This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.

This study demonstrates that the Grünwald–Letnikov fractional proportional–integral–derivative (GPID) controller outperforms traditional PID controllers in adaptive cruise control systems, while conventional PID controllers struggle with nonlinearities, dynamic uncertainties, and stability, the GPID enhances robustness and provides more precise control across various driving conditions. Simulation results show that the GPID improves the accuracy, reducing errors better than the PID controller. Additionally, the GPID maintains a more consistent speed and reaches the target speed faster, demonstrating superior speed control. The GPID’s performance across different fractional orders highlights its adaptability to changing road conditions, which is crucial for ensuring safety and comfort. By leveraging fractional calculus, the GPID also improves acceleration and deceleration profiles. These findings emphasize the GPID’s potential to revolutionize adaptive cruise control, significantly enhancing driving performance and comfort. Numerical results obtained in $\alpha =0.99$ from the GPID controller have shown better accuracy and speed consistency, adapting to road conditions for improved safety and comfort. The GPID also demonstrated faster stabilization of speed at 60 km/h with smaller errors and reduced the error to 0.59 km/h at 50 s compared to 0.78 km/h for the PID.

Over the last several years, there has been a considerable improvement in the possible methods for solving fractional-order chaotic systems; however, achieving high accuracy remains a challenge. This work proposes a new precise numerical technique for fractional-order chaotic systems. Through simulations, we obtain new types of complex and previously undiscussed dynamic behaviors.These phenomena, not recognized in prior numerical results or theoretical estimations, underscore the unique dynamics present in fractional systems. We also study the effects of the fractional parameters ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ on the system’s behavior, comparing them to integer-order derivatives. It has been demonstrated via the findings that the suggested technique is consistent with conventional numerical methods for integer-order systems while simultaneously providing an even higher level of precision. It is possible to demonstrate the efficacy and precision of this technique through simulations, which demonstrates that this method is useful for the investigation of complicated chaotic models.

This work pioneers a novel approach in image edge detection through the utilization of the generalized fractional operator. By harnessing the global attributes inherent in fractional derivatives, it aims to enhance the extraction of intricate edge details.This is accomplished by creating the mask by using fractional derivative and adapt the mask by another parameter, yielding compelling and informative edge representations, as validated by experimental results. This advancement not only augments computer vision and image analysis techniques but also holds promise for refining image processing methodologies. Future endeavors may explore its adaptability across diverse imaging domains like medical and satellite imagery, while integration into deep learning frameworks could elevate its potential for advanced feature extraction and deeper image understanding. Additionally, optimizing its computational efficiency would broaden its scope for real-time deployment in fields such as robotics and autonomous systems.

Fractional calculus extends conventional differentiation and integration to non-integer orders, facilitating a more suitable modeling framework for complicated dynamical processes characterized by memory and long-range dependence. The fractional and variable-order fractional Genesio-Tesi systems have recently attracted significant interest owing to their rich nonlinear dynamics and the added flexibility introduced by variable fractional orders. This work comparatively studies the stability and chaotic behavior of constant-order versus variable-order formulations of the fractional Genesio–Tesi system. The study of the system dynamics is carried out by numerical simulations, including time series, bifurcation diagrams, Lyapunov exponents, and phase portraits. We identify further stability boundaries and chaotic regimes through analytical investigations based on the Jacobian eigenvalue spectrum. It is found that variable-order derivatives intensify sensitivity and transient responses, disclosing chaotic patterns that contribute to a more profound understanding of fractional nonlinear dynamics.

This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations.