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Asymptotic stability of solutions for a diffusive epidemic model

Bouaziz Khelifa, Douaifia Redouane, Abdelmalek Salem

Demonstratio Mathematica

2022

Vol. 55, nr 1

553--573

The aim of this paper is to study the existence and the asymptotic stability of solutions for an epidemiologically emerging reaction-diffusion model. We show that the model has two types of equilibrium points to resolve the proposed system for a fairly broad class of nonlinearity that describes the transmission of an infectious disease between individuals. The model is analyzed by using the basic reproductive number R0 . Finally, we present the numerical examples simulations that clarifies and confirms the results of the study throughout the paper.

Development of adaptive methods for reaction-diffusion and other transport problems arising in electrochemistry

Bieniasz L.

Computer Methods in Materials Science

2009

Vol. 9, No. 2

296-301

Development of adaptive methods for reaction-diffusion and other transport problems arising in electrochemistry Lesław K. Bieniasz Institute of Physical Chemistry of the Polish Academy of Sciences, Department of Complex Systems and Chemical Processing of Information, ul. Niezapominajek 8, 30-239 Cracow, Poland. Tel. (+48 12) 639 52 12, Fax. (+48 12) 425 19 23, E-mail: nbbienia@cyf-kr.edu.pl, URL: http://www.cyf-kr.edu.pl/~nbbienia, and Institute of Teleinformatics, Faculty of Electrical and Computer Engineering, Cracow University of Technology, ul. Warszawska 24, 31-155 Cracow, Poland. Computational modelling of reaction-diffusion and other reactive transport phenomena presents a challenging task in many areas of science and technology related to chemistry and biology, materials science not being excluded. Problems of this kind become particularly difficult to solve when the governing equations (for example partial or ordinary differential equations) are singularly perturbed, so that their solutions possess local layers or moving fronts. In such cases, adaptive methods that detect such difficult local solution structures, and appropriately concentrate the computational effort on resolving them, are necessary. Another reason for developing the adaptive methods is the modern trend towards automation of computational procedures: users of the simulation software want to obtain solutions having a guaranteed prescribed accuracy, independently of the location, extension and duration of the local solution structures, which may well not be known a priori. For the past 15 years, the present author has been developing finite-difference adaptive approaches to the numerical solution of reaction-diffusion and other reactive transport equations occurring in electrochemistry. The work concentrated on initial boundary value problems for partial differential equations in one-dimensional geometry [1-17], boundary value problems for ordinary differential equations [18-19], and recently on integral equations [20]. In the present communication the results of this work will be briefly summarized. Experiments with the patch-adaptive grid strategy [1-17] and with the local grid node insertion/deletion [18-19] will be used to demonstrate the advantages and disadvantages of the various methods. Some conclusions of potential interest to modellers in other areas will be attempted. In particular, it will be argued that much more work still has to be done to design satisfactory methods, even for spatially one-dimensional equations, despite the fact that for such problems the adaptive methodology is currently regarded to be mature.

W ciągu ponad 15 lat pracy autora nad rozwojem adaptacyjnych metod różnic skończonych dla zagadnień reakcji-dyfuzji oraz innych zjawisk transportu reakcyjnego występujących w elektrochemii nagromadziło się wiele doświadczeń, które mogą być interesujące dla badaczy zajmujących się modelowaniem w innych dziedzinach, łącznie z nauką o materiałach. Wyniki tych badań zostaną krótko przedstawione, ze wskazaniem na wady i zalety różnych metod. Przedstawione zostaną argumenty na rzecz tezy, że potrzeba znacznie więcej pracy aby zaprojektować zadowalające metody adaptacyjne, nawet dla równań w przestrzeni jednowymiarowej, mimo iż obecnie uważa się, że metodologia adaptacyjna dla takich problemów osiągnęła stan dojrzały.