In this work, the efficacy of fractional models under Atangana–Baleanu–Caputo, Caputo–Fabrizio, and Caputo is compared to the performance of integer-order models in the forecasting of weekly influenza cases using data from the Kingdom of Saudi Arabia. The suggested fractional influenza model was effectively verified using fractional calculus. Our investigation uncovered the topic’s essential properties and deepened our understanding of disease progression. Furthermore, we analyzed the numerical scheme’s positivity, limitations, and symmetry. The fractional-order models demonstrated superior accuracy, producing smaller root mean square error (RMSE) and mean absolute error (MAE) than the classical model. The novelty of this work lies in introducing the Atangana–Baleanu–Caputo fractional model to influenza forecasting to incorporate memory of an epidemic, which leads to higher accuracy than traditional models. These models effectively captured the peak and drop of influenza cases. Based on these findings, it can be concluded that fractional-order models perform better than typical integer-order models when predicting influenza dynamics. These insights should illuminate the importance of fractional calculus in addressing epidemic threats.

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.

Edge detection is an essential image processing act that is crucial for many computer vision applications such as object detection, image segmentation, face recognition, text recognition, medical imaging, and autonomous vehicles. Deep learning is the most advanced and widely used tool of them all. In this paper, we present a novel deep learning model and use image datasets to test it. Our model uses a fractional calculus tool, which could enhance gradient approaches’ performances. Specifically, we approximate the fractional-order derivative-order neural network (GLFNet) using a Grünwald–Letnikov fractional definition. First, the original dataset is subjected to a Grünwald–Letnikov fractional order. After that, the CNN model is updated with the new dataset, concluding the standard CNN procedure. The training rate, the improvement in the F-measure for identifying the effective edge while maintaining the CNN model’s memory consumption, and the values of the loss errors between the prediction and training processes were all tested using the MNIST dataset. Our experiments show that GLFNet considerably enhances edge detection. GLFNet outperformed CNN with an average loss error ratio of 15.40, suggesting fewer loss mistakes. The F1-measure ratio of 0.81 indicates that GLFNet can compete with CNN in terms of precision and recall. The training time for GLFNet was lowered by an average ratio of 1.14 when compared to CNN, while inference time was faster with a ratio of 1.14, indicating increased efficiency. These findings demonstrate the efficacy of introducing Grünwald–Letnikov fractional convolution into deep learning models, resulting in more precise and reliable edge detection while preserving comparable memory utilization.

This paper aims to develop the scenario to find out how two approaches can be employed in estimating the numerical results in systems. The first technique is the application of the Caputo Fabrizio fractional schema, which is a method for overcoming the challenges of singular calculus while providing a generalized approach in a more flexible framework. The second approach involves the Laplace transform and the Homotopy analysis method (LHAM) to give an analytical solution that inherits good features of both methods, capable of tackling nonlinearity and yielding fast convergence. In the present work, we use these techniques to investigate the inverted Rössler system where the equations are defined within the Caputo Fabrizio derivative. By applying this system, which is associated with behaviour analysis using both methods, we conclude that Caputo Fabrizio is more appropriate for deriving solutions, whereas LHAM is preferable since it converges rapidly and is computationally efficient. Comparisons are provided for the validity of the outlined approaches, which is corroborated by the potential use of such methods in solving systems that recognize ‘chaotic’ patterns and in extending the applications of the forecasting approaches identified to other branches of science and engineering. Besides, an additional analysis focuses on the strengths and weaknesses of each method to enable the user to understand how the method works and how it produces results.

This paper provides a solution to the new fractional-order Lorenz–Stenflo model using the adaptive predictor–corrector approach and the $\rho$-Laplace New Iterative Method ($L_{\rho }NIM$), representing an extensive comparison between both techniques with RK4 related to accuracy and error analysis. The results show that the suggested approaches allow one to be more accurate in analyzing the dynamics of the system. These techniques also produce results that are comparable to the results of other approximate techniques. The techniques can, thus, be used on a wider class of systems in order to provide more accurate results. These techniques also appropriately identify chaotic attractors in the system. These techniques can be applied to solve various numerical problems arising in science and engineering in the future.

This paper presents an in-depth analysis of fractional nonlinear dispersive K(m,n,1) equations using two innovative mathematical techniques: These two are the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM). These methods are used to formulate approximate analytical solutions in the framework of the Caputo operator. The MTIM offers an iterative type solution for fractional differential equations, and the MRPSM gives a series-based solution where nonlinearities are addressed efficiently. The effectiveness of these methods are illustrated through tables and figures, which can impress with their demonstrable result. They also demonstrate that our approaches of MTIM and MRPSM are adequate for solving the fractional nonlinear dispersive equations, which are enhanced compared with the conventional methods. This work significantly enriches the system's knowledge of applying sophisticated mathematical tools to interdisciplinary fractional differential equation problems with diverse bearings in numerous scientific and engineering disciplines.