Search Results (5)

Blast-furnace staves serve as critical protective components in ironmaking, requiring synergistic optimization of slag-coating behavior and self-protection capability to extend furnace lifespan and reduce energy consumption. Traditional integer-order heat transfer models, constrained by assumptions of homogeneous materials and instantaneous heat conduction, fail to accurately capture the cross-scale thermal memory effects and non-local diffusion characteristics in multiphase heterogeneous blast-furnace systems, leading to substantial inaccuracies in predicting dynamic slag-layer evolution. This review synthesizes recent advancements across three interlinked dimensions: first, analyzing design principles of zonal staves and how refractory material properties influence slag-layer formation, proposing a “high thermal conductivity–low thermal expansion” material matching strategy to mitigate thermal stress cracks through optimized synergy; second, developing a mechanistic model by introducing the Caputo fractional derivative to construct a non-Fourier heat-transfer framework (i.e., a heat-transfer model that accounts for thermal memory effects and non-local diffusion, beyond the instantaneous heat conduction assumption of Fourier’s law), which effectively describes fractal heat flow in micro-porous structures and interfacial thermal relaxation, addressing limitations of conventional models; and finally, integrating industrial case studies to validate the improved prediction accuracy of the fractional-order model and exploring collaborative optimization of cooling intensity and slag-layer thickness, with prospects for multiscale interfacial regulation technologies in long-life, low-carbon stave designs. Full article

Bone remodeling is a dynamic biological process that preserves bone strength and structure through the coordinated actions of osteoblasts, osteoclasts, osteocytes, and bone mass density. Traditional models based on ordinary differential equations often fail to capture the memory-dependent nature of these interactions. In this study, we propose a novel mathematical model of bone remodeling using the Atangana–Baleanu–Caputo fractional derivative, which accounts for the non-local and hereditary characteristics of biological systems. The model introduces fractional-order dynamics into a previously established ODE framework while maintaining the intrinsic symmetry between bone-forming and bone-resorbing mechanisms, as well as the balance mediated by porosity-related feedback. We establish the existence, uniqueness, and positivity of solutions, and analyze the equilibrium points and their global stability using a Lyapunov function. Numerical simulations under various fractional orders demonstrate symmetric convergence toward equilibrium across all biological variables. The results confirm that fractional-order modeling provides a more accurate and balanced representation of bone remodeling and reveal the underlying symmetry in the regulation of bone tissue. This work contributes to the growing use of fractional calculus in modeling physiological processes and highlights the importance of symmetry in both mathematical structure and biological behavior. Full article

Neural networks, mimicking the structural and functional aspects of the human brain, have found widespread applications in diverse fields such as pattern recognition, control systems, and information processing. A critical phenomenon in these systems is synchronization, where multiple neurons or neural networks harmonize their dynamic behaviors to a common rhythm, contributing significantly to their efficient operation. However, the inherent complexity and nonlinearity of neural networks pose significant challenges in understanding and controlling this synchronization process. In this paper, we focus on the synchronization of a class of fractional-order, delayed, and non-autonomous neural networks. Fractional-order dynamics, characterized by their ability to capture memory effects and non-local interactions, introduce additional layers of complexity to the synchronization problem. Time delays, which are ubiquitous in real-world systems, further complicate the analysis by introducing temporal asynchrony among the neurons. To address these challenges, we propose a straightforward yet powerful global synchronization framework. Our approach leverages novel state feedback control to derive an analytical formula for the synchronization controller. This controller is designed to adjust the states of the neural networks in such a way that they converge to a common trajectory, achieving synchronization. To establish the asymptotic stability of the error system, which measures the deviation between the states of the neural networks, we construct a Lyapunov function. This function provides a scalar measure of the system’s energy, and by showing that this measure decreases over time, we demonstrate the stability of the synchronized state. Our analysis yields sufficient conditions that guarantee global synchronization in fractional-order neural networks with time delays and Caputo derivatives. These conditions provide a clear roadmap for designing neural networks that exhibit robust and stable synchronization properties. To validate our theoretical findings, we present numerical simulations that demonstrate the effectiveness of our proposed approach. The simulations show that, under the derived conditions, the neural networks successfully synchronize, confirming the practical applicability of our framework. Full article

Numerous studies have observed and analyzed the dynamics of language change from a diffusion perspective. As a complex and changeable system, the process of language change is characterized by a long memory that conforms to ultraslow diffusion. However, it is not perfectly suited for modeling with the traditional diffusion model. The Caputo nonlocal structural derivative is a further development of the classic Caputo fractional derivative. Its kernel function, characterized as an arbitrary function, proves highly effective in dealing with ultraslow diffusion. In this study, we utilized an extended logarithmic function to formulate a Caputo nonlocal structural derivative diffusion model for qualitatively analyzing the evolution process of language. The mean square displacement that grows logarithmically was derived through the Tauberian theorem and the Fourier–Laplace transform. Its effectiveness and credibility were verified by the appearance of already popular words on Japanese blogs. Compared to the random diffusion model, the Caputo nonlocal structural derivative diffusion model proves to be more precise in simulating the process of language change. The microscopic mechanism of ultraslow diffusion was explored using the continuous time random walk model, which involves a logarithmic function with a long tail. Both models incorporate memory effects, which can provide useful guidance for modeling diffusion behavior in other social phenomena. Full article

One of the tools for building new fixed-point results is the use of symmetry in the distance functions. The symmetric property of metrics is particularly useful in constructing contractive inequalities for analyzing different models of practical consequences. A lot of important invariant point results of crisp mappings have been improved by using the symmetry of metrics. However, more than a handful of fixed-point theorems in symmetric spaces are yet to be investigated in fuzzy versions. In accordance with the aforementioned orientation, the idea of Presic-type intuitionistic fuzzy stationary point results is introduced in this study within a space endowed with a symmetrical structure. The stability of intuitionistic fuzzy fixed-point problems and the associated new concepts are proposed herein to complement their corresponding concepts related to multi-valued and single-valued mappings. In the instance where the intuitionistic fuzzy-set-valued map is reduced to its crisp counterparts, our results complement and generalize a few well-known fixed-point theorems with symmetric structure, including the main results of Banach, Ciric, Presic, Rhoades, and some others in the comparable literature. A significant number of consequences of our results in the set-up of fuzzy-set- and crisp-set-valued as well as point-to-point-valued mappings are emphasized and discussed. One of our findings is utilized to assess situations from the perspective of an application for the existence of solutions to non-convex fractional differential inclusions involving Caputo fractional derivatives with nonlocal boundary conditions. Some nontrivial examples are constructed to support the assertions and usability of our main ideas. Full article