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Partial Differential Equations Commons

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All Articles in Partial Differential Equations

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A High Order Finite Difference Method To Solve The Steady State Navier-Stokes Equations, Nihal J. Siriwardana, Saroj P. Pradhan 2021 Prairie View A&M University

A High Order Finite Difference Method To Solve The Steady State Navier-Stokes Equations, Nihal J. Siriwardana, Saroj P. Pradhan

Applications and Applied Mathematics: An International Journal (AAM)

In this article, we develop a fourth order finite difference method to solve the system of steady state Navier-Stokes equations and apply it to the benchmark problem known as the square cavity flow problem. The numerical results of 𝑢-velocity components and 𝑣-velocity components obtained at the center of the cavity are compared with the results obtained by the method developed by Greenspan and Casulli to solve the time dependent system of Navier-Stokes equations. The method described in this article is easy to implement and it has been shown to be more efficient and stable than the method by Greenspan and ...


Hybrid Algorithm For Singularly Perturbed Delay Parabolic Partial Differential Equations, Imiru T. Daba, Gemechis F. Duressa 2021 Wollega University

Hybrid Algorithm For Singularly Perturbed Delay Parabolic Partial Differential Equations, Imiru T. Daba, Gemechis F. Duressa

Applications and Applied Mathematics: An International Journal (AAM)

This study aims at constructing a numerical scheme for solving singularly perturbed parabolic delay differential equations. Taylor’s series expansion is applied to approximate the shift term. The obtained result is approximated by using the implicit Euler method in the temporal discretization on a uniform step size with the hybrid numerical scheme consisting of the midpoint upwind method in the outer layer region and the cubic spline method in the inner layer region on a piecewise uniform Shishkin mesh in the spatial discretization. The constructed scheme is an ε−uniformly convergent accuracy of order one. Some test examples are considered ...


Qualitative Analysis Of A Modified Leslie-Gower Predator-Prey Model With Weak Allee Effect Ii, Manoj K. Singh, B. S. Bhadauria 2021 Babasaheb Bhimrao Ambedkar University

Qualitative Analysis Of A Modified Leslie-Gower Predator-Prey Model With Weak Allee Effect Ii, Manoj K. Singh, B. S. Bhadauria

Applications and Applied Mathematics: An International Journal (AAM)

The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for ...


A Component-Wise Approach To Smooth Extension Embedding Methods, Vivian Montiforte 2021 The University of Southern Mississippi

A Component-Wise Approach To Smooth Extension Embedding Methods, Vivian Montiforte

Dissertations

Krylov Subspace Spectral (KSS) Methods have demonstrated to be highly scalable methods for PDEs. However, a current limitation of these methods is the requirement of a rectangular or box-shaped domain. Smooth Extension Embedding Methods (SEEM) use fictitious domain methods to extend a general domain to a simple, rectangular or box-shaped domain. This dissertation describes how these methods can be combined to extend the applicability of KSS methods, while also providing a component-wise approach for solving the systems of equations produced with SEEM.


The Effect Of Initial Conditions On The Weather Research And Forecasting Model, Aaron D. Baker 2021 Stephen F Austin State University

The Effect Of Initial Conditions On The Weather Research And Forecasting Model, Aaron D. Baker

Electronic Theses and Dissertations

Modeling our atmosphere and determining forecasts using numerical methods has been a challenge since the early 20th Century. Most models use a complex dynamical system of equations that prove difficult to solve by hand as they are chaotic by nature. When computer systems became more widely adopted and available, approximating the solution of these equations, numerically, became easier as computational power increased. This advancement in computing has caused numerous weather models to be created and implemented across the world. However a challenge of approximating these solutions accurately still exists as each model have varying set of equations and variables to ...


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 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 ...


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.


A Development Of A Polyhedron In The Galilean Space, Abdulaziz Artykbaev, Jasur Sobirov 2021 Tashkent State Transport University

A Development Of A Polyhedron In The Galilean Space, Abdulaziz Artykbaev, Jasur Sobirov

Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

In this paper, we study the development of a polyhedron in the Galilean space. A development of a polyhedron is an isometric mapping of a polyhedron to a plane, in which the gluing sides are indicated. Since the motion of the Galilean space differs significantly from the motion of the Euclidean space, the development of a polyhedron of the Galilean space will also differ from the development of a polyhedron of the Euclidean space. We prove that the total angle around the vertex of the polyhedral angle is preserved in the development. We also give illustrations of the developments for ...


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 ...


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