A mathematical model for the spread of the COVID-19 disease based on a fractional Atangana–Baleanu operator is studied. Some fixed point theorems and generalized Gronwall inequality through the AB fractional integral are applied to obtain the existence and stability results. The fractional Adams–Bashforth is used to discuss the corresponding numerical results. A numerical simulation is presented to show the behavior of the approximate solution in terms of graphs of the spread of COVID-19 in the Chinese city of Wuhan. We simulate our table for the data of Wuhan from February 15, 2020 to April 25, 2020 for 70 days. Finally, we present a debate about the followed simulation in characterizing how the transmission dynamics of infection can take place in society.

In this paper, we study some new properties of Sadik transform such as integration, time delay, initial value theorem, and final value theorem. Moreover, we prove the theorem of Sadik transform for Caputo fractional derivative and we also establish sufficient conditions for the existence of the Sadik transform of Caputo fractional derivatives. At the end, the fractional-order dynamical systems in control theory as application of this transform is discussed, in addition, some numerical examples to justify our results.

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The purpose of this research was to discover a novel method to recover k-fractional integral inequalities of the Pólya–Szegö-type. We employ these generalized inequalities to investigate some new fractional integral inequalities of the Grüss-type. More precisely, we generalize the proportional fractional operators with respect to another strictly increasing continuous function ψ. Then, we state and prove some of its properties and special cases. With the help of this generalized operator, we investigate some Pólya–Szegö- and Grüss-type fractional integral inequalities. The functions used in this work are bounded by two positive functions to obtain Pólya–Szegö- and Grüss-type k-fractional integral inequalities in a new sense. Moreover, we discuss some new special cases of the Pólya–Szegö- and Grüss-type inequalities through this work.

The given paper describes the implicit fractional differential equation with nonlinear integral boundary conditions in the frame of Caputo-Katugampola fractional derivative. We obtain an analogous integral equation of the given problem and prove the existence and uniqueness results of such a problem using the Banach and Krasnoselskii fixed point theorems. To show the effectiveness of the acquired results, convenient examples are presented.