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

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.…”

“…It would also be a formidable task to derive a mathematical theory of all aspects of the present method. Therefore, an experimental route was taken, by continuing the approach adopted in previous works [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. As a result, a practical algorithm has been constructed, which is likely to work, provided that a number of relatively straightforward assumptions is satisfied.…”