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1

On a Robin (p, q)-equation with a logistic reaction

Papageorgiou Nikolaos S., Vetro Calogero, Vetro Francesca

Opuscula Mathematica

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2019

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Vol. 39, no. 2

227--245

EN

We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a p-Laplacian and of a q-Laplacian ((p,q)-equation) plus an indefinite potential term and a parametric reaction ol logistic type (superdiffusive case). We prove a bilurcation-type result describing the changes in the set ol positive solutions as the parameter λ > 0 varies. Also, we show that lor every admissible parameter λ > 0, the problem admits a smallest positive solution. Keywords: positive solutions, superdiffusive reaction, local minimizers, maximum principle, min

2

Fractional heat conduction in a sphere under mathematical and physical Robin conditions

Kukla S., Siedlecka U.

Journal of Theoretical and Applied Mechanics

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2018

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Vol. 56 nr 2

339--349

EN

In this paper, the effect of a fractional order of time-derivatives occurring in fractional heat conduction models on the temperature distribution in a composite sphere is investigated. The research concerns heat conduction in a sphere consisting of a solid sphere and a spherical layer which are in perfect thermal contact. The solution of the problem with a classical Robin boundary condition and continuity conditions at the interface in an analytical form has been derived. The fractional heat conduction is governed by the heat conduction equation with the Caputo time-derivative, a Robin boundary condition and a heat flux continuity condition with the Riemann-Liouville derivative. The solution of the problem of non-local heat conduction by using the Laplace transform technique has been determined, and the temperature distribution in the sphere by using a method of numerical inversion of the Laplace transforms has been obtained.

3

Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Akbari H.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 1

17--27

EN

We consider a general elliptic Robin boundary value problem. Using orthogonal Coifman wavelets (Coiflets) as basis functions in the Galerkin method, we prove that the rate of convergence of an approximate solution to the exact one is O(2-nN) in the H1 norm, where n is the level of approximation and N is the Coiflet degree. The Galerkin method needs to evaluate a lot of complicated integrals. We present a structured approach for fast and effective evaluation of these integrals via trivariate connection coefficients. Due to the fast convergence rate, very good approximations are found at low levels and with low Coiflet degrees, hence the size of corresponding linear systems is small. Numerical experiments confirm these claims.

4

Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment

Piechór K.

Archives of Mechanics

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2012

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Vol. 64, nr 5

477-477

EN

The model we consider treats a cell or a group of cells as a viscoelastic medium whose stress tensor has a term - the traction- representing the stresses generated in the medium by the actomyosin molecules. We consider three kinds of domains (“shapes” of cells): the thin circular cylinder mimicking a long cell, the thin slab being a cari-cature of a tissue, and the unbounded space. We assume that the viscous effects are much weaker than the elastic ones and consider two extreme cases: either the body force is negligible or it is strong. This leads to three pairs, one pair for each domain, of approximations for the dilatation. We interpolate between the approximated ex-pressions forming one pair and as the result we obtain a single calcium conservation equation and a system of buffer equations. Using the rapid buffering approximation we reduce the problem to a single reaction-diffusion equation. We study the travelling wave solutions to these equations. We show that not only the high affinity buffers but also the mechanical effects alone can prevent the formation and propagation of the waves if the supply of calcium is not sufficiently substantial.