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2 2020

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The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

Duru Hakkı, Güneş Baransel

Journal of Applied Mathematics and Computational Mechanics

2020

Vol. 19, nr 4

45--56

In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.

A study on efficiency of 3D partial differential diffusive model of presynaptic processes

Bielecki Andrzej, Gierdziewicz Maciej, Kalita Piotr

Biocybernetics and Biomedical Engineering

2020

Vol. 40, no. 1

100--110

The verification of efficiency of the diffusive-type model of neurotransmitter transport dynamics inside the presynaptic bouton, including the spatial aspect of the transport, is the main objective of the simulations described in this paper. Finite element and finite difference methods were applied to solve the model numerically. The bouton was represented by a 3D tetrahedral mesh. Some biological parameters have been taken from scientific literature whereas some other remain unknown. Therefore the simulations were performed for various values of the unknown parameters. Such an approach allowed us to assess the proper order of magnitude of those parameters by comparison of the dynamics of the process implied by the model with the observed post-synaptic potentials. The effect of synaptic depression has been captured in the simulations. The presented approach allowed us to verify to what extent the presynaptic transport can be explained by the diffusive mechanism. Furthermore, the sensitivity of the model to release rate and the synthesis rate is considered. Moreover, the algebraic models of the diminishment of released neurotransmitter amounts were proposed and tested, showing a good accuracy. Various geometrical aspects of the dynamics were studied. In particular, the influence of the geometry of the synthesis region was examined. In the paper, the aspects of numerical simulations are studied as well. Namely, the quality of the generated mesh was discussed. The results of the simulations were compared with the results of measurements and observations and will be later contrasted with more complex models.