Partial Differential Equations Commons^™

Multi-Scale Cardiovascular Flow Analysis By An Integrated Meshless-Lumped Parameter Model, Leonardo A. Bueno, Eduardo A. Divo, Alain J. Kassab

Publications

A computational tool that integrates a Radial basis function (RBF)-based Meshless solver with a Lumped Parameter model (LPM) is developed to analyze the multi-scale and multi-physics interaction between the cardiovascular flow hemodynamics, the cardiac function, and the peripheral circulation. The Meshless solver is based on localized RBF collocations at scattered data points which allows for automation of the model generation via CAD integration. The time-accurate incompressible flow hemodynamics are addressed via a pressure-velocity correction scheme where the ensuing Poisson equations are accurately and efficiently solved at each time step by a Dual-Reciprocity Boundary Element method (DRBEM) formulation that takes advantage …

A Wasserstein Gradient Flow Approach To Poisson-Nernst-Planck Equations, David Kinderlehrer, Leinard Monsaingeon, Xiang Xu

Mathematics & Statistics Faculty Publications

The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational scheme is then set up and is the starting point of the construction of global weak solutions in a unified framework for the cases of both linear and non-linear diffusion. The proof of the main results relies on the derivation of additional estimates based on the flow interchange technique developed by Matthes et al. in [D. Matthes, R.J. McCann and G. Savare, Commun. Partial …

A Numerical Scheme For Generalized Fractional Optimal Control Problems, N. Singha, C. Nahak

Applications and Applied Mathematics: An International Journal (AAM)

This paper introduces a generalization of the Fractional Optimal Control Problem (GFOCP). Proposed generalizations differ in terms of explaining the constraint involved in the dynamical system of the control problem. We assume the constraint as an arbitrary function of fractional derivatives and fractional integrals. By this assumption the restriction on constraint, to be of some prescribed function of fractional operators, is removed. Deduction of necessary optimality conditions followed by particular cases and examples has been provided. Additionally, we construct a solution scheme for the suggested class of (GFOCP)’s. The formulation of this scheme is done by implementing the Adomian decomposition …

Heat Source Thermoelastic Problem In A Hollow Elliptic Cylinder Under Time-Reversal Principle, Pravin Bhad, Vinod Varghese, Lalsingh Khalsa

Applications and Applied Mathematics: An International Journal (AAM)

The article investigates the time-reversal thermoelasticity of a hollow elliptical cylinder for determining the temperature distribution and its associated thermal stresses at a certain point using integral transform techniques by unifying classical orthogonal polynomials as the kernel. Furthermore, by considering a circle as a special kind of ellipse, it is seen that the temperature distribution and the comparative study of a circular cylinder can be derived as a special case from the present mathematical solution. The numerical results obtained are accurate enough for practical purposes.

Complex Solutions Of The Time Fractional Gross-Pitaevskii (Gp) Equation With External Potential By Using A Reliable Method, Nasir Taghizadeh, Mona N. Foumani

Applications and Applied Mathematics: An International Journal (AAM)

In this article, modified (G'/G )-expansion method is presented to establish the exact complex solutions of the time fractional Gross-Pitaevskii (GP) equation in the sense of the conformable fractional derivative. This method is an effective method in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The present approach has the potential to be applied to other nonlinear fractional differential equations. Based on two transformations, fractional GP equation can be converted into nonlinear ordinary differential equation of integer orders. In the end, we will discuss the solutions of the fractional GP equation with external potentials.

Spreading Speeds Along Shifting Resource Gradients In Reaction-Diffusion Models And Lattice Differential Equations., Jin Shang

Electronic Theses and Dissertations

A reaction-diffusion model and a lattice differential equation are introduced to describe the persistence and spread of a species along a shifting habitat gradient. The species is assumed to grow everywhere in space and its growth rate is assumed to be monotone and positive along the habitat region. We show that the persistence and spreading dynamics of a species are dependent on the speed of the shifting edge of the favorable habitat, c, as well as c^*(∞) and c^*(−∞), which are formulated in terms of the dispersal kernel and species growth rates in both directions. When …

Impact Of Permeable Lining Of The Wall On The Peristaltic Flow Of Herschel Bulkley Fluid, G. C. Sankad, Asha Patil

Applications and Applied Mathematics: An International Journal (AAM)

The peristaltic motion is modeled for the Herschel Bulkley fluid, considered to flow in a non-uniform inclined channel. The channel wall is supposed to be lined with a non-erodible porous material. The flow is considered to be moving in a wave frame of reference moving with same velocity as of the sinusoidal wave. Low Reynolds number and long wave length assumptions are made to solve the model. Analytical solution is obtained for the pressure difference and also for the frictional force. Graphs are plotted, using Mathematica software, for both the results of pressure difference and frictional force against time average …

On The Propagation Of Atmospheric Gravity Waves In A Non-Uniform Wind Field: Introducing A Modified Acoustic-Gravity Wave Equation, Ahmad Talaei

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Atmospheric gravity waves play fundamental roles in a broad-range of dynamical processes extending throughout the Earth’s neutral atmosphere and ionosphere. In this paper, we present a modified form for the acoustic-gravity wave equation and its dispersion relationships for a compressible and non-stationary atmosphere in hydrostatic balance. Importantly, the solutions have been achieved without the use of the well-known Boussinesq approximation which have been used extensively in previous studies.

We utilize the complete set of governing equations for a compressible atmosphere with non-uniform airflows to determine an equation for vertical velocity of possible atmospheric waves. This intricate wave equation is simplified …

Fast Method Of Particular Solutions For Solving Partial Differential Equations, Anup Raja Lamichhane

Dissertations

Method of particular solutions (MPS) has been implemented in many science and engineering problems but obtaining the closed-form particular solutions, the selection of the good shape parameter for various radial basis functions (RBFs) and simulation of the large-scale problems are some of the challenges which need to overcome. In this dissertation, we have used several techniques to overcome such challenges.

The closed-form particular solutions for the Matérn and Gaussian RBFs were not known yet. With the help of the symbolic computational tools, we have derived the closed-form particular solutions of the Matérn and Gaussian RBFs for the Laplace and biharmonic …

How Steep Is Steep? Learning Curves In Training Undergraduates To Do Fluid-Structure Interaction Modeling, Nicholas A. Battista

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Mathematical Hybrid Models For Image Segmentation., Carlos M. Paniagua Mejia

Electronic Theses and Dissertations

Two hybrid image segmentation models that are able to process a wide variety of images are proposed. The models take advantage of global (region) and local (edge) data of the image to be segmented. The first one is a region-based PDE model that incorporates a combination of global and local statistics. The influence of each statistic is controlled using weights obtained via an asymptotically stable exponential function. Through incorporation of edge information, the second model extends the capabilities of a strictly region-based variational formulation, making it able to process more general images. Several examples are provided showing the improvements of …

An Algorithm For The Machine Calculation Of Minimal Paths, Robert Whitinger

Electronic Theses and Dissertations

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R³, but also to the general case of finding minimal functionals on hypersurfaces in Rⁿ associated with an arbitrary metric.

Krylov Subspace Spectral Method With Multigrid For A Time-Dependent, Variable-Coefficient Partial Differential Equation, Haley Renee Dozier

Master's Theses

Krylov Subspace Spectral (KSS) methods are traditionally used to solve time-dependent, variable-coefficient PDEs. They are high-order accurate, component-wise methods that are efficient with variable input sizes.

This thesis will demonstrate how one can make KSS methods even more efficient by using a Multigrid-like approach for low-frequency components. The essential ingredients of Multigrid, such as restriction, residual correction, and prolongation, are adapted to the timedependent case. Then a comparison of KSS, KSS with Multigrid, KSS-EPI and standard Krylov projection methods will be demonstrated.

Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, Mikhail Khenner

Mathematics Faculty Publications

A basic kinetic model that incorporates a coupled dynamics of the carbon atoms and dimers ona copper surface is used to compute growth of a single-layer graphene island. The speed of theisland's edge advancement on Cu[111] and Cu[100] surfaces is computed as a function of the growthtemperature and pressure. Spatially resolved concentration pro les of the atoms and dimers aredetermined, and the contributions provided by these species to the growth speed are discussed.Island growth in the conditions of a thermal cycling is studied.

An Averaging Method For Advection-Diffusion Equations, Nicholas Spizzirri

Dissertations, Theses, and Capstone Projects

Many models for physical systems have dynamics that happen over various different time scales. For example, contrast the everyday waves in the ocean with the larger, slowly moving global currents. The method of multiple scales is an approach for approximating the solutions of differential equations by separating out the dynamics at slower and faster time scales. In this work, we apply the method of multiple scales to generic advection-diffusion equations (both linear and non-linear, and in arbitrary spatial dimensions) and develop a method for 'averaging out' the faster scale phenomena, giving us an 'effective' solution for the slower scale dynamics. …

Approximate Analytical Solution Of Boussinesq Equation In Homogeneous Medium With Leaky Base, Rajeev K. Bansal

Applications and Applied Mathematics: An International Journal (AAM)

Approximate analytical solutions of Boussinesq equation are widely used for approximation of subsurface seepage flow in confined and unconfined aquifers under varying hydrological conditions. In this paper, we use a 2-dimensional linearized Boussinesq equation to simulate the water table fluctuations in an isotropic aquifer overlying a semi pervious bed under multiple localized recharge and withdrawal. The unconfined aquifer is considered to be in contact with two water bodies of constant water head along opposite cost lines, while the remaining two faces have no flow condition. The mathematical model is solved analytically using finite Fourier sine transform and the application of …

A Robust Uniform B-Spline Collocation Method For Solving The Generalized Phi-Four Equation, W. K. Zahra, W. A. Ouf, M. S. El-Azab

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we develop a numerical solution based on cubic B-spline collocation method. By applying Von-Neumann stability analysis, the proposed technique is shown to be unconditionally stable. The accuracy of the presented method is demonstrated by a test problem. The numerical results are found to be in good agreement with the exact solution.

A New Approach For Solving System Of Local Fractional Partial Differential Equations, Hossein Jafari, Hassan K. Jassim

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we apply a new method for solving system of partial differential equations within local fractional derivative operators. The approximate analytical solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm. The obtained results show that the introduced approach is a promising tool for solving system of linear and nonlinear local fractional differential equations. Furthermore, we show that local fractional Laplace variational iteration method is able …

Upper, Lower Solutions And Analytic Semigroups For A Model With Diffusion, Yannick T. Kouakep

Applications and Applied Mathematics: An International Journal (AAM)

In this discussion we consider an autonomous parabolic epidemic 2-dimensional system modelling the dynamics of transmission of immunizing diseases for a closed population into bounded regular domain. Our model takes into account diffusion of population with external influx as well as one class of infected individuals. We study the well-posedness two-component diffusion equations including external supplies with Neumann conditions using upper/lower solutions and analytic semigroups. In case of constant population or not, with non-oscillatory solution and constant diffusion, this problem admits travelling wave solutions whose minimum wave speed is surveyed here.

Mathematical Models Of Biofilm For Antimicrobial Persistence, Jia Zhao

Biology and Medicine Through Mathematics Conference

No abstract provided.