Numerical Approximation of Atangana–Baleanu Differentiation

Abstract

Recently, a new contribution was made in the field of fractional calculus where new differential operators are with non-singular and non-local kernel. The new kernel introduced is the well-known generalized Mittag–Leffler function and the properties of this function enable the new operators to have some interesting properties that are observed in real-world situation, for instance, the crossover of the mean square displacement and scaling variant. The fractional differential operators have been applied intensively in several fields since were suggested in 2016. Due to their wider applicability, these operators gave birth to fractional differential equations with no artificial singularities as in the case of Riemann–Caputo derivatives, but with non-local behaviour. We have also seen an interest of these operators in the field of numerical analysis. Thus, to accommodate readers interesting in applying these derivatives to numerical analysis, we present in this chapter some numerical scheme in connection with the Atangana–Baleanu fractional differential operators.