The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $\varsigma =1$, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.

This study presents a mathematical model capturing Malaria transmission dynamics in Yemen, incorporating a social hierarchy structure. Piecewise Caputo-Fabrizio derivatives are utilized to effectively capture intricate dynamics, discontinuities, and different behaviors. Statistical data from 2000 to 2021 is collected and analyzed, providing predictions for Malaria cases in Yemen from 2022 to 2024 using Eviews and Autoregressive Integrated Moving Average models. The model investigates the crossover effect by dividing the study interval into two subintervals, establishing existence, uniqueness, positivity, and boundedness of solutions through fixed-point techniques and fractional-order properties of the Laplace transformation. The basic reproduction number is computed using a next-generation technique, and numerical solutions are obtained using the Adams-Bashforth method. The results are comprehensively discussed through graphs. The obtained results can help us to better control and predict the spread of the disease.

In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.