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.

本文考虑了在求解分数阶偏微分方程时出现的在Caputo意义上计算Atangana-Baleanu导数值的问题。在这种情况下，需要随着时间或空间变量值的增加，以递归方式计算导数的值。我们考虑了Mittag-Leffler函数的级数展开和积分表示，该函数是Atangana-Baleanu导数中的积分内核。将Mittag-Leffler函数的自变量扩展为级数，并应用变量分离技术，可在自变量的值增加时获得高效的递归计算算法。与在计算Atangana-Baleanu导数的值时执行直接数值积分的算法的线性复杂度相比，该算法的计算复杂度接近恒定。对于提出的算法，我们提出了准确性和性能估计，检查了它们的计算Mittag-Leffler函数的积分，并使用一些有限差分方案求解了时间分数对流扩散方程。