José Augusto Ferreira

Qualitative behavior of numerical traveling solutions for reaction–diffusion equations with memory

In this article the qualitative properties of numerical traveling wave solutions for integro- differential equations, which generalize the well known Fisher equation are studied. The integro-differential equation is replaced by an equivalent hyperbolic equation which allows us to characterize the numerical velocity of traveling wave solutions. Numerical results are presented.

On the supraconvergence of elliptic finite difference schemes

Abstract This paper deals with the supraconvergence of elliptic finite difference schemes on variable grids for second order elliptic boundary value problems subject to Dirichlet boundary conditions in two-dimensional domains. The assumptions in this paper are less restrictive than those considered so far in the literature allowing also variable coefficients, mixed derivatives and polygonal domains. The nonequidistant grids we consider are more flexible than merely rectangular ones such that, e.g., local grid refinements are covered. The results also develop a close relation between supraconvergent finite difference schemes and piecewise linear finite element methods. It turns out that the finite difference equation is a certain nonstandard finite element scheme on triangular girds combined with a special form of quadrature. In extension to what is known for the standard finite element scheme, here also the gradient is shown to be convergent of second order, and so our result is also a superconvergence result for the underlying finite element method.

Memory effects and random walks in reaction-transport systems

In this article, we study continuous and discrete models to describe reaction transport systems with memory and long range interaction. In these models the transport process is described by a non-Brownian random walk model and the memory is induced by a waiting time distribution of the gamma type. Numerical results illustrating the behavior of the solution of discrete models are also included.

Qualitative analysis of a delayed non-Fickian model

This article focusses on the mathematical analysis of a delayed integro-differential model in which flux does not obey the classical Ficks law. The well-posedness of the integro-differential model in the Hadamards sense is established. The dependence on the delay parameter of the total amount of desorpted/sorpted mass is studied. Numerical results that show the effectiveness of the model are included.

Mathematical modeling of efficient protocols to control glioma growth

In this paper we propose a mathematical model to describe the evolution of glioma cells taking into account the viscoelastic properties of brain tissue. The mathematical model is established considering that the glioma cells are of two phenotypes: migratory and proliferative. The evolution of the migratory cells is described by a diffusion-reaction equation of non Fickian type deduced considering a mass conservation law with a non Fickian migratory mass flux. The evolution of the proliferative cells is described by a reaction equation. A stability analysis that leads to the design of efficient protocols is presented. Numerical simulations that illustrate the behavior of the mathematical model are included.

A priori estimates for the zeros of interval polynomials

The aim of this paper is to study the zeros of interval polynomials. The characterization of such zeros is given as a function of the interval coefficients and estimates for such zeros are established. Interval polynomials of degree two, three and four are considered.

Integro-differential models for percutaneous drug absorption

In this paper we propose new mathematical models for percutaneous absorption of a drug. The new models are established by introducing, in the classical Ficks law, a memory term being the advection–diffusion equations of the classical models replaced by integro-differential equations. The well-posedness of the models is studied with Dirichlet, Neumann and natural boundary conditions. Methods for the computation of numerical solutions are proposed. Stability and convergence of the introduced methods are studied. Finally, numerical simulations illustrating the behaviour of the model are included.

Numerical simulation of aqueous humor flow: From healthy to pathologic situations

A mathematical model which simulates drug delivery through the cornea, from a therapeutic lens to the anterior chamber of the eye, is proposed. The model consists of three coupled systems of partial differential equations linked by interface conditions: drug diffusion in the therapeutic lens; diffusion and metabolic consumption in the cornea; diffusion, convection and metabolic consumption in the anterior chamber of the eye. The dependence of intraocular pressure on the obstruction of the trabecular mesh and the production rate of aqueous humor by the ciliary body is modeled. The therapeutic effects of drugs that act on the trabecular mesh or on the ciliary body are analysed. Comparisons between topical administration and drug delivery from a therapeutic lens are included.

On the convergence on nonrectangular grids

The convergence properties of a full discrete approximations to the convection-diffusion equation is the subject of this paper. The full discrete scheme considered is of Lagrangian type: Euler Implicit on time and centered finite difference on space, and is defined using nonrectangular grids. We analyse this scheme under smoothness conditions on nonrectangular space-time grid. The main result establish the convergence of the approximations and we prove that the assumptions on the discrete spatial nodes movement are achieved if we consider the equidistribution principle.

Analytical and numerical study of a coupled cardiovascular drug delivery model

A two dimensional coupled model of drug delivery in the cardiovascular tissue using biodegradable drug eluting stents is developed. Qualitative behavior, stability analysis as well as simulations of the model have been presented. Numerical results computed with an implicit-explicit finite element method show a complete agreement with the expected physical behavior.