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Travelling Waves in Partially Degenerate Reaction-Diffusion Systems

B. Kazmierczak, V. Volpert (2010)

Mathematical Modelling of Natural Phenomena

We study the existence and some properties of travelling waves in partially degenerate reaction-diffusion systems. Such systems may for example describe intracellular calcium dynamics in the presence of immobile buffers. In order to prove the wave existence, we first consider the non degenerate case and then pass to the limit as some of the diffusion coefficient converge to zero. The passage to the limit is based on a priori estimates of solutions independent of the values of the diffusion coefficients....

Triangular stochastic matrices generated by infinitesimal elements

Inheung Chon, Hyesung Min (1999)

Czechoslovak Mathematical Journal

We show that each element in the semigroup S n of all n × n non-singular upper (or lower) triangular stochastic matrices is generated by the infinitesimal elements of S n , which form a cone consisting of all n × n upper (or lower) triangular intensity matrices.

Tumour angiogenesis model with variable vessels' effectiveness

Jan Poleszczuk, Iwona Skrzypczak (2011)

Applicationes Mathematicae

We propose a model of vascular tumour growth, which generalises the well recognised model formulated by Hahnfeldt et al. in 1999. Our model is based on the same idea that the carrying capacity for any solid tumour depends on its vessel density but it also incorporates vasculature quality which may be lost during angiogenesis as recognised by Jain in 2005. In the model we assume that the loss of vessel quality affects the diffusion coefficient inside the tumour. We analyse basic mathematical properties...

Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

Svatoslav Staněk (2013)

Open Mathematics

We investigate the fractional differential equation u″ + A c D α u = f(t, u, c D μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives....

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