*Article* **Improvement of Mathematical Model for Sedimentation Process**

**Ivan Pavlenko 1, Marek Ochowiak 2,\*, Praveen Agarwal 3, Radosław Olszewski 4, Bernard Michałek <sup>4</sup> and Andzelika Krupi ´ ˙ nska <sup>2</sup>**


**Abstract:** In this article, the fractional-order differential equation of particle sedimentation was obtained. It considers the Basset force's fractional origin and contains the Riemann–Liouville fractional integral rewritten as a Grunwald–Letnikov derivative. As a result, the general solution of the proposed fractional-order differential equation was found analytically. The belonging of this solution to the real range of values was strictly theoretically proven. The obtained solution was validated on a particular analytical case study. In addition, it was proven numerically with the approach based on the S-approximation method using the block-pulse operational matrix. The proposed mathematical model can be applied for modeling the processes of fine particles sedimentation in liquids, aerosol deposition in gas flows, and particle deposition in gas-dispersed systems.

**Keywords:** particle sedimentation; resistance force; fractional-order integro-differential equation; laplace transform; Mittag–Leffler function; block-pulse operational matrix
