This article describes the corona virus spread in a population under certain assumptions with the help of a fractional order mathematical model. The fractional order derivative is the well-known fractal fractional operator. We have given the existence results and numerical simulations with the help of the given data in the literature. Our results show similar behavior as the classical order ones. This characteristic shows the applicability and usefulness of the derivative and our numerical scheme.

In this paper, a new collocation method based on Haar wavelet is developed for numerical solution of Riccati type differential equations with non-integer order. The fractional derivatives are considered in the Caputo sense. The method is applied to one test problem. The maximum absolute estimated error functions are calculated, and the performance of the process is demonstrated by calculating the maximum absolute estimated error functions for a distinct number of nodal points. The results show that the method is applicable and efficient.

In this paper, existence, uniqueness, and Hyers-Ulam stability for the solution of second-order fuzzy differential equations (FDEs) are studied. To deal a physical model, it is required to insure whether unique solution of the model exists. The natural transform has the speciality to converge to both Laplace and Sumudu transforms only by changing the variables. Therefore, this method plays the rule of checker on the Laplace and Sumudu transforms. We use natural transform to obtain the solution of the proposed FDEs. As applications of the established results, some nontrivial examples are provided to show the authenticity of the presented work.

The red palm weevil (Rhynchophorus ferrugineus, RPW) is one of the most destructive pests of date palms worldwide. Its larvae feed inside the stem, so damage is often irreversible before any symptoms are visible. Early detection and rapid removal of infested palms are therefore essential. We develop a five-compartment fractional-order mathematical model that tracks healthy palms, infested palms, larvae, adult weevils, and pheromone traps. The model uses fractional-order (Caputo) derivatives, which allow past infestation history to influence current dynamics. We use this memory as a phenomenological representation of delayed and cumulative effects, such as the 2–4 month larval development cycle, and we test it against integer-order, Erlang-distributed, and fixed-delay alternatives. We provide mathematical proofs that the model is well posed and identify a threshold, the basic reproduction number ${\mathcal{R}}_0$, that separates eradication from persistent infestation. The model is calibrated to two UAE datasets: the 1994/95 Al Ain survey of about $1.1\times {10}^6$ palms (${\mathcal{R}}_0$ > 1) and the 2025 national program outcomes (${\mathcal{R}}_0$ < 1), using the reported first and last counts of each series and a reconstructed monthly trajectory between them. This drop in ${\mathcal{R}}_0$ corresponds to three control improvements: sensor-based early detection, endotherapy (localized insecticide injection into the stem), and preventive stem injection. Model comparison supports a fractional order of about $0.95$ only for the 1994/95 epoch, whereas the 2025 data are described equally well without memory. Sensitivity analysis shows that faster removal of infested palms, lower RPW-to-palm contact rates, and reduced larval production are among the most effective control tactics. Model simulations of suspended and restored control suggest, as a hypothesis-generating scenario derived from hypothesized data, that a lapse in control can allow a transient resurgence, which supports continued monitoring. These results give quantitative guidance for planning pest management programs.