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The skin is the largest organ of the human body, offering an accessible interface for the administration of drugs. The main obstacle to transdermal drug delivery is the skin's barrier properties, d...

Convection enhanced drug delivery (CED) is a technique used to make therapeutic agents reach, through a catheter, sites of difficult access. The name of this technique comes from the convective flow originated by a pressure gradient induced at the tip of the catheter. This flow enhances passive diffusion and allows a more efficient spread of the agents by the target site. CED is particularly useful in the treatment of diseases that affect the central nervous system, where the blood-brain barrier prevents the diffusion of most therapeutic agents from the cerebral blood vessels to the brain interstitial space. In this work we deal with the numerical analysis of a coupled system of partial differential equations that can be used to simulate CED in an elastic medium like brain tissue. The model variables are the fluid velocity, the pressure, the tissue deformation, and the agents concentration. We prove the stability of the coupled problem and from the numerical point of view we propose a fully discrete piecewise linear finite element method (FEM). The convergence analysis shows that the method has second order convergence for the pressure, displacement, and concentration. Numerical experiments illustrating the theoretical convergence rates and the behavior of the system are also given. UIDB/00324/2020; UIDB/04621/2020 info:eu-repo/semantics/publishedVersion

Following the implantation of indwelling medical devices, bacteria inoculated during the surgery or coming from a preexistent focus of infection race for the medical surface where they attach. Adaptation to survive is a common feature of life, and microorganisms are not an exception. Bacteria form, in short periods of time, a habitat-the biofilm-where they develop multiresistance and tolerance to antibiotics and to the host immune system. To avoid its formation, researchers in the biomedical sciences showed evidence that coating medical devices with antibacterial agents- antibiotics-is a promising strategy. We present a mathematical model to simulate the action of an antibiotic, released from a medical surface, to fight bacterial infection. The model is composed by a system of partial differential equations that describe the distribution of drug and the evolution of a bacterial population. The preexistence of infection focus, the inoculation of bacteria during the surgery, the race for the medical surface, the resistance and tolerance of the population are taken into account. Analytical estimates of the bacterial density show the crucial importance of aseptic surgical procedures and of timely detection of preexisting infection focus. Numerical simulations illustrate several scenario.

The ability of numerical models to deal with wave breaking processes and dry areas is of paramount importance for applications in coastal zones and dam breaks. The mathematical models commonly used in such real problems are usually based on Boussinesq-type equations and, to a small extend, on Serre equations. However, these standard models are weakly dispersive and must be appropriately modified to deal with breaking waves and dry areas. Indeed, nearshore and dam break problems involve complex wave dynamics and highly dispersive wave processes can easily arise. In those cases, it is well known that weakly dispersive models like the ones based on classical Boussinesq or Serre equations are unreliable for an accurate simulation of the phenomena involved. In this work we extend the applicability of an improved Serre model, herein denoted by Serre α , β , to include the wave breaking process, the broken waves propagation, and dry areas. We provide a comprehensive set of numerical examples involving wave propagation over exposed and submerged structures, as well as dam break problems. The numerical experiments show the accuracy and robustness of the proposed model. Particular attention is given to bottom friction modeling, where the standard Manning’s assumption is compared with a more realistic formulation. Also noteworthy is the simulation of wave breaking problems with highly dispersive effects. The advantages of the Serre α , β model over the standard Serre model for these challenging cases are clear.

In this paper, we study a system of partial differential equations defined in a moving domain. This system is defined by a heat equation and a diffusion equation for a concentration of non-Fickian type whose diffusion coefficient depends on the temperature, completed with suitable initial and boundary conditions. The non-Fickian mass flux is established considering the viscoelastic properties of the medium where the strain depends on the temperature and on the concentration. The initial boundary value problem (IBVP) analyzed can be used to describe the drying of viscoelastic materials where the internal structure offers a resistance to the movement of the moisture molecules and a consequent delay in the moisture removal. Due to heat transference into the materials and moisture removal, shrinkage of the medium occurs. The stability of the IBVP defined in a moving domain is analyzed and its qualitative behavior is numerically studied.

The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serreα, β, presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios. A careful development of the Serreα, β model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serreα, β model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.

We consider a polymeric spherical platform containing a solid dispersed drug that is in contact with a solvent fluid. While swelling, a non-Fickian sorption of the solvent molecules occurs induced by the effect of the viscoelastic properties of the polymer. The solid drug in contact with the solvent fluid dissolves and a Fickian release of dissolved drug takes place. The fluid entrance, the drug dissolution, and the drug release to an external environment are described by a system of PDEs complemented with an equation for the swelling front, initial, and boundary conditions. The model includes the two major factors that govern a swelling process of a polymeric platform within a release medium: the cross-link density and the concentration of the external medium. Energy estimates for the mass of solvent fluid and of undissolved and dissolved drug in the polymeric platform are established. Numerical simulations that illustrate the theoretical results are also included.

Efficacious drug delivery to the posterior chamber of the eye is a very challenging problem due to the many physiological barriers that protect the eye against the entry of exogenous substances. To...

In this paper we consider general linear damped wave equations with memory. We establish energy estimates that under the assumption of exponentially bounded kernels, induce exponential decaying solutions. Numerical waves that mimic their continuous counterpart are also introduced using a finite element approach.

The main motivation of the present work is the numerical study of a system of Partial Differential Equations that governs drug transport, through a target tissue or organ, when enhanced by the simultaneous action of an electric field and a temperature rise. The electric field, while forcing charged drug molecules through the tissue or the organ, thus creating a convection field, also leads to a rise in temperature that affects drug diffusion. The differential system is composed by a nonlinear elliptic equation, describing the potential of the electric field, and by two parabolic equations: a diffusion–reaction equation for temperature and a convection–diffusion–reaction for drug concentration. The temperature and the concentration equations are coupled with the potential equation via a reaction term and the convection and diffusion terms respectively. As the parabolic equations depend directly on the potential and its gradient, the central question is the design and mathematical study of an accurate method for the elliptic equation and its gradient. We propose a finite difference method, which is equivalent to a fully discrete piecewise linear finite element method, with superconvergent/supercloseness properties. The method is second order convergent with respect to a H 1 -discrete norm for the elliptic problem, and with respect to a L 2 -discrete norm for the two parabolic problems. The stability properties of the method are also analyzed. Numerical experiments illustrating the drug transport for different electrical protocols are also included.

The aim of this paper is the mathematical study of the interactions between bacterial populations, materials they colonize, and drugs delivered from surfaces where they adhere. Bacteria can cause infections, which are common events in different types of medical implants as, for example, orthopedic prosthesis, and are often responsible for rejection. A controlled drug delivery to fight bacterial adhesion is crucial in reducing infection rates. A strategy recently adopted to address the problem is to deliver therapeutic agents locally by dispersing them into polymeric implant coatings. The mathematical model is composed of a system of three partial differential equations that describe the drug release from a biodegradable polymeric coating and by an ordinary differential equation that governs the density of a bacterial population. The link between the system of partial differential equations and the ordinary differential equation is defined by an integral that represents the mass of drug that is released by the polymeric coating at time t. Quasi-sharp estimates for the bacterial density that give insight into its dependence on the polymeric properties and the drug characteristics are established. Numerical experiments illustrating the behavior of the density of bacteria, depending on the characteristics of the drug-polymeric coating system, are included.

The aim of this paper is to propose an iterative method to compute the dominant zero of a quaternionic unilateral polynomial. We prove that the method is convergent in the sense that it generates a sequence of quaternions that converges to the dominant zero of the polynomial. The idea subjacent to this method is the well known Sebastiao e Silva’s method, proposed in “Sur une methode d’approximation semblable a celle de Graffe”, Portugaliae Mathematica, 1941, to approximate the dominant zero of complex polynomials.

Diffusion processes have traditionally been modeled using the classical parabolic advection-diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non-Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro-differential models (IDMs) have been also proposed to describe non-Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection-diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two-dimensional IDM composed by a integro-differential equation for the concentration and Darcy's law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh. Copyright © 2015 John Wiley & Sons, Ltd.

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.

In this paper diffusion through a viscoelastic biodegradable material is studied. The phenomenon is described by a set of three coupled partial differential equations that take into account passive diffusion, stress driven diffusion and the degradation of the material. The stability properties of the model are studied. Erodible viscoelastic materials, as biodegradable polymers, have a huge range of applications in medicine to make drug eluting implants. Using the mathematical model the behavior of a particular ocular drug eluting implant which describes drug delivery into the vitreous chamber of the eye is presented. The model consists of coupled systems of partial differential equations linked by interface conditions. The chemical structure, the viscoelastic properties and the diffusion in the implant as well as the transport in the vitreous are taken into account to simulate the evolution in vivo of released drug. The dependence of the delivery profile on the properties of the material is addressed. Numerical simulations that illustrate the interplay between these phenomena are included.

Diffusion processes are usually simulated using the classical diffusion equation. In certain scenarios, such equation induces anomalous behavior and consequently several improvements were introduced in the literature to overcome them. One of the most popular was the replacement of the diffusion equation by an integro-differential equation. Such equation can be established considering a modification of Fick's mass flux where a delay in time is introduced. In this article, we consider mathematical models for diffusion processes that take into account a memory effect in time and space. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1589–1602, 2015

Simulations show that the kinetics of permeant fluids in viscoelastic matrices depends on the rheological and chemical properties of the material. Fick's law fails to describe transport through viscoelastic materials because of the stress exerted on the incoming fluid which causes a delay. Reversible binding to immobilizing sites also retards permeation of molecules. The effects of mechanical properties and chemical affinities of materials on the transport of solutes are studied. An integro-partial-differential equation is used to model the transport. While the differential part of the equation is represented by an elliptic operator, the integral part describes the contributions of stress and reversible binding. The stability of the model is investigated. The steady-state flux and effective time constant are calculated. The lag time is also studied using multiple integration. Subsequent analyses reveal the dependence of the steady-state flux, the effective time constant, and the lag time on the Young modulus, the viscosity, and the binding/unbinding rates. The results presented in this paper make it possible to tune the mechanical and chemical properties to achieve a desired transport profile.

We present a multi-layer mathematical model to describe the the transdermal drug release from an iontophoretic system. The Nernst-Planck equation describe the basic convection-diffusion process, with the electric potential obtained by Laplace equation. These equations are complemented with suitable interface and boundary conditions in a multi-domain. The stability of the mathematical problem is discussed in different scenarios and a finite-difference method is used to solve the coupled system. Numerical experiments are included to illustrate the drug dynamics under different conditions., Comment: In Mathematical Medicine and Biology, online, 2016

In this paper the effect of plaque composition, on the accumulation of drug released by a drug eluting stent, is analyzed. The mathematical model is represented by two coupled systems of partial differential equations that describe the pharmacokinetics of drug in the stent coating and in the arterial wall. The influence of the stiffness and porosity of soft and hard plaques is studied. A case study based on optical coherence tomography images is also included.

In this paper we propose a numerical scheme to approximate the solution of a non-Fickian coupled model that describes, e.g., miscible transport in porous media. The model is defined by a system of a quasilinear elliptic equation, which governs the fluid pressure, and a quasilinear integro-differential equation, which models the convection–diffusion transport process. The numerical scheme is based on a conforming piecewise linear finite element method for the discretization in space. The fully discrete approximations is obtained with an implicit–explicit method. Estimates for the continuous in time and the fully discrete methods are derived, showing that the numerical approximation for the concentrations and the pressure are second order convergent in a discrete $$L^2$$ -norm and in a discrete $$H^1$$ -norm, respectively.

In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L 2 -norm.

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.

The dynamics of diffusive and stress-induced transport in polymeric delivery systems was investigated. Partial and ordinary differential equations were first written to describe drug release behaviors in Maxwell and Maxwell–Voigt materials. The time constants governing the flux and concentration responses of a permeating species were determined from a Laplace transform solution of the original model. A “tracking strategy”, based on the estimated characteristic times, was proposed to estimate the delivery rate and the concentration near the exit side of the membrane. The methodology was more efficient at times greater than the time constant and the prediction error decreased further as the process approached steady state. Numerical illustrations and comparisons made with published data show the effectiveness of the proposed approach in describing the influence of the Young modulus, viscosity and relaxation time on the transient regime.

In this paper we study numerical methods for integro-differential initial boundary value problems that arise, naturally, in many applications such as heat conduction in materials with memory, diffusion in polymers and diffusion in porous media. Here, we propose finite difference methods to compute approximations for the continuous solutions of such problems. We analyze stability and study convergence for those methods. Supraconvergent estimates are obtained. As such methods can be seen as lumped mass methods, our supraconvergent result corresponds to a superconvergent property in the context of finite element methods. Numerical results illustrating the theoretical results are included.

The Fisher's equation is established combining the Fick's law for the flux and the mass conservation law with a reaction term evaluated at the present time. If this term depends on the solution at some past time, a delay parameter is introduced and the delay Fisher's equation is obtained. Modifying the Fick's law for the flux considering a time memory term, integro-differential equations of Volterra type are established. In this paper we study reaction-diffusion equations obtained combining the two modifications: a time memory term in the flux and a delay parameter in the reaction term. The delay integro-differential equations also known as delay Volterra integro-differential equations, are studied in the theoretical view point: stability estimates are established. Numerical methods which mimic the theoretical models are analysed. Numerical experiments illustrating the established results are also included.

This article focusses on the mathematical analysis of a delayed integro-differential model in which flux does not obey the classical Fick's law. The well-posedness of the integro-differential model in the Hadamard's 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.

A two dimensional coupled nonlinear non-Fickian model for drug release from a biodegradable drug eluting stent into the arterial wall is studied. The influence of porosity and degradation of the polymer as well as the dissolution rate of the drug are analyzed. Numerical simulations that illustrate the kind of dependence of drug profiles on these properties are included.

A coupled non-Fickian model of a cardiovascular drug delivery system using a biodegradable drug eluting stent is proposed. Energy estimates are used to study the qualitative behaviour of the model. The numerical results are obtained using an IMEX finite element method. The influence of vessel stiffness in the sorp- tion of drug eluted from the stent is analyzed. The results presented in this paper open new perspectives to adapt the drug delivery profile to the needs of the patient. Keywords: Non-Fickian coupled model, Cardiovascular drug delivery, Drug eluting stent, Numerical simulation. Mathematics Subject Classification (2010): 65M60, 92-08.

In this paper we propose new mathematical models for percutaneous absorption of a drug. The new models are established by introducing, in the classical Fick's 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. http://www.informaworld.com/10.1080/00207160701210091

In this paper we study numerical methods for solving integro-differential equations which generalize the well-known Fisher equation. The numerical methods are obtained considering the MOL (Method of Lines) approach. The stability and convergence of the methods are studied. Numerical results illustrating the theoretical results proved are also included.

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. http://www.informaworld.com/10.1080/00036810500048277

In this paper we study the convergence of a centred finite difference scheme on a non-uniform mesh for a ID elliptic problem subject to general boundary conditions. On a non-uniform mesh, the scheme is, in general, only first-order consistent. Nevertheless, we prove for s ∈ (1/2, 2] order O(h s )-convergence of solution and gradient if the exact solution is in the Sobolev space H 1+s (0, L),i.e. the so-called supraconvergence of the method. It is shown that the scheme is equivalent to a fully discrete linear finite-element method and the obtained convergence order is then a superconvergence result for the gradient. Numerical examples illustrate the performance of the method and support the convergence result.

The aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations. This nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem. The method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations. http://www.sciencedirect.com/science/article/B6TYH-4DTKSMT-1/1/3d2e49a86d0876f5ec457fcba0a29a24

In this paper a non linear mathematical model to describe absorption phenomena in polymers is proposed. The model is established assuming that the diffusing penetrant causes a deformation which induces a viscoelastic stress responsible for a convective field. This convective field is defined to represent an opposition of the polymer to the Fickian diffusion. Several numerical examples show the effectiveness of the model.

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.

The aim of this work is to present a nonstandard linear finite element method for a planar elasticity problem. The error for the solution computed with this method is estimated with respect to H1×H1-norm and second-order convergence is shown. http://www.sciencedirect.com/science/article/B6TYD-42MFD5J-5/1/b7fb435ba1f05aa7afa768ee9308a46a

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 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.

A three-dimensional mathematical model for sorption/desorption by a cylindrical polymeric matrix with dispersed drug is proposed. The model is based on a system of partial differential equations coupled with boundary conditions over a moving boundary. We assume that the penetrant diffuses into a swelling matrix and causes a deformation, which induces a stressdriven diffusion and consequently a non-Fickian mass flux. A physically sound nonlinear dependence between strain and penetrant concentration is considered and introduced in a Boltzmann integral with a kernel computed from a Maxwell–Wiechert model. Numerical simulations show how the mechanistic behavior can have a role in drug delivery design.

In this paper initial boundary value problems, defined using quasilinear diffusion equations of Volterra type, are considered. These equations arise for instance to describe diffusion processes in viscoelastic media whose behavior is represented by a Voigt–Kelvin model or a Maxwell model. A finite difference discretization defined on a general non-uniform grid with second order convergence order in space is proposed. The analysis does not follow the usual splitting of the global error using the solution of an elliptic equation induced by the integro-differential equation. The new approach enables us to reduce the smoothness required to the theoretical solution when the usual split technique is used. Non-singular and singular kernels are considered. Numerical simulations which show the effectiveness of the method are included.

The aim of this paper is to study a finite difference method for quasilinear coupled problems of partial differential equations that presents numerically an unexpected second order convergence rate. The error analysis presented allows us to conclude that the finite difference method is supraconvergent. As the method studied in this paper can be seen as a fully discrete piecewise linear finite element method, we conclude the supercloseness of our approximations.

In this paper attention is focused on convergence properties on nonuniform grids of numerical discretizations of first- and second-order spatial derivatives which occur, for example, in the transport and heat equations. The direct study of the discretization error equations lead us to the establishment of expressions for the global discretization error, which depend on the grid properties. From these expressions we conclude the convergence of the discretizations and also outline a grid selection strategy.

In certain cases, quasilinear convection-diffusion-reaction equations range from parabolic to almost hyperbolic, depending on the ratio between convection and diffusion coefficients. From a numerical point of view, two main difficulties can arise related to the existence of layers and/or the non-smoothness of the coefficients of such equations. In this paper we study the steady-state solution of a convection-dominated problem. We present a new numerical method based on the idea of solving an associated modified problem, whose solution corresponds to a lifting of the solution of the initial problem. The method introduced here avoids an a priori knowledge of the layer(s) location and allows an efficient handling of the lack of smoothness of the coefficients. Numerical simulations that show the effectiveness of our approach are included.

The classical convection-diffusion-reaction equation has the unphysical property that if a sudden change in the dependent variable is made at any point, it will be felt instantly everywhere. This phenomena violate the principle of causality. Over the years, several authors have proposed modifications in an effort to overcome the propagation speed defect. The purpose of this paper is to study, from analytical and numerical point of view a modification to the classical model that take into account the memory effects. Besides the finite speed of propagation, we establish an energy estimate to the exact solution. We also present a numerical method which has the same qualitative property of the exact solution. Finally we illustrate the theoretical results with some numerical simulations. http://www.sciencedirect.com/science/article/B6TYD-4S3G3SF-1/1/62545cd460b5e040aa2f285075df6b90

The pulp and paper industry plays an important role in European economies. The chemical reactions that transform wood chips in pulp occur mainly in a complex moving bed reactor, the digester. Nowadays the use of mathematical models to simulate the transient behaviour of the digester in terms of temperature and compound concentrations represents a real need for industry because it allows simulation of experiments that can not be afforded or that might be very risky. The digester - the most critical piece of the equipment of a pulp mill - is a heterogeneous reactor with an almost cylindrical shape, where wood chips react with an aqueous solution of sodium hydroxide and sodium sulfide, to remove the lignin from the cellulose fibers. From a mathematical point of view the dynamical behaviour of the reactor can be represented by a system of hyperbolic nonlinear partial differential equations. In this system, with 15 equations, we can identify three main types: the equations that describe the temperature and the concentration respectively of the solid, entrapped liquid and free liquid phase. Each of these type of equations present a certain complexity, its numerical simulation being a hard task. In this sense we point out the high nonlinearity of the functions that represent the chemical reactions; the discontinuities induced by the extraction and injection of the free liquor; the discontinuities in the convection velocity of the free liquor - positive where the liquid flown downwards and negative where the free liquid flows upwards. Numerical methods based on operator splitting, nonuniform refinement and some particular techniques to smooth discontinuities, are studied from a qualitative and quantitative viewpoint. Several simulations on temperature and concentrations of organic and inorganic compounds are presented. Special attention will be devoted to the effects induced in the process by discontinuities of wood chips composition. http://www.sciencedirect.com/science/article/B6TYC-4P2J0CT-1/1/eeeb9d8e5610d110621fbd0ad424387b

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. http://www.informaworld.com/10.1080/00036810601110638

In this paper we consider a hyperbolic equation, with a memory term in time, which can be seen as a singular perturbation of the heat equation with memory. The qualitative properties of the solutions of the initial boundary value problems associated with both equations are studied. We propose numerical methods for the hyperbolic and parabolic models and their stability properties are analysed. Finally, we include numerical experiments illustrating the performance of those methods. Centre for Mathematics of University of Coimbra

The aim of this paper is to study the role of explicitness, implicitness and order in the stability and qualitative properties of splitting methods for solving advection-reaction equations. Numerical pathologies produced by simulations are identified which allow the correction of wrong numerical reactive flows. Several numerical examples which show the effectiveness of our approach are presented.

In this paper we study the convergence properties of semi-discrete approximations for parabolic problems on two-dimensional polygonal domains. The semi-discretizations are obtained by using the non-standard piecewise linear finite element method that was introduced by Grigorieff and Ferreira (1998). Main features of that method are the superconvergence on certain non-uniform meshes, as well as that the usual strong coersivity condition of the associated bilinear form is relaxed. Moreover, the method is equivalent to a finite difference scheme that is, in turn, supraconvergent. Here, we will prove that all the properties that are of interest in the stationary case, are also present in the semi-discretizations of parabolic problems.

The aim of this paper is to study discretizations of convection-diffusion-reaction equations using splitting methods. Estimates for the physical splitting errors and the numerical splitting errors are established. These estimates lead to the selection of optimal sequences and coupling of physical phenomena and adequate use of implicitness and explicitness. Numerical simulations of two chemical industry problems are included.