Numerical methods for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation

Branco, J. R.; Ferreira, J. A.; Oliveira, Paula de

doi:10.1016/j.apnum.2006.01.002

Cited by 55 publications

(30 citation statements)

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“…In what concerns the numerical methods for the integro-differential problem we consider the semi-discretization (14)- (15) and the fully discrete scheme (25)-(26). The stability properties and the convergence of both discretizations were shown.…”

Section: Discussionmentioning

confidence: 99%

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Integro-differential models for percutaneous drug absorption

Barbeiro

Ferreira

2007

International Journal of Computer Mathematics

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Section: Discussionmentioning

confidence: 99%

“…The approach followed in this paper was considered in other contexts, for instance in reaction-diffusion equations with memory that were studied in [14,15], and in the study of the effect of memory terms in the heat equation which was analysed in [16].…”

Section: Bc(l T) = C(l T)mentioning

confidence: 99%

Integro-differential models for percutaneous drug absorption

Barbeiro

Ferreira

2007

International Journal of Computer Mathematics

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“…The theoretical and numerical analysis of the Volterra integro-differential equations (9) and (10) were consider for instance in [2]- [5], [8], [13], [15]- [17], [24], [30], [31], [33]. Existence results for the solution of IBVP defined with the retarded Volterra integro-differential equations (11) were established in [9], [10].…”

Section: Introductionmentioning

confidence: 99%

Non-Fickian delay reaction–diffusion equations: Theoretical and numerical study

Branco

Ferreira

Silva

2010

Applied Numerical Mathematics

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The Fisher's equation is established combining the Fick's law for the flux and the mass conservation law. Assuming that the reaction 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 temporal memory term, integro-differential equations of Volterra type were introduced in the literature.In these paper we study reaction-diffusion equations obtained combining the two modifications: a temporal memory term in the flux and a delay in the reaction term. The delay integro-differential equations, also known as delay Volterra integrodifferential equations, are studied in the theoretical view point: stability estimates are established. Numerical methods which mimic the theoretical models are studied. Numerical experiments illustrating the established results are also included.

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“…The previous deduction can be avoided defining the new flux by (7). In fact we can argue that the flux is not proportional to the gradient of the drug concentration but it is proportional to the "average in time" of the gradient of the concentration.…”

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Coupled vehicle–skin models for drug release

Barbeiro

Ferreira

2009

Computer Methods in Applied Mechanics and Engineering

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Please cite this article as: S. Barbeiro, J.A. Ferreira, Coupled vehicle-skin models for drug release, Comput. Methods Appl. Mech. Engrg. (2009), doi: 10.1016/j.cma.2009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT AbstractPercutaneous absorption of a drug delivered by a vehicle source is usually modeled by using diffusion Fick's law. In this case, the model consists in a system of partial differential equations of diffusion type with a compatibility condition on the transition boundary between the vehicle and the skin. Using this model, the fractional drug release in both componentsvehicle and skin -is proportional to the square root of the release time. Often experimental results show that the predicted drug concentration distribution in the vehicle and in the skin by the Fick's model does not agree with experimental data. In this paper we present a nonFickian mathematical model for the introduced percutaneous absorption problem. In this new model the Fick's law for the flux is modified by introducing a non-Fickian contribution defined with a relaxation parameter related to the properties of the components. Combining the flux equation with the mass conservation law, a system of integro-differential equations is established with a compatibility condition on the boundary between the two components of the physical model. The stability analysis is presented. In order to simulate the mathematical model, its discrete version is introduced. The stability and convergence properties of the discrete system are studied. Numerical experiments are also included.

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Numerical methods for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation

Cited by 55 publications

References 9 publications

Integro-differential models for percutaneous drug absorption

Integro-differential models for percutaneous drug absorption

Non-Fickian delay reaction–diffusion equations: Theoretical and numerical study

Coupled vehicle–skin models for drug release

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