We proposed an original hybrid approach that combined a continuous modeling framework based on Caputo fractional differential equations with an Earliest Deadline First–Ant Colony Optimization (EDF–ACO) algorithm for solving the parallel machine scheduling problem with precedence constraints. Unlike most existing works, where the fractional order is usually fixed at its classical value, we investigated the influence of the fractional order $\alpha$ in the interval $(0,1]$ and analyzed its impact on the optimization process. The results showed that intermediate values of $\alpha$ allowed the effective incorporation of memory effects, leading to improved numerical stability, smoother convergence, and a reduction of oscillatory behavior. The fractional evaluation mechanism was coupled with the pheromone update strategy of the EDF–ACO algorithm, providing a more stable guidance for the search process. The existence and uniqueness of the solution to the fractional model were established using Banach's fixed point theorem, ensuring the consistency of the proposed continuous evaluation framework. Numerical experiments confirmed the effectiveness of the approach in terms of solution quality and convergence stability across different scheduling configurations.

In this work, we present existence results for some problems which arise in image processing namely image restoration. Our essential tools are Picard's fixed point theorem for a strict contraction and Mountain-pass Theorem for critical point.

The goal of this work is to study the existence of a unique solution and the Ulam-Hyers stability of a coupled system of generalized hybrid pantograph equations with fractional deformable derivatives. Our main tool is Banach's contraction principle. The paper ends with an example to support our results.

The Department of Economic and Social Affairs of the United Nations has released seventeen goals for sustainable development and SDG No. 3 is “Good Health and Well-being”, which mainly emphasizes the strategies to be adopted for maintaining a healthy life. The Monkeypox Virus disease was first reported in 1970. Since then, various health initiatives have been taken, including by the WHO. In the present work, we attempt a fractional model of Monkeypox virus disease, which we feel is crucial for a better understanding of this disease. We use the recently introduced $𝒜ℬ𝒞$ fractional derivative to closely examine the Monkeypox virus disease model. The evaluation of this model determines the existence of two equilibrium states. These two stable points exist within the model and include a disease-free equilibrium and endemic equilibrium. The disease-free equilibrium has undergone proof to demonstrate its stability properties. The system remains stable locally and globally whenever the effective reproduction number remains below one. The effective reproduction number becoming greater than unity makes the endemic equilibrium more stable both globally and locally than unity. To comprehensively study the model’s solutions, we employ the Picard-Lindelof approach to investigate their existence and uniqueness. We investigate the Ulam-Hyers and UlamHyers Rassias stability of the fractional order nonlinear framework for the Monkeypox virus disease model. Furthermore, the approximate solutions of the $𝒜ℬ𝒞$ fractional order Monkeypox virus disease model are obtained with the help of a numerical technique combining the Lagrange polynomial interpolation and fundamental theorem of fractional calculus with the $𝒜ℬ𝒞$ fractional derivative.