-second order convergent estimates for non-Fickian models

Barbeiro, Sílvia; Ferreira, J. A.; Pinto, Luís

doi:10.1016/j.apnum.2010.09.005

Cited by 14 publications

(16 citation statements)

References 32 publications

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“…In fact, in iontophoresis or electroporation applications, electric fields are used to enhance drug diffusion and absorption by the target tissue. The wave equations governing the electric potential and the electric field, equations (4) and (5), respectively, are particular cases of the general equation (1). In future work we intent to address this more complex problem which is obtained coupling a equation of type (1) with a properly defined parabolic equation for the drug concentration.…”

Section: Discussionmentioning

confidence: 99%

“…Without being exhaustive we refer to [6,14,20]. However, to obtain an accurate description of the drug evolution in a more general setting, it is necessary to construct an accurate approximation for the electric potential V defined by (4) or electric field defined by (5). It is desirable to compute a second order approximation for the gradient of the potential with respect to a discrete L 2 -norm, that is, a second order approximation for the potential with respect to a discrete H 1 -norm.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, to reduce the smoothness assumptions on the solution of the IBVP (1)-(3), we consider the approach introduced in [4] for one dimensional problems and in [16] for two dimensional elliptic equations and consider later in different contexts: in [5,17,19] for non-Fickian diffusion problems, and in [18] for diffusion problems in porous media. Avoiding the analytical difficulties that arise from the application of this technique we prove the same convergence result provided that u…”

Section: Introductionmentioning

confidence: 99%

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Second order approximations for kinetic and potential energies in Maxwell's wave equations

Ferreira

Jordão

Pinto

2017

Applied Numerical Mathematics

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Second order approximations for kinetic and potential energies in Maxwell's wave equations

Ferreira

Jordão

Pinto

2017

Applied Numerical Mathematics

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“…where M is the number of observations, u m is the measured concentration at time t m , and u m h is the estimated concentration at time t m . The minimization of the RMSE for Equations (1) and (10) was obtained using the freely available CTRW toolbox [30], while for Equation (9), we used the built-in routines of Matlab (version 7.9.0 (R2009b)), where the approximation u m h is given by the onedimensional version of the numerical method studied in [20,24], and [25]. It should be stressed that the outlet boundary condition is imposed far enough from the grid point where the BTC is evaluated.…”

Section: Breakthrough Curve Analysismentioning

confidence: 99%

“…The mathematical and numerical analyses of initial boundary value problems (IBVPs) based on integro-differential equations of this type were studied, for example, in [20][21][22][23][24][25] and [26]. The ability of the integro-differential model (IDM) (9) to capture the dynamics of tracer transport has already been tested by the authors in [27] by fitting the model to experimental breakthrough curves (BTCs).…”

Section: Introductionmentioning

confidence: 99%

An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

Ferreira

Pinto

2015

Math Methods in App Sciences

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Communicated by J. Vigo-AguiarDiffusion 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.where the total mass flux J is given by where J adv is the flux due to convective transport J adv D vu, and by assuming that the diffusion-dispersion mass flux J dis satisfies the so-called Fick's law,Equation (1), being of the parabolic type, induces a pathologic behavior in the concentration u, namely, it presents infinite speed of propagation, which has no physical meaning. Other limitations associated with the definition of D.v/ have also been reported [6][7][8].Regarding the use of such an equation to simulate tracer transport, several gaps have been observed when simulation results were compared with laboratory experiments. These observations have been extensively reviewed [9][10][11][12][13][14][15][16][17][18].Several approaches have been developed in order to overcome the limitations associated with Equation (1); we refer, for example, to [7] and [8]. A common approach consists of the introduction of a hyperbolic or non-Fickian correction term. A particular model of this type can be obtained by assuming that the diffusion-dispersion mass flux J dis satisfies the following differential equation:where is a delay parameter [19]. Note that the left-hand side of Equation (5) is a first-order approximation of the left-hand side of J dis .x, t C / D D.v/ru.x, t/, which means that the dispersion mass flux at the point x and time t C depends on the gradient of the concentration at the same point but at a delayed time. Equation (5) leads towith the kernel term K er .t/ D 1 e t , provided that J dis .0/ D 0. Combining the partition (4) with Equation (3), where J dis is given by Equation (6), we obtain the integro-differential equationMoreover, if we assume that the diffusion-dispersion mass flux has both a Fickian component J F , and a non-Fickian component J NF , defined byandThe intent of this section is...

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Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

Kumar

Sista

Sreenadh

2020

Math Methods in App Sciences

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The aim of this paper is to study parabolic integro‐differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.

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-second order convergent estimates for non-Fickian models

Cited by 14 publications

References 32 publications

Second order approximations for kinetic and potential energies in Maxwell's wave equations

Second order approximations for kinetic and potential energies in Maxwell's wave equations

An integro‐differential model for non‐Fickian tracer transport in porous media: validation and numerical simulation

Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

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