scholarly journals Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 91
Author(s):  
Costică Moroşanu ◽  
Silviu Pavăl
Keyword(s):  
Mathematical Model ◽  
Image Segmentation ◽  
A Priori Estimates ◽  
A Priori ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Second Order ◽  
Reaction Diffusion Model ◽  
Anisotropic Reaction Diffusion ◽  
Physical Phenomena

In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.

scholarly journals Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions

2021 ◽  
Vol 38 (1) ◽  
pp. 95-116
Author(s):  
MITROFAN M. CHOBAN ◽  
◽  
COSTICĂ N. MOROȘANU ◽  
Keyword(s):  
Boundary Conditions ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Second Order ◽  
Well Posedness ◽  
Dynamic Boundary Conditions ◽  
Dynamic Boundary ◽  
Anisotropic Reaction Diffusion

The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

Dynamics of an infection age-space structured cholera model with Neumann boundary condition

2021 ◽  
pp. 1-30
Author(s):  
WEIWEI LIU ◽  
JINLIANG WANG ◽  
RAN ZHANG
Keyword(s):  
Vibrio Cholerae ◽  
Disease Transmission ◽  
Global Attractivity ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Human Populations ◽  
Global Dynamics ◽  
Infection Age ◽  
Reaction Diffusion Model ◽  
Disease Persistence

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝ n . We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

A mathematical model for the third-body concept

2017 ◽  
Vol 23 (3) ◽  
pp. 420-432 ◽  
Author(s):  
Pavel Krejčí ◽  
Adrien Petrov
Keyword(s):  
Mathematical Model ◽  
A Priori Estimates ◽  
A Priori ◽  
Dynamical Problem ◽  
Sobolev Embedding ◽  
Third Body ◽  
Embedding Theory ◽  
The Third ◽  
Galerkin Approximations ◽  
Body Concept

The third-body concept is a pragmatic tool used to understand the friction and wear of sliding materials. The wear particles play a crucial role in this approach and constitute the main part of the third-body. This paper aims to introduce a mathematical model for the motion of a third-body interface separating two surfaces in contact. This model is written in accordance with the formalism of hysteresis operators as solution operators of the underlying variational inequalities. The existence result for this dynamical problem is obtained by using a priori estimates established for Faedo–Galerkin approximations, and some more specific techniques such as anisotropic Sobolev embedding theory.

A reaction-diffusion-advection competition model with a free boundary

2021 ◽  
Vol 65 (3) ◽  
pp. 25-37
Keyword(s):  
Free Boundary ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Competition Model ◽  
Baseline Data ◽  
The Other ◽  
Quasilinear System ◽  
Priori Estimates ◽  
Competitive Diffusion

In this paper, we study a competitive diffusion quasilinear system with a free boundary. First, we investigate the mathematical questions of the problem. A priori estimates of Schauder type are established, which are necessary for the solvability of the problem. One of two competing species is an invader, which initially exists on a certain sub-interval. The other is initially distributed throughout the area under consideration. Examining the influence of baseline data on the success or failure of the first invasion. We conclude that there is a dichotomy of spread and extinction and give precise criteria for spread and extinction in this model.

Data Highways

2012 ◽  
pp. 223-241
Author(s):  
Karina Mabell Gomez ◽  
Daniele Miorandi ◽  
David Lowe
Keyword(s):  
Network Topology ◽  
Ad Hoc ◽  
A Priori ◽  
Reaction Diffusion ◽  
Theoretical Work ◽  
Ad Hoc Wireless Networks ◽  
Global Network ◽  
Wireless Sensor ◽  
A Priori Knowledge ◽  
Reaction Diffusion Model

The design of efficient routing algorithms is an important issue in dense ad hoc wireless networks. Previous theoretical work has shown that benefits can be achieved through the creation of a set of data “highways” that carry packets across the network, from source(s) to sink(s). Current approaches to the design of these highways however require a–priori knowledge of the global network topology, with consequent communications burden and scalability issues, particularly with regard to reconfiguration after node failures. In this chapter, we describe a bio–inspired approach to generating these data highways through a distributed reaction–diffusion model that uses localized convolution with activation–inhibition filters. The result is the distributed emergence of data highways that can be tuned to provide appropriate highway separation and connection to data sinks. In this chapter, we present the underlying models, algorithms, and protocols for generating data highways in a dense wireless sensor network. The proposed methods are validated through extensive simulations performed using OMNeT++.

scholarly journals On the Neumann Problem for Monge-Ampére Type Equations

2016 ◽  
Vol 68 (6) ◽  
pp. 1334-1361 ◽  
Author(s):  
Feida Jiang ◽  
Neil S. Trudinger ◽  
Ni Xiang
Keyword(s):  
Global Regularity ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Neumann Boundary ◽  
Natural Conditions ◽  
Priori Estimates ◽  
The Neumann Problem ◽  
Monge Ampere ◽  
Oblique Boundary

AbstractIn this paper, we study the global regularity for regular Monge-Ampère type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. The techniques build upon the delicate and intricate treatment of the standard Monge-Ampère case by Lions, Trudinger, and Urbas in 1986 and the recent barrier constructions and second derivative bounds by Jiang, Trudinger, and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

scholarly journals Simulation of the gastric digestion of proteins of meat bolus using a reaction–diffusion model

2018 ◽  
Vol 9 (12) ◽  
pp. 6455-6469 ◽  
Author(s):  
Jason Sicard ◽  
Pierre-Sylvain Mirade ◽  
Stéphane Portanguen ◽  
Sylvie Clerjon ◽  
Alain Kondjoyan
Keyword(s):  
Mathematical Model ◽  
Diffusion Model ◽  
Reaction Diffusion ◽  
Gastric Digestion ◽  
Physiological Factors ◽  
Reaction Diffusion Model ◽  
Meat Proteins

A mathematical model predicts the gastric digestion of meat proteins and quantifies the impacts of physiological factors on digestibility.

A priori estimates of solutions of nonlinear boundary value problems for singular in a phase variable second order differential inequalities

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ivan Kiguradze
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Second Order ◽  
Phase Variable ◽  
Differential Inequalities ◽  
Nonlinear Boundary Value Problems ◽  
Estimates Of Solutions ◽  
Priori Estimates

Abstract.For singular in a phase variable second order differential inequalities, a priori estimates of positive solutions, satisfying nonlinear nonlocal boundary conditions, are established.

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