The role of power decay, exponential decay and Mittag-Leffler function’s waiting time distribution: Application of cancer spread

Abstract

The use of fractional differential operators (Riemann–Liouvillend Caputo) to model real world problems leads to a waiting time distribution governs by the power law. This operators obeys the so-called index law of the classical mechanic and also possess an artificial singularity around the initial or starting point of the process. There are failures or limitations attached to the power-law fractional differential operator: its capacity of obeying the index-law leads waiting time distribution with no crossover behavior which is a limitation for describing real world with statistical resetting and also spread or diffusion of some material in different scales, or some kind of anomalous spread like that of cancer. The second big problem is the singularity induced by the power law decay function, this singularity leads bad memory at the zero starting point. At this point, the prediction suggested by the model are not reliable as they can sometime misleading due to influence of the artificial singularity. The introduction of non-singular kernel is welcome in the field of fractional calculus as they brought news weapons in this field to accurately model real world problems, in particular diffusion or spread in different states. We welcome them in fractional calculus for two reasons: They do not lead to inclusion of artificial singularity into the mathematical models, in this case the history of the dynamical process. The second reason is their ability of describing the two different waiting times distribution, which is an ideal waiting time distribution as such is observed in many biological phenomena such as the spread of cancer. This crossover behavior of this new fractional differential operators is due to their capacity of not obeying the index law imposed in fractional calculus. The numerical simulations presented in this paper are in perfect agreement with the conclusion of the study done in Atangana (2018) where it was proved that, the index law is irrelevant in fractional calculus as the most important fact is the effect of the fractional differential operator to the associate evolution equation. We conclude in this paper that fractional differential operators not obeying index law are suitable mathematical tools to model real world problems.