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.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

scholarly journals Spatial behaviour of thermoelasticity with microtemperatures and microconcentrations

2020 ◽  
Vol 34 ◽  
pp. 02001
Author(s):  
Adina Chirilă ◽  
Marin Marin
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Integrated Circuits ◽  
Boundary Conditions ◽  
Chemical Potential ◽  
Lateral Boundary ◽  
Spatial Behaviour ◽  
Cross Sectional ◽  
Lateral Boundary Conditions ◽  
The Mathematical Model

We consider a thermoelastic material with microtemperatures and microconcentrations. The mathematical model is represented by a system of partial differential equations with the coupling of the displacement, temperature, chemical potential, microconcentrations and microtemperatures fields. The processes of heat and mass diffusion play an important role in many engineering applications, such as satellite problems, manufacturing of integrated circuits or oil extractions. We study the spatial behaviour in a prismatic cylinder occupied by an anisotropic and inhomogeneous material. We impose final prescribed data that are proportional, but not identical, to their initial values. Moreover, we have zero body forces and zero lateral boundary conditions. The spatial behaviour is analysed in terms of some cross-sectional integrals of the solution that depend on the axial variable.

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.

On Geometrical Rigidity of Surfaces.

1997 ◽  
Vol 07 (04) ◽  
pp. 507-555 ◽  
Author(s):  
Daniel Choi
Keyword(s):  
Asymptotic Behavior ◽  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Goursat Problem ◽  
Elastic Shells ◽  
Rigidity Result ◽  
Linear Elastic ◽  
Linear Partial Differential Equations ◽  
Numerical Studies ◽  
Constant Angle

In this paper we first present a panorama about geometrical rigidity and inextensional displacements (also called infinitesimal bendings) for surfaces with kinematic boundary conditions and for surfaces with edges (in the sense of folds or junctions). This theory is fundamental for thin linear elastic shells, as it rules their asymptotic behavior when the thickness tends to zero. This behavior enlights some difficulties encountered in numerical studies of very thin elastic shells. Our approach is based on the introduction of a nonclassical space denoted by R(S) and related to inextensional displacements. It permits us to obtain new results concerning developable surfaces and hyperbolic surfaces, with one or two edges (most of them assumed to keep constant angle), including a theorem of rigid edge when the edge is an asymptotic line of the surface. By applying these results, we are able to exhibit a new example of sensitive problem for a shell with hyperbolic mean surface and with two edges keeping constant angle. In the Appendix, we give a nonclassical variant of Goursat problem for hyperbolic linear partial differential equations system, used in the proof of a rigidity result.

Adding police to a mathematical model of burglary

2010 ◽  
Vol 21 (4-5) ◽  
pp. 401-419 ◽  
Author(s):  
ASHLEY B. PITCHER
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuum Limit ◽  
The Other ◽  
Residential Burglary ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
The Continuum

We review the Short model of urban residential burglary derived from taking the continuum limit of two difference equations – one of which models the attractiveness of individual houses to burglary, and the other of which models burglar movement. This leads to a system of non-linear partial differential equations. We propose a change to the Short model and also add deterrence caused by the presence of uniformed officers to the model. We solve the resulting system of non-linear partial differential equations numerically and present results both with and without deterrence.

scholarly journals The Cauchy Problem For Linear Partial Differential Equations With Restricted Boundary Conditions

1956 ◽  
Vol 8 ◽  
pp. 426-431 ◽  
Author(s):  
E. P. Miles ◽  
Ernest Williams
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Cauchy Data ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
The Cauchy Problem ◽  
Image Position

We shall discuss solutions of linear partial differential equations of the form1where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;2j = 1,2, ߪ,s − 1,whenever the function P is annihilated by a finite iteration of the operator Φ.

scholarly journals A Method of Deriving Solutions of a Class of Boundary Value Problems

1959 ◽  
Vol 11 (3) ◽  
pp. 147-152 ◽  
Author(s):  
F. J. Lockett
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Direct Application ◽  
Standard Methods ◽  
Linear Partial Differential Equations ◽  
Finite Extent ◽  
Infinite Range

In many branches of applied mathematics there exists a class of problems which depend for their solution upon the integration of a set of simultaneous linear partial differential equations subject to certain boundary conditions. In all but the simplest cases it is not practicable to deal with these equations by standard methods. For problems involving infinite regions, solutions can often be found by the use of integral transforms. However, in many problems we are concerned with media of finite extent, so that if we are to make a direct application of this method, we shall have to use finite transforms, and under certain conditions these are much more difficult to apply than transforms over an infinite range.

The creeping motion of a non-Newtonian fluid past a sphere

1962 ◽  
Vol 13 (3) ◽  
pp. 417-426 ◽  
Author(s):  
B. Caswell ◽  
W. H. Schwarz
Keyword(s):  
Reynolds Number ◽  
Flow Stress ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Slow Flow ◽  
Linear Partial Differential Equations ◽  
Creeping Motion ◽  
Flow Past A Sphere

The equations of motion and continuity are solved together with the slow-flow stress equations for an incompressible Rivlin-Ericksen fluid. The boundary conditions for slow flow past a sphere are satisfied by matching inner (Stokes) and outer (Oseen) Reynolds-number expansions of the stream function. The terms in the inner expansion are the solutions of non-linear partial differential equations which are solved approximately by expanding in terms of a non-Newtonian parameter λ. The drag force on the sphere is obtained from the solution.

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