Partial Differential Equations Commons^™

The Exact Factorization Equations For One- And Two-Level Systems, Bart Rosenzweig

Theses and Dissertations

Exact Factorization is a framework for studying quantum many-body problems. This decomposes the wavefunctions of such systems into conditional and marginal components. We derive corresponding evolution equations for molecular systems whose conditional electronic subsystems are described by one or two Born-Oppenheimer levels and develop a program for their mathematical study.

An Examination Of Fontan Circulation Using Differential Equation Models And Numerical Methods, Vanessa Maybruck

Honors Student Research

Certain congenital heart defects can lead to the development of only a single pumping chamber, or ventricle, in the heart instead of the usual two ventricles. Individuals with this defect undergo a corrective, three-part surgery, the third step of which is the Fontan procedure, but as the patients age, their cardiovascular health will likely deteriorate. Using computational fluid dynamics and differential equations, Fontan circulation can be modeled to investigate why the procedure fails and how Fontan failure can be maximally prevented. Borrowing from well-established literature on RC circuits, the differential equation models simulate systemic blood flow in a piecewise, switch-like …

Five-Wave Resonances In Deep Water Gravity Waves: Integrability, Numerical Simulations And Experiments, Dan Lucas, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, Miguel D. Bustamante

Articles

In this work we consider the problem of finding the simplest arrangement of resonant deep water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wave vectors K₁ + K₂ = K₃ (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wave packets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction …

Improving The Temporal Accuracy Of Turbulence Models And Resolving The Implementation Issues Of Fluid Flow Modeling, Kyle J. Schwiebert

Dissertations, Master's Theses and Master's Reports

A sizeable proportion of the work in this thesis focuses on a new turbulence model, dubbed ADC (the approximate deconvolution model with defect correction). The ADC is improved upon using spectral deferred correction, a means of constructing a higher order ODE solver. Since both the ADC and SDC are based on a predictor-corrector approach, SDC is incorporated with essentially no additional computational cost. We will show theoretically and using numerical tests that the new scheme is indeed higher order in time than the original, and that the benefits of defect correction, on which the ADC is based, are preserved.

The …

Exact Solutions Of Two Nonlinear Space-Time Fractional Differential Equations By Application Of Exp-Function Method, Elahe M. Eskandari, Nasir Taghizadeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we discuss on the exact solutions of the nonlinear space-time fractional Burgerlike equation and also the nonlinear fractional fifth-order Sawada-Kotera equation with the expfunction method.We use the functional derivatives in the sense of Riemann-Jumarie derivative and fractional convenient variable transformation in this study. Further, we obtain some exact analytical solutions including hyperbolic function.

Accurate Spectral Algorithms For Solving Variable-Order Fractional Percolation Equations, M. A. Abdelkawy

Applications and Applied Mathematics: An International Journal (AAM)

A high accurate spectral algorithm for one-dimensional variable-order fractional percolation equations (VO-FPEs) is considered.We propose a shifted Legendre Gauss-Lobatto collocation (SL-GLC) method in conjunction with shifted Chebyshev Gauss-Radau collocation (SC-GR-C) method to solve the proposed problem. Firstly, the solution and its space fractional derivatives are expanded as shifted Legendre polynomials series. Then, we determine the expansion coefficients by reducing the VO-FPEs and its conditions to a system of ordinary differential equations (SODEs) in time. The numerical approximation of SODEs is achieved by means of the SC-GR-C method. The under-study’s problem subjected to the Dirichlet or non-local boundary conditions is presented …

Explicit Formulas For Solutions Of Maxwell’S Equations In Conducting Media, Valery Yakhno

Applications and Applied Mathematics: An International Journal (AAM)

A new explicit presentation of the fundamental solution of the time-dependent Maxwell’s equations in conducting isotropic media is derived by Hadamard techniques through the fundamental solution of the telegraph operator. This presentation is used to obtain explicit formulas for generalized solutions of the initial value problem for Maxwell’s equations. A new explicit Kirchhoff’s formula for the classical solution of the initial value problem for the Maxwell equations in conducting media is derived. The obtained explicit formulas can be used in the boundary integral method, Green’s functions method and for computation of electric and magnetic fields in conducting media and materials.

From Wave Propagation To Spin Dynamics: Mathematical And Computational Aspects, Oleksii Beznosov

Mathematics & Statistics ETDs

In this work we concentrate on two separate topics which pose certain numerical challenges. The first topic is the spin dynamics of electrons in high-energy circular accelerators. We introduce a stochastic differential equation framework to study spin depolarization and spin equilibrium. This framework allows the mathematical study of known equations and new equations modelling the spin distribution of an electron bunch. A spin distribution is governed by a so-called Bloch equation, which is a linear Fokker-Planck type PDE, in general posed in six dimensions. We propose three approaches to approximate solutions, using analytical and modern numerical techniques. We also present …

Numerical Simulations Of Nonlinear Waves And Their Stability: Stokes Waves And Nonlinear Schroedinger Equation, Anastassiya Semenova

Mathematics & Statistics ETDs

The present work offers an investigation of dynamics and stability of nonlinear waves in Hamiltonian systems. The first part of the manuscript discusses the classical problem of water waves on the surface of an ideal fluid in 2D. We demonstrate how to construct the Stokes waves, and how to apply a continuation method to find waves in close vicinity to the limiting Stokes wave. We provide new insight into the stability of the Stokes waves by identifying previously inaccessible branches of instability in the equations of motion for the fluid. We provide numerical evidence that pairs of unstable eigenvalues of …

Parameter Estimation For Tear Film Thinning, Rayanne Luke

Biology and Medicine Through Mathematics Conference

No abstract provided.

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 …

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.

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.

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.

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.

Comparative Analysis On Angular Flow And Mass Transfer In Haemodialysis, J. K. Misra, Pradeep K. Singh, Naseem Ahmad, Pankaj Sharma

Applications and Applied Mathematics: An International Journal (AAM)

Healthy kidney cleans blood and removes unwanted materials in the form of urine. When the kidney does not work properly, dialysis is one of the best solutions. Dialysis required if unhealthy kidney does not remove enough wastes and fluid from the blood. This usually happens when only 10 - 15 % of kidney’s function left. A dialyzer is used to clean blood. In an attempt to address clinical and experimental discrepancies, compartmental theoretical models have been used. Noda et al. (1979) were among the first to introduce a theoretical model on mass transfer using countercurrent flows. Their proposed model assumes …

Analytical Wave Solutions Of The Space Time Fractional Modified Regularized Long Wave Equation Involving The Conformable Fractional Derivative, M. Hafiz Uddin, Md. Ashrafuzzaman Khan, M. Ali Akbar, Md. Abdul Haque

Karbala International Journal of Modern Science

The space time fractional modified regularized long wave equation is a model equation to the gravitational water waves in the long-wave occupancy, shallow waters waves in coastal seas, the hydro-magnetic waves in cold plasma, the phonetic waves in dissident quartz and phonetic gravitational waves in contractible liquids. In nonlinear science and engineering, the mentioned equation is applied to analyze the one way tract of long waves in seas and harbors. In this study, the closed form traveling wave solutions to the above equation are evaluated due to conformable fractional derivatives through double (G'⁄G,1⁄G)-expansion method and the Exp-function method. The existence …

Optimal Homotopy Asymptotic Solution For Thermal Radiation And Chemical Reaction Effects On Electrical Mhd Jeffrey Fluid Flow Over A Stretching Sheet Through Porous Media With Heat Source, Gossaye Aliy, Naikoti Kishan

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the problem of thermal radiation and chemical reaction effects on electrical MHD Jeffrey fluid flow over a stretching surface through a porous medium with the heat source is presented. We obtained the approximate analytical solution of the nonlinear differential equations governing the problem using the Optimal Homotopy Asymptotic Method (OHAM). Comparison of results has been made with the numerical solutions from the literature, and a very good agreement has been observed. Subsequently, effects of governing parameters of the velocity, temperature and concentration profiles are presented graphically and discussed.

Equatorial Wave–Current Interactions, Adrian Constantin, Rossen Ivanov

Articles

We study the nonlinear equations of motion for equatorial wave–current interactions in the physically realistic setting of azimuthal two-dimensional inviscid flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface. We derive a Hamiltonian formulation for the nonlinear governing equations that is adequate for structure-preserving perturbations, at the linear and at the nonlinear level. Linear theory reveals some important features of the dynamics, highlighting differences between the short- and long-wave regimes. The fact that ocean energy is concentrated in the long-wave propagation modes motivates the pursuit of in-depth nonlinear analysis in the long-wave …

Surface Waves Over Currents And Uneven Bottom, Alan Compelli, Rossen Ivanov, Calin I. Martin, Michail D. Todorov

Articles

The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution …

Similarity Analysis Of Three Dimensional Nanofluid Flow By Deductive Group Theoretic Method, Hemangini Shukla, Hema C. Surati, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

The objective of this paper is to obtain similarity solution of three-dimensional nanofluid flow over flat surface stretched continuously in two lateral directions. Two independent variables from governing equations are reduced by applying deductive two parameter group theoretical method. Partial differential equations with boundary conditions are converted into ordinary differential equations with appropriate boundary conditions. Obtained equations are solved for temperature and velocity. The effect of nanoparticles volume fraction on temperature and velocity profile is investigated.

Linear Stability Analysis With Solution Patterns Due To Varying Thermal Diffusivity For A Convective Flow In A Porous Medium, Dambaru Bhatta

Applications and Applied Mathematics: An International Journal (AAM)

Here we investigate the effect of the vertical rate of change in thermal diffusivity due to a hydrothermal convective flow in a horizontal porous medium. The continuity equation, the heat equation and the momentum-Darcy equation constitute the governing system for the flow in a porous medium. Assuming a vertically varying basic state, we derive the linear system and from this linear system, we compute the critical Rayleigh and wave numbers. Using fourth-order Runge-Kutta and shooting methods, we obtain the marginal stability curves and linear solutions to analyze the solution pattern for different diffusivity parameters.

Conformable Derivative Operator In Modelling Neuronal Dynamics, Mehmet Yavuz, Burcu Yaşkıran

Applications and Applied Mathematics: An International Journal (AAM)

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical …

Two Applications Of High Order Methods: Wave Propagation And Accelerator Physics, Oleksii Beznosov

Shared Knowledge Conference

Numerical simulations of partial differential equations (PDE) are used to predict the behavior of complex physics phenomena when the real life experiments are expensive. Discretization of a PDE is the representation of the continuous problem as a discrete problem that can be solved on a computer. The discretization always introduces a certain inaccuracy caused by the numerical approximation. By increasing the computational cost of the numerical algorithm the solution can be computed more accurately. In the theory of numerical analysis this fact is called the convergence of the numerical algorithm. The idea behind high order methods is to improve the …

Generalized Problem Of Thermal Bending Analysis In The Cartesian Domain, V. S. Kulkarni, Vinayaki Parab

Applications and Applied Mathematics: An International Journal (AAM)

This is an attempt for mathematical formulation and general analytical solution of the most generalized thermal bending problem in the Cartesian domain. The problem has been formulated in the context of non-homogeneous transient heat equation subjected to Robin’s boundary conditions. The general solution of the generalized thermoelastic problem has been discussed for temperature change, displacements, thermal stresses, deflection, and deformation. The most important feature of this work is any special case of practical interest may be readily obtained by this most generalized mathematical formulation and its analytical solution. There are 729 such combinations of possible boundary conditions prescribed on parallelepiped …

Finite Element Solution Of The Two-Dimensional Incompressible Navier-Stokes Equations Using Matlab, Endalew G. Tsega, V. K. Katiyar

Applications and Applied Mathematics: An International Journal (AAM)

The Navier–Stokes equations are fundamental in fluid mechanics. The finite element method has become a popular method for the solution of the Navier-Stokes equations. In this paper, the Galerkin finite element method was used to solve the Navier-Stokes equations for two-dimensional steady flow of Newtonian and incompressible fluid with no body forces using MATLAB. The method was applied to the lid-driven cavity problem. The eight-noded rectangular element was used for the formulation of element equations. The velocity components were located at all of 8 nodes and the pressure variable is located at 4 corner of the element. From location of …

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

The Advection-Diffusion Equation And The Enhanced Dissipation Effect For Flows Generated By Hamiltonians, Michael Kumaresan

Dissertations, Theses, and Capstone Projects

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices …