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All Articles in Partial Differential Equations

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A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Gamage, Yan Peng 2021 Old Dominion University

A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Gamage, Yan Peng

College of Sciences Posters

Interface problems have many applications in fluid dynamics, molecular biology, electromagnetism, material science, heat distribution in engines, and hyperthermia treatment of cancer. Mathematically, interface problems commonly lead to partial differential equations (PDE) whose in- put data are discontinuous or singular across the interfaces in the solution domain. Many standard numerical methods designed for smooth solutions poorly work for interface problems as solutions of the interface problems are mostly non-smoothness or discontinuous. Moving interface problems depends on the accuracy of the gradient of the solution at the interface. Therefore, it became essential to derive a method for interface problems that gives ...


Nonlocal Boundary Value Problem For A System Of Mixed Type Equations With A Line Of Degeneration, Kudratillo Fayazov, Ikrombek Khajiev 2021 Turin Polytechnic University in Tashkent

Nonlocal Boundary Value Problem For A System Of Mixed Type Equations With A Line Of Degeneration, Kudratillo Fayazov, Ikrombek Khajiev

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

This work is devoted to the study of a nonlocal boundary value problem for a system of two-dimensional parabolic equations with changing direction of time. A priori estimate is obtained for the solution of the problem under consideration, and theorems on stability and conditional stability are proved depending on the parameters of the nonlocal condition. As a result, the uniqueness of the solution to the problem is presented.


Nonlocal Problems For A Fractional Order Mixed Parabolic Equation, Azizbek Mamanazarov 2021 Fergana State University

Nonlocal Problems For A Fractional Order Mixed Parabolic Equation, Azizbek Mamanazarov

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In the present work nonlocal problems with Bitsadze-Samarskii type conditions, with the first and the second kind integral conditions for mixed parabolic equation involving Riemann-Liouville fractional differential operator have been formulated and investigated. The uniqueness and the existence of the solution of the considered problems were proved. To do this, considered problems are equivalently reduced to the problems with nonlocal conditions with respect to the trace of the unknown function and its space-derivatives. Then using the representation of the solution of the second kind of Abel's integral equation, it was found integral representations of the solutions of these problems ...


Principles For Determining The Motion Of Blood Through Arteries, Sylvio R. Bistafa 2021 University of Sao Paulo

Principles For Determining The Motion Of Blood Through Arteries, Sylvio R. Bistafa

Euleriana

Translation of Principia pro motu sanguinis per arterias determinando (E855). This work of 1775 by L. Euler is considered to be the first mathematical treatment of circulatory physiology and hemodynamics.


Hyperbolic Quadrature Method Of Moments For The One-Dimensional Kinetic Equation, Rodney O. Fox, Frédérique Laurent 2021 Iowa State University

Hyperbolic Quadrature Method Of Moments For The One-Dimensional Kinetic Equation, Rodney O. Fox, Frédérique Laurent

Chemical and Biological Engineering Publications

A solution is proposed to a longstanding open problem in kinetic theory, namely, 5 given any set of realizable velocity moments up to order 2n, a closure for the moment of order 2n+1 is 6 constructed for which the moment system found from the free-transport term in the one-dimensional 7 (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic 8 quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth 9 order (n ≤ 2). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The 10 HyQMOM closure ...


Multicomponent Fokas-Lenells Equations On Hermitian Symmetric Spaces, Vladimir Gerdjikov, Rossen Ivanov 2021 Bulgarian Academy of Sciences

Multicomponent Fokas-Lenells Equations On Hermitian Symmetric Spaces, Vladimir Gerdjikov, Rossen Ivanov

Articles

Multi-component integrable generalizations of the Fokas-Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.


A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui 2021 Colby College

A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui

Honors Theses

In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under ...


The Revised Nim For Solving The Non-Linear System Variant Boussinesq Equations And Comparison With Nim, Oday Ahmed Jasim 2020 University of Mosul, Mosul

The Revised Nim For Solving The Non-Linear System Variant Boussinesq Equations And Comparison With Nim, Oday Ahmed Jasim

Karbala International Journal of Modern Science

This research aims to guide researchers to use a new method, and it is the Revised New Iterative Method (RNIM) to solve partial differential equation systems and apply them to solve problems in various disciplines such as chemistry, physics, engineering and medicine. In this paper, the numerical solutions of the nonlinear Variable Boussinesq Equation System (VBE) were obtained using a new modified iterative method (RNIM); this was planned by (Bhaleker and Datterder-Gejj). A numerical solution to the Variable Boussinesq Equation System (VBE) was also found using a widely known method, a new iterative method (NIM). By comparing the numerical solutions ...


Rapid Implicit Diagonalization Of Variable-Coefficient Differential Operators Using The Uncertainty Principle, Carley Walker 2020 The University of Southern Mississippi

Rapid Implicit Diagonalization Of Variable-Coefficient Differential Operators Using The Uncertainty Principle, Carley Walker

Master's Theses

We propose to create a new numerical method for a class of time-dependent PDEs (second-order, one space dimension, Dirichlet boundary conditions) that can be used to obtain more accurate and reliable solutions than traditional methods. Previously, it was shown that conventional time-stepping methods could be avoided for time-dependent mathematical models featuring a finite number of homogeneous materials, thus assuming general piecewise constant coefficients. This proposed method will avoid the modeling shortcuts that are traditionally taken, and it will generalize the piecewise constant case of energy diffusion and wave propagation to work for an infinite number of smaller pieces, or a ...


Asymptotic Analysis Of Radial Point Rupture Solutions For Elliptic Equations, Attou Miloua 2020 Illinois State University

Asymptotic Analysis Of Radial Point Rupture Solutions For Elliptic Equations, Attou Miloua

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.


Analysis, Control Of Efsb Pest Population Using Graph Theoretic Approach And Pattern Formation In The Pest Model, Pankaj Gulati 2020 Illinois State University

Analysis, Control Of Efsb Pest Population Using Graph Theoretic Approach And Pattern Formation In The Pest Model, Pankaj Gulati

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.


Assess The Impacts Of Human Mobility Change On Covid-19 Using Differential Equations With Google Community Mobility Data, Nao Yamamoto 2020 Illinois State University

Assess The Impacts Of Human Mobility Change On Covid-19 Using Differential Equations With Google Community Mobility Data, Nao Yamamoto

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.


Deep Learning With Physics Informed Neural Networks For The Airborne Spread Of Covid-19 In Enclosed Spaces, Udbhav Muthakana, Padmanabhan Seshaiyer, Maziar Raissi, Long Nguyen 2020 George Mason University

Deep Learning With Physics Informed Neural Networks For The Airborne Spread Of Covid-19 In Enclosed Spaces, Udbhav Muthakana, Padmanabhan Seshaiyer, Maziar Raissi, Long Nguyen

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.


From Wave Propagation To Spin Dynamics: Mathematical And Computational Aspects, Oleksii Beznosov 2020 University of New Mexico

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 ...


Controlling Aircraft Yaw Movement By Interval Type-2 Fuzzy Logic, Yamama Shafeek, Laith Majeed, Rasha Naji 2020 University of Technology, Iraq

Controlling Aircraft Yaw Movement By Interval Type-2 Fuzzy Logic, Yamama Shafeek, Laith Majeed, Rasha Naji

Emirates Journal for Engineering Research

Aircraft yaw movement is essential in maneuvering; it has been controlled by some methods which achieved tracking but not fast enough. This paper performs the dynamic modeling of aircraft yaw movement and develops PI and PI-like interval type-2 fuzzy logic controller for the model. The mathematical model is derived by inserting the parameters values of single-engine Navion aircraft into standard equations. Using Matlab/ Simulink platform, the controllers' effectivity is tested and verified in two different cases; system without disturbance and when system is disturbed by some wind gust to investigate the system robustness. Simulation results show that PI controller response ...


A Phase-Field Approach To Diffusion-Driven Fracture, Friedrich Wilhelm Alexander Dunkel 2020 Louisiana State University and Agricultural and Mechanical College

A Phase-Field Approach To Diffusion-Driven Fracture, Friedrich Wilhelm Alexander Dunkel

LSU Doctoral Dissertations

In recent years applied mathematicians have used modern analysis to develop variational phase-field models of fracture based on Griffith's theory. These variational phase-field models of fracture have gained popularity due to their ability to predict the crack path and handle crack nucleation and branching.

In this work, we are interested in coupled problems where a diffusion process drives the crack propagation. We extend the variational phase-field model of fracture to account for diffusion-driving fracture and study the convergence of minimizers using gamma-convergence. We will introduce Newton's method for the constrained optimization problem and present an algorithm to solve ...


Analytical And Computational Modelling Of The Ranque-Hilsch Vortex Tube, Nolan J. Dyck 2020 The University of Western Ontario

Analytical And Computational Modelling Of The Ranque-Hilsch Vortex Tube, Nolan J. Dyck

Electronic Thesis and Dissertation Repository

The Ranque-Hilsch vortex tube (RHVT) is a simple mechanical device with no moving parts capable of separating a supply of compressed fluid into hot and cold streams through a process called temperature separation. The overall aim is to develop models which can be used to assess the temperature separation mechanisms in the RHVT, leading to a better overall understanding of the underlying physics. The introductory chapter contains a thermodynamic analysis and introduction to the flow physics, alongside three miniature literature reviews and critiques identifying research gaps.

The body of the thesis contains three articles. The first article studies the flow ...


Numerical Approach To Non-Darcy Mixed Convective Flow Of Non-Newtonian Fluid On A Vertical Surface With Varying Surface Temperature And Heat Source, Ajaya Prasad Baitharu, Sachidananda Sahoo, Gauranga Charan Dash 2020 Department of Mathematics, College of Engineering and Technology,Bhubaneswar-751029, Odisha, INDIA

Numerical Approach To Non-Darcy Mixed Convective Flow Of Non-Newtonian Fluid On A Vertical Surface With Varying Surface Temperature And Heat Source, Ajaya Prasad Baitharu, Sachidananda Sahoo, Gauranga Charan Dash

Karbala International Journal of Modern Science

An analysis is performed on non-Darcy mixed convective flow of non-Newtonian fluid past a vertical surface in the presence of volumetric heat source originated by some electromechanical or other devices. Further, the vertical bounding surface is subjected to power law variation of wall temperature, but the numerical solution is obtained for isothermal case. In the present non-Darcy flow model, effects of high flow rate give rise to inertia force. The inertia force in conjunction with volumetric heat source/sink is considered in the present analysis. The Runge-Kutta method of fourth order with shooting technique has been applied to obtain the ...


Heat And Mass Transfer Of Mhd Casson Nanofluid Flow Through A Porous Medium Past A Stretching Sheet With Newtonian Heating And Chemical Reaction, Lipika Panigrahi, Jayaprakash Panda, Kharabela Swain, Gouranga Charan Dash 2020 Veer Surenrda Sai University of Technology, Burla, India

Heat And Mass Transfer Of Mhd Casson Nanofluid Flow Through A Porous Medium Past A Stretching Sheet With Newtonian Heating And Chemical Reaction, Lipika Panigrahi, Jayaprakash Panda, Kharabela Swain, Gouranga Charan Dash

Karbala International Journal of Modern Science

An analysis is made to investigate the effect of inclined magnetic field on Casson nanofluid over a stretching sheet embedded in a saturated porous matrix in presence of thermal radiation, non-uniform heat source/sink. The heat equation takes care of energy loss due to viscous dissipation and Joulian dissipation. The mass transfer and heat equation become coupled due to thermophoresis and Brownian motion, two important characteristics of nanofluid flow. The convective terms of momentum, heat and mass transfer equations render the equations non-linear. This present flow model is pressure gradient driven and it is eliminated with the help of potential ...


Diagonalization Of 1-D Schrodinger Operators With Piecewise Constant Potentials, Sarah Wright 2020 The University of Southern Mississippi

Diagonalization Of 1-D Schrodinger Operators With Piecewise Constant Potentials, Sarah Wright

Master's Theses

In today's world our lives are very layered. My research is meant to adapt current inefficient numerical methods to more accurately model the complex situations we encounter. This project focuses on a specific equation that is used to model sound speed in the ocean. As depth increases, the sound speed changes. This means the variable related to the sound speed is not constant. We will modify this variable so that it is piecewise constant. The specific operator in this equation also makes current time-stepping methods not practical. The method used here will apply an eigenfunction expansion technique used in ...


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