An adaptive Huber method with local error control, for the numerical solution of the first kind Abel integral equations

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

“…Computational details referring to the programming language, machine accuracy, and computer environment, have been the same as in former studies [4][5][6][7][8][9][10]. All results presented here refer to u = 20, and method parameters [6]: starting integration step h start = 10 À2 , and maximum integration step h max = 1.…”

“…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:…”

“…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:…”

“…As the series can be effectively summed up only for t < u, all errors reported below (together with related quantities such as the number N of integration steps and computational time ct), refer to simulations performed within t < u. Figure 3 demonstrates the proportionality of the largest modulus of the absolute true errors of c(t), to the absolute error tolerance parameter tol of the adaptive algorithm [4][5][6][7][8][9][10]. The proportionality implies that a requested accuracy is indeed well achieved by the adaptive Huber method, in the whole range of a, except for the a values close to zero.…”

“…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…”