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A finite difference method for fractional diffusion equations with Neumann boundary conditions

100%

Szekeres B. J. , Izsák F.

Open Mathematics

2015

tom 13

nr 1

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald–Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

Implicit a posteriori error estimation using patch recovery techniques

100%

Horváth T. , Izsák F.

Open Mathematics

2012

tom 10

nr 1

55-72

We develop implicit a posteriori error estimators for elliptic boundary value problems. Local problems are formulated for the error and the corresponding Neumann type boundary conditions are approximated using a new family of gradient averaging procedures. Convergence properties of the implicit error estimator are discussed independently of residual type error estimators, and this gives a freedom in the choice of boundary conditions. General assumptions are elaborated for the gradient averaging which define a family of implicit a posteriori error estimators. We will demonstrate the performance and the favor of the method through numerical experiments.

An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

80%

Faragó I. , Izsák F. , Szabó T. , Kriston Á.

Open Mathematics

2013

tom 11

nr 4

746-759

An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.

Stochastic cellular automata modeling of excitable systems

70%

Szakály T. , Lagzi I. , Izsák F. , Roszol L. , Volford A.

Open Physics

2007

tom 5

nr 4

471-486

A stochastic cellular automaton is developed for modeling waves in excitable media. A scale of key features of excitation waves can be reproduced in the presented framework such as the shape, the propagation velocity, the curvature effect and spontaneous appearance of target patterns. Some well-understood phenomena such as waves originating from a point source, double spiral waves and waves around some obstacles of various geometries are simulated. We point out that unlike the deterministic approaches, the present model captures the curvature effect and the presence of target patterns without permanent excitation. Spontaneous appearance of patterns, which have been observed in a new experimental system and a chemical lens effect, which has been reported recently can also be easily reproduced. In all cases, the presented model results in a fast computer simulation.