This paper introduces a novel class of neural networks called Incommensurate Fractional Recurrent Neural Networks (FRNNs) , inspired by incommensurate fractional difference systems. Unlike classical RNNs, FRNNs incorporate learnable fractional orders to capture heterogeneous temporal dependencies. To address the failure of back-propagation caused by cross-layer and cross-time coupling, we propose a forward-forward recursive gradient computation method combined with mini-batch SGD and ADAM optimization. Extensive experiments on multiple time-series datasets, including synthetic sinusoidal sequences, chaotic sequences, and real-world financial and energy data, demonstrate that FRNN achieves lower prediction error compared to RNN. These results confirm the effectiveness of fractional memory dynamics for sequential data processing.

Abstract Monkeypox virus is one of the major causes of both smallpox and cowpox infection in our society. It is typically located next to tropical rain forests in remote villages in Central and West Africa. The disease is brought on by the monkeypox virus, a member of the Orthopoxvirus genus and the Poxviridae family. For analysis and the dynamical behaviour of the monkeypox virus infection, we developed a fractional order model with the Mittag‐Leffler kernel. The uniqueness, positivity, and boundedness of the model are treated with fixed point theory results. A Lyapunov function is used to construct both local and global asymptotic stability of the system for both endemic and disease‐free equilibrium points. Finally, numerical simulations are carried out using the effective numerical scheme with an extended Mittag‐Leffler function to demonstrate the accuracy of the suggested approaches.

Abstract The goal of this research is to design the Gudermannian neural networks (GNNs) to solve a type of two-point nonlinear singular boundary value problems (TPN-SBVPs) that arise within thermal-explosion theory. The results of these investigation are provided for different neurons (4, 12 and 20), as well as absolute error along with the time complexity. For solving the TPN-SBVPs, a genetic algorithm (GA) and sequential quadratic programming (SQP) are used to optimize the error function. The accuracy of designed GNNs is provided by using a hybrid GA–SQP combination, which is based on a comparison of obtained and actual solutions. Furthermore, statistical analysis of the data is proposed in order to establish the competence as well as effectiveness of designed and the efficacy of the designed computing framework for solving the TPN-SBVPs.

Abstract The aim of the current study is to present the numerical solutions of a nonlinear second-order coupled Emden–Fowler equation by developing a neuro-swarming-based computing intelligent solver. The feedforward artificial neural networks (ANNs) are used for modelling, and optimization is carried out by the local/global search competences of particle swarm optimization (PSO) aided with capability of interior-point method (IPM), i.e., ANNs-PSO-IPM. In ANNs-PSO-IPM, a mean square error-based objective function is designed for nonlinear second-order coupled Emden–Fowler (EF) equations and then optimized using the combination of PSO-IPM. The inspiration to present the ANNs-PSO-IPM comes with a motive to depict a viable, detailed and consistent framework to tackle with such stiff/nonlinear second-order coupled EF system. The ANNs-PSO-IP scheme is verified for different examples of the second-order nonlinear-coupled EF equations. The achieved numerical outcomes for single as well as multiple trials of ANNs-PSO-IPM are incorporated to validate the reliability, viability and accuracy.

The fractional logistic map holds rich dynamics and is adopted to generate chaotic series. A watermark image is then encrypted and inserted into the original images. Since the encryption image takes the fractional order within (0, 1], it increases the key space and becomes difficult to attack. This study provides a robust watermark method in the protection of the copyright of hardware, images, and other electronic files.

Recently Caputo and Fabrizio suggested new definition of fractional derivative that the new kernel has no singularity. In this paper, an analytical method for solving Riccati differential equation with a new fractional derivative is reported. We present numerical results of solving the fractional R iccati differential equations by using the variational iteration method and its modification. The obtained results of two methods demonstrate the efficiency and simplicity of the MVIM that gives good approximations for a larger interval.

Based on concepts for semigroup theory, fractional calculus, Banach contraction principle and Krasnoselskii fixed point theorem (FPT), this manuscript is principally involved with existence results of Riemann-Liouville (RL) fractional neutral integro-differential systems (FNIDS) with nonlocal conditions (NLCs) in Banach spaces. An example is offered to demonstrate the theoretical concepts.

The main study of this paper is focused on the solutions of a class of fuzzy sequential fractional differential equations in the form of (0Dxβy)'(x) = b(x)y(x), where (0Dxβy)(x) is the fuzzy Riemann-Liouville derivative of order β ∈ (0, 1). On this subject, a new fuzzy complete metric space is introduced. Finally, we proof the existence and uniqueness of our solution using the contraction principle.