P. de Oliveira

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.

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.

Two adaptive grid methods for fixed bed systems simulation

Abstract In this paper, two adaptive gridding algorithms, based on the Method of Lines, are proposed for the numerical solution of the PDE which describes the energy balance of a fixed bed system. A standard time-step control of the stiff solver—based on Richardson extrapolation—is coupled, in both algorithms, with a spatial adapting strategy. In one of the algorithms, this procedure consists of increasing the number of grid points where a certain measure of the error indicates that refinement is needed. In the other, the locally refined solutions are used to compute approximations for the spatial truncation error which are injected in the discretized equation. Both algorithms lead to highly accurate solutions on relatively coarse grids and present substantial savings in computing time and memory.

A converging finite volume scheme for hyperbolic conservation laws with source terms

In this paper we study convergence of numerical discretizations of hyperbolic nonhomogeneous scalar conservation laws. Particular attention is devoted to point source problems. Standard numerical methods, obtained by a direct discretization of the differential form, fail to converge, even in the linear case. We consider the equation in integral form in order to construct a class of convergent accurate methods. Numerical examples are included.

Reaction-diffusion in viscoelastic materials

In this paper we study initial boundary value problems that describe reaction-diffusion phenomena in viscoelastic materials. The mathematical model, represented by an integro-differential equation coupled with an ordinary differential equation, is analyzed from theoretical and numerical viewpoints.

Simulation of the transient behaviour of a digester used in the pulp and paper industry

Abstract The development of mathematical models that describe industrial processes is playing an increasing role in industrial context, because such models can replace, in some cases, experimental simulation, in a more inexpensive and flexible way. In this paper is presented a preliminary study of a transient model of a continuous moving bed reactor - the digester - used in the pulp and paper industry. In this complex and heterogeneous digester a moving bed of wood chips, containing cellulose, hemicellulose and lignin, reacts with sodium hydroxide and sodium sulfide - dissolved in a liquid phase - to remove the lignin from the cellulose fibers. The mathematical model studied here has been developed by Fernandes in a forthcoming Ph.D Thesis. The model is derived from the fundamental principles of mass and energy balance and is represented by a system of 15 partial differential equations (PDE) of convection-reaction type. The numerical methods used in the discretization of this PDE system are based on operator splitting which essentially consists in considering separately convection and reaction phenomena. This approach allows the use of methods with different properties - explicitness, implicitness and order - for each subprocess. The final global class of methods represents in some sense a “patching of tailored methods” well adapted to the feature of individual phenomena. Numerical simulations of the concentrations of organics and inorganics are presented.

Some simple characterizations of contractivity regions

Abstract We present necessary and sufficient conditions for a multiderivative multistep method to be contractive in a sector, and in disk of the complex plane.

Transdermal iontophoresis—A quantitative and qualitative study

Abstract The use of enhancers to increase the drug molecules penetration into target tissues is a usual technique in drug delivery. In transdermal drug delivery, electric fields are often used to increase the drug transport through the skin. In this paper we study a drug delivery mechanism from a reservoir which is in contact with the skin. We assume that the drug transport in the coupled system is enhanced by a small electric field that induces a convective field. We establish energy estimates for the coupled system and we propose a semi-analytical discrete coupled model that mimics the continuous model. The qualitative behaviour of the system is illustrated.

Monotonicity on nonuniform grids

In this paper we study a class of numerical methods used to solve two-point boundary value problems on nonuniform grids. Particular attention is devoted to positive solutions, i.e. conditions under which the solutions of the problem are positive. Applications to steady states of air pollution problems are also referred to.