This paper derives a computationally efficient and fast-running solver for the approximate solution of fractional differential equations with impulsive effects. In this connection, for approximating the fractional-order integral operator, a B-spline version of interpolation by corresponding equal mesh points is adopted. An illustrative example illustrates the accuracy of the new solver results as compared with those of the previous study. The proposed solver’s performance is evaluated by the fractional Rössler and susceptible-exposed-infectious impulsive systems. Moreover, the effect of impulsive behaviors is shown for various values of impulsive.

The benzimidazoles skeleton, a category of heterocyclic compounds have important biological and pharmaceutical properties such as antihelminthic, antiulcer, diuretic, anticonvulsant, analgesic, antiulcer, antihypertensive, anticoagulant, anticancer, anti-inflammatory, antimicrobial, antiviral, antiparasitic and antioxidant. On the other hand, graphene oxide due to their operational simplicity, easy work up and inherent non-toxic, and possessing a wide variety of functional groups, such as epoxy, hydroxyl, and carboxyl, pays an important role in organic synthesis. Therefore, in this chapter graphene oxide was investigated as an efficient catalytic system for the preparation of benzimidazole compounds via the condensation reaction of o-phenylenediamines with aromatic aldehydes in very good yields at ambient temperature. The results in this method were compared with previously reported in literature respectively. Also graphene oxide was recovered and reused without decreasing in its efficacy for 3 th runs.

This paper proposes an effective numerical scheme for solving impulsive fractional differential equations. For this purpose, Hermite interpolation is used to approximate fractional-order integrals. The proposed methods convergence analysis is studied in detail by bounding the approximation error. Finally, the application and performance of the presented method are illustrated in two practical examples, including the impulsive control of the family of Lorenz systems, and the obtained results are compared with an existing method.