide Arrays, Jovan Petrovic, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Emerging optical and quantum computers require hardware capable of coherent transport of and operations on quantum states. Here, we investigate finite optical waveguide arrays with linear coupling as means of efficient and compact coherent state transfer. Coherent transfer with periodic state revivals is enabled by engineering coupling coefficients between neighbouring waveguides to yield commensurate eigenvalue spectrum. Particular cases of finite arrays have been actively studied to achieve the perfect state transfer by mirroring the input into the output state.

We explore a much wider scope of coherent propagation and revivals of both the state amplitude and phase. We analytically solve …

Rainich-Type Conditions For Perfect Fluid Spacetimes, Dionisios Krongos, Charles G. Torre

Research Vignettes

In this worksheet we describe and illustrate a relatively simple set of new Rainich-type conditions on an n-dimensional spacetime which are necessary and sufficient for it to define a perfect fluid solution of the Einstein field equations. Procedures are provided which implement these Rainich-type conditions and which reconstruct the perfect fluid from the metric. These results provide an example of the idea of geometrization of matter fields in general relativity, which is a purely geometrical characterization of matter fields via the Einstein field equations.

Scaling Group Analysis On Mhd Free Convective Heat And Mass Transfer Over A Stretching Surface With Suction / Injection, Heat Source/Sink Considering Viscous Dissipation And Chemical Reaction Effects, Hunegnaw Dessie, Naikoti Kishan

Applications and Applied Mathematics: An International Journal (AAM)

This paper concerns with scaling group analysis on MHD free convective heat and mass transfer over stretching surface considering effects of thermal-diffusion and diffusion-thermo with suction /injection, heat source/sink and chemical reaction by taking into account viscous dissipation. Scaling group transformations are used to convert the partial differential equations of governing equations into ordinary differential equation and are solved numerically by Keller Box Method. Numerical results obtained for different parameters are drawn graphically and their effects on velocity, temperature and concentration profiles are discussed and shown graphically. Skin-friction coefficient, Nusselt number and Sherwood number are presented in table. It is …

The Investigation Of Surplus Of Energy And Signal Propagation At Time-Domain Waveguide Modes, Özlem Işık, Zeynep F. Koçak, Emre Eroğlu

Applications and Applied Mathematics: An International Journal (AAM)

Classical waveguide theory has been developed bearing on Bernoulli’s product method which results in separation of space and time variables in Maxwell’s equations. The time-harmonic waveguide modes have been stated mathematically for transmitting signals along the waveguides. As a starting point, present studies on transverse-electric (TE) and transverse-magnetic (TM) waveguide modes with previous results are taken and exhibited in an advanced form. They have been obtained within the framework of an evolutionary approach to solve Maxwell’s equations with time derivative. As a result every modal field is obtained in the form of a product of vector functions of transverse coordinates …

Centered-Difference Applications For Schrödinger's Equation, Matthew Thomas Murachver

Physics

This project enumerates methods utilizing discretized centered-difference approximations on the second order differential equation for quantum particles known as Schrodinger’s Equation. An eigenvalue-eigenfunction scheme is developed to sieve for valid solutions to The Time Independent Schrodinger Equation. Additionally the Crank-Nicolson method is applied to the Time Dependent Schrodinger Equation to describe wavefunction (eigenfunction) time evolution. The validity of these methods is discussed with applications to several fundamental pedagogical introductory quantum mechanic systems.

Isotropic Oscillator Under A Magnetic And Spatially Varying Electric Field, David L. Frost Mr., Frank Hagelberg

Undergraduate Honors Theses

We investigate the energy levels of a particle confined in the isotropic oscillator potential with a magnetic and spatially varying electric field. Here we are able to exactly solve the Schrodinger equation, using matrix methods, for the first excited states. To this end we find that the spatial gradient of the electric field acts as a magnetic field in certain circumstances. Here we present the changes in the energy levels as functions of the electric field, and other parameters.

One-Dimensional Weakly Nonlinear Model Equations For Rossby Waves, David Henry, Rossen Ivanov

Articles

In this study we explore several possibilities for modelling weakly nonlinear Rossby waves in fluid of constant depth, which propagate predominantly in one direction. The model equations obtained include the BBM equation, as well as the integrable KdV and Degasperis-Procesi equations.

Vector Spaces (Appendix B), Charles G. Torre

Foundations of Wave Phenomena

Throughout this text we have noted that various objects of interest form a vector space. Here we outline the basic structure of a vector space. You may find it useful to refer to this Appendix when you encounter this concept in the text.

Taylor’S Theorem And Taylor Series (Appendix A), Charles G. Torre

Foundations of Wave Phenomena

Taylor’s theorem and Taylor’s series constitute one of the more important tools used by mathematicians, physicists and engineers. They provides a means of approximating a function in terms of polynomials.

References And Suggestions For Further Reading (Appendix C), Charles G. Torre

Foundations of Wave Phenomena

References and Suggestions for Further Reading (Appendix C)

A Soft Condensed Matter Approach Towards Mathematical Modelling Of Mass Transport And Swelling In Food Grains, Michael Chapwanya, N. Misra

Articles

Soft condensed matter (SCM) physics has recently gained importance for a large class of engineering materials. The treatment of food materials from a soft matter perspective, however, is only at the surface and is gaining importance for understanding the complex phenomena and structure of foods. In this work, we present a theoretical treatment of navy beans from a SCM perspective to describe the hydration kinetics. We solve the transport equations within a porous matrix and employ the Flory–Huggin’s equation for polymer–solvent mixture to balance the osmotic pressure. The swelling of the legume seed is modelled as a moving boundary with …

Development And Applications Of The Expanded Equivalent Fluid Method, Bharath Kumar Kandula

Dissertations

Ocean acoustics is the study of sound in the oceans. Electromagnetic waves attenuate rapidly in the water medium. Sound is the best means to transmit information underwater. Computational numerical simulations play an important role in ocean acoustics. Simulations of acoustic propagation in the oceans are challenging due to the complexities involved in the ocean environment. Different methods have been developed to simulate underwater sound propagation. The Parabolic-Equation (PE) method is the best choice in several ocean acoustic problems. In shallow water acoustic experiments, sound loses some of its energy when it interacts with the bottom. An equivalent fluid technique was …

A Women In Mathematics, Computer Science, And Physics Course, Jim Crumley, Kristen Nairn, Lynn Ziegler, Pamela L. Bacon, Yu Zhang

MapCores Faculty Publications

Increasing women's participation is a concern in disciplines beyond
physics. As part of our Mathematics, Physics, Computer Science
Research Scholars (MapCores) program, we teach a women in science
class covering these three areas. Our course is a special version of
our college's first year seminar, which is a course designed to
prepare our students to read, write, and speak at a college-level. We
structure our FYS to promote academic confidence and interest in our
disciplines for the women in MapCores. It covers not only contributions
that women have made and barriers that women face in these
disciplines, but also research …

Variable Viscosity Condition In The Modeling Of A Slider Bearing, Kedar Nath Uprety, S.C. Mancas

Publications

To reduce tear and wear of machinery lubrication is essential. Lubricants form a layer between two surfaces preventing direct contact and reduce friction between moving parts and hence reduce wear. In this short letter the lubrication of two slider bearings with parallel and nonparallel is studied. First, we show that bearings with parallel plates cannot support any load. For bearings with nonparallel plates we are interested on how constant and temperature dependent viscosity affects the properties of the bearings. Also, a critical temperature for which the bearings would fail due to excess in temperature is found for both latter cases. …

Modelling The Flow Of Aqueous Humor In Schlemm’S Canal In The Eye, Ram Avtar, Swati Srivastava, Rashmi Srivastava

Applications and Applied Mathematics: An International Journal (AAM)

A simple mathematical model for the transient flow of aqueous humor in the canal of Schlemm is developed to investigate the acceleration effects of a sudden elevation in the intraocular pressure on the flow characteristics of the aqueous humor in the canal. The model treats a canal segment as a tube of elliptic cross-section. Exact analytical solution to the model is obtained using separation of variables method. The effects of some important model parameters on the maximum and minimum shear stresses exerted on the Schlemm’s canal epithelial cells (wall) by flowing aqueous humor are investigated for the steady-state flow.

Smarticles: A Method For Identifying And Correcting Instability And Error Caused By Explicit Integration Techniques In Physically Based Simulations, Susan Aileen Marano

Master's Theses

Using an explicit integration method in physically based animations has many advantages including conceptual and computational simplicity, however, it re- quires small time steps to ensure low numerical instability. Simulations with large numbers of individually interacting components such as cloth, hair, and fluid models, are limited by the sections of particles most susceptible to error. This results in the need for smaller time steps than required for the majority of the system. These sections can be diverse and dynamic, quickly changing in size and location based on forces in the system. Identifying and handling these trou- blesome sections could allow …

Analysis Of A Partial Differential Equation Model Of Surface Electromigration, Selahittin Cinar

Masters Theses & Specialist Projects

A Partial Differential Equation (PDE) based model combining surface electromigration and wetting is developed for the analysis of the morphological instability of mono-crystalline metal films in a high temperature environment typical to operational conditions of microelectronic interconnects. The atomic mobility and surface energy of such films are anisotropic, and the model accounts for these material properties. The goal of modeling is to describe and understand the time-evolution of the shape of film surface. I will present the formulation of a nonlinear parabolic PDE problem for the height function h(x,t) of the film in the horizontal …

A Numerical Model For Nonadiabatic Transitions In Molecules, Devanshu Agrawal

Undergraduate Honors Theses

In molecules, electronic state transitions can occur via quantum coupling of the states. If the coupling is due to the kinetic energy of the molecular nuclei, then electronic transitions are best represented in the adiabatic frame. If the coupling is instead facilitated through the potential energy of the nuclei, then electronic transitions are better represented in the diabatic frame. In our study, we modeled these latter transitions, called ``nonadiabatic transitions.'' For one nuclear degree of freedom, we modeled the de-excitation of a diatomic molecule. For two nuclear degrees of freedom, we modeled the de-excitation of an ethane-like molecule undergoing cis-trans …

Numerical Solutions For Problems With Complex Physics In Complex Geometry, Yifan Wang

Doctoral Dissertations

In this dissertation, two high order accurate numerical methods, Spectral Element Method (SEM) and Discontinuous Galerkin method (DG), are discussed and investigated. The advantages of both methods and their applicable areas are studied. Particular problems in complex geometry with complex physics are investigated and their high order accurate numerical solutions obtained by using either SEM or DG are presented. Furthermore, the Smoothed Particle Hydrodynamics (SPH) (a mesh-free weighted interpolation method) is implemented on graphics processing unit (GPU). Some numerical simulations of the complex flow with a free surface are presented and discussed to show the advantages of SPH method in …