scholarly journals Global Bounded Classical Solutions for a Gradient-Driven Mathematical Model of Antiangiogenesis in Tumor Growth

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Xiaofei Yang ◽  
Bo Lu
Keyword(s):  
Mathematical Model ◽  
Endothelial Cells ◽  
Partial Differential Equations ◽  
Global Existence ◽  
Tumor Growth ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Angiogenic Factors ◽  
Classical Solutions ◽  
Parabolic Partial Differential Equations

In this paper, we consider a gradient-driven mathematical model of antiangiogenesis in tumor growth. In the model, the movement of endothelial cells is governed by diffusion of themselves and chemotaxis in response to gradients of tumor angiogenic factors and angiostatin. The concentration of tumor angiogenic factors and angiostatin is assumed to diffuse and decay. The resulting system consists of three parabolic partial differential equations. In the present paper, we study the global existence and boundedness of classical solutions of the system under homogeneous Neumann boundary conditions.

Stability Criterion For Linear Oscillatory Parabolic Systems

1992 ◽  
Vol 114 (1) ◽  
pp. 175-178 ◽  
Author(s):  
Keum S. Hong ◽  
Joseph Bentsman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Simulations ◽  
Stability Criterion ◽  
Neumann Boundary ◽  
Parabolic Systems ◽  
Neumann Boundary Conditions ◽  
Distributed Parameter ◽  
Parabolic Partial Differential Equations

This paper presents a stability criterion for a class of distributed parameter systems governed by linear oscillatory parabolic partial differential equations with Neumann boundary conditions. The results of numerical simulations that support the criterion are presented as well.

scholarly journals Global existence and boundedness of solutions in a Lotka-Volterra reaction-diffusion system of predator-prey model with nonlinear prey-taxis

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 5023-5035
Author(s):  
Demou Luo
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Boundedness Of Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Global Classical Solutions ◽  
Prey Model

In this paper, we investigate a diffusive Lotka-Volterra predator-prey model with nonlinear prey-taxis under Neumann boundary conditions. This system describes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which lead to the collection of u). We apply some methods to overcome the substantial difficulty of the existence of nonlinear prey-taxis term and prove that the unique global classical solutions of Lotka-Volterra predator-prey model are globally bounded.

Global existence and boundedness of classical solutions to the Owen–Sherratt model

2016 ◽  
Vol 09 (01) ◽  
pp. 1650004
Author(s):  
Khadijeh Baghaei ◽  
Mahmoud Hesaaraki
Keyword(s):  
Mathematical Model ◽  
Global Existence ◽  
Classical Solutions ◽  
The Mathematical Model

In this paper, we study the mathematical model proposed by Owen and Sherratt in 1997. We prove that the classical solutions to this model are uniformly-in-time bounded.

CONTINUOUS NEURAL NETWORKS APPLIED TO IDENTIFY A CLASS OF UNCERTAIN PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 01 (04) ◽  
pp. 485-508
Author(s):  
R. FUENTES ◽  
A. POZNYAK ◽  
I. CHAIREZ ◽  
M. FRANCO ◽  
T. POZNYAK
Keyword(s):  
Neural Networks ◽  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Parametric Modeling ◽  
Parabolic Partial Differential Equations ◽  
Qualitative Behavior ◽  
Compact Set ◽  
Large Set ◽  
Partial Differential

There are a lot of examples in science and engineering that may be described using a set of partial differential equations (PDEs). Those PDEs are obtained applying a process of mathematical modeling using complex physical, chemical, etc. laws. Nevertheless, there are many sources of uncertainties around the aforementioned mathematical representation. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set. If the continuous mathematical model is incomplete or partially known, the methodology based on Differential Neural Network (DNN) provides an effective tool to solve problems on control theory such as identification, state estimation, trajectories tracking, etc. In this paper, a strategy based on DNN for the no parametric identification of a mathematical model described by parabolic partial differential equations is proposed. The identification solution allows finding an exact expression for the weights' dynamics. The weights adaptive laws ensure the "practical stability" of DNN trajectories. To verify the qualitative behavior of the suggested methodology, a no parametric modeling problem for a couple of distributed parameter plants is analyzed: the plug-flow reactor model and the anaerobic digestion system. The results obtained in the numerical simulations confirm the identification capability of the suggested methodology.

The influence of a lipid reservoir on the tear film formation

2020 ◽  
Vol 37 (3) ◽  
pp. 363-388
Author(s):  
Kara L Maki ◽  
Richard J Braun ◽  
Gregory A Barron
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Film Formation ◽  
Tear Film ◽  
Lipid Concentration ◽  
Linear Partial Differential Equations ◽  
Lipid Dynamics ◽  
Analytical Approaches ◽  
Thinning Parameter

Abstract We present a mathematical model to study the influence of a lipid reservoir, seen experimentally, at the lid margin on the formation and relaxation of the tear film during a partial blink. Applying the lubrication limit, we derive two coupled non-linear partial differential equations characterizing the evolution of the aqueous tear fluid and the covering insoluble lipid concentration. Departing from prior works, we explore a new set of boundary conditions (BCs) enforcing hypothesized lipid concentration dynamics at the lid margins. Using both numerical and analytical approaches, we find that the lipid-focused BCs strongly impact tear film formation and thinning rates. Specifically, during the upstroke of the eyelid, we find specifying the lipid concentration at the lid margin accelerates thinning. Parameter regimes that cause tear film formation success or failure are identified. More importantly, this work expands our understanding of the consequences of lipid dynamics near the lid margins for tear film formation.

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