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

Abstract

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.

An adaptive Huber method for nonlinear systems of Volterra integral equations with weakly singular kernels and solutions

“…Figure 4 demonstrates that for the examples studied the convergence order is very close to 2 (for first kind VIEs it is practically equal 2; for second kind VIEs it seems to be a bit closer to 1.8). This result is again in accord with previous findings for nonsingular solutions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] (cf., in particular, discussions of Fig. 5 in Ref.…”

“…5, reveal that the computing time ct needed for solving examples 1-9 varies between about 10 −2 second, and about 10 3 seconds, when tol varies between 10 −3 and 10 −6 − 10 −8 , depending on the examples and equation parameters. This is somewhat more than was previously observed for nonsingular solutions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], but the previous calculations were on a faster computer. The present timings may also not be directly comparable with the previous ones, because of a number of modifications introduced into the adaptive code, referring to the data structures used and C++ class hierarchies.…”

“…[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] relies on replacing the unknown solutions U (t) by piecewise linear spline functions of t, using a dynamically selected grid of nodes t n (n = 0, 1, . .…”

Automatic simulation of electrochemical transients assuming finite diffusion space at planar interfaces, by the adaptive Huber method for Volterra integral equations

“…Years ago, the present author started a long-term research programme [4], aimed at automating theoretical modelling and simulation procedures of electro-analytical chemistry [5,6], and creating related problem-solving software [7,8]. In particular, in a series of recent studies [9][10][11][12][13][14][15][16][17][18][19][20] an adaptive method has been developed for simulating controlled-potential transient electrochemical experiments [21] by means of the classical integral equation (IE) approach [22]. The IE approach is semianalytical.…”

A Review of Non-Cottrellian Diffusion Towards Micro- and Nano-Structured Electrodes

“…In order to enable automatic simulation of electrochemical transient experiments performed under conditions of anomalous diffusion in the framework of the formalism of integral equations, the adaptive Huber method has been extended onto integral transformation kernel representing fractional diffusion (Bieniasz, 2011).…”