LAUSR

Abstract The paper considers the problem of computing the values of the Atangana-Baleanu derivative in Caputo sense which arises while solving fractional partial differential equations. In such case the values of the derivative are needed to be calculated in recurrent manner with increasing value of time or space variable. We consider series expansion and integral representations of the Mittag-Leffler function that is an integral kernel in the Atangana-Baleanu derivative. Expanding into series the argument of the Mittag-Leffler function and applying a variable separation technique we obtain an efficient recurrent computational algorithm when the value of argument increases. The proposed algorithm has close-to-constant computational complexity comparing to linear complexity of the algorithms that perform direct numerical integration while computing the values of the Atangana-Baleanu derivative. For the proposed algorithm we present accuracy and performance estimates checking them computing integrals of the Mittag-Leffler function and solving time-fractional convection-diffusion equation using some finite-difference scheme.