Initialisation of the adaptive Huber method for solving the first kind Abel integral equation

Bieniasz, Lesław K.

doi:10.1007/s00607-008-0020-9

Cited by 25 publications

(61 citation statements)

References 11 publications

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“…From the experimental point of view this response is uninteresting, and is normally ignored. For this reason, and also because the adaptive Huber method cannot compute singular solutions [4][5][6][7][8][9][10], we subtracted the singularity from c(t), by replacing Equation 14 by the IE:…”

Section: Resultsmentioning

confidence: 99%

“…Readers are referred to References [4][5][6][7][8][9][10] for details of the adaptive Huber method. The kernel-dependent part of the method is fully characterized by a number of coefficients, specifically:…”

Section: Methods Extensionmentioning

confidence: 99%

“…Specifically, we develop an extension of the recently described adaptive Huber method [4][5][6][7][8][9][10] for the automatic solution of the weakly singular Volterra integral equations (IEs) arising in the theory of controlled-potential transient methods such as, in particular, linear potential sweep and cyclic voltammetry [3]. In References [4][5][6][7][8][9][10] the discussion and testing of the adaptive Huber method was focused on IEs involving integrals…”

Section: Introductionmentioning

confidence: 99%

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Extension of the Adaptive Huber Method for Solving Integral Equations Occurring in Electroanalysis, onto Kernel Function Representing Fractional Diffusion

Bieniasz

2011

Electroanalysis

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Electroanalytical transient experiments performed under conditions of anomalous diffusion have recently attracted some attention. In order to enable automatic simulation of such experiments in the framework of the formalism of integral equations, the adaptive Huber method, recently elaborated by the present author, is extended onto integral transformation kernel function K(t,t) = (tÀt) a/2À1 (where 0 < a 1), representing fractional diffusion. The extended method is tested on a model integral equation describing cyclic voltammetry for a reversible charge transfer reaction. The performance of the method is found similar to the case of ordinary diffusion.

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Section: Resultsmentioning

confidence: 99%

Section: Methods Extensionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extension of the Adaptive Huber Method for Solving Integral Equations Occurring in Electroanalysis, onto Kernel Function Representing Fractional Diffusion

Bieniasz

2011

Electroanalysis

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“…(1), were obtained by modifying the procedures described in Refs. [8,12,13]. Hybrid algorithms, combining incomplete CR with LU decompositions were also tried, but we do not elaborate on them here, since they were not found more efficient under conditions of the present implementation.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For an overview of the literature related to the sequential algorithms of solving equations similar to Eq. (1), the Reader is referred to Bieniasz [12,13]. In contrast to those serial algorithms, the numerical algorithm to be described here is an adaptation of the CR method for tridiagonal matrices, first described by Hockney [14] and Buzbee et al [15] for block-tridiagonal matrices, and attributed to Golub and Hockney.…”

Section: Introductionmentioning

confidence: 99%

A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices

Bieniasz

2017

J Math Chem

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Extensions have been developed, of several variants of the stride of two cyclic reduction method. The extensions refer to quasi-tridiagonal linear equation systems involving two additional nonzero elements in the first and last rows of the equation matrix, adjacent to the main three diagonals. Equations of this kind arise, for example, in the simulations of biosensors or other electrochemical systems by solving relevant ordinary or partial differential equations by finite difference methods, when boundary derivatives are approximated by one-sided, multipoint finite differences. The correctness of the algorithms developed has been verified using example matrices with pseudo-random coefficients, under conditions of both sequential and parallel execution.

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Numerical Solution Methods

Bieniasz

2014

Monographs in Electrochemistry

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Initialisation of the adaptive Huber method for solving the first kind Abel integral equation

Cited by 25 publications

References 11 publications

Extension of the Adaptive Huber Method for Solving Integral Equations Occurring in Electroanalysis, onto Kernel Function Representing Fractional Diffusion

Extension of the Adaptive Huber Method for Solving Integral Equations Occurring in Electroanalysis, onto Kernel Function Representing Fractional Diffusion

A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices

Numerical Solution Methods

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