Search Results (106)

In this paper, the new fractal modified Zakharov–Kuznetsov equation (fmZKe) defined on Cantor sets is investigated. The fmZKe is a non-differentiable model that arises naturally in mathematical physics, nonlinear wave theory, and plasma physics. The extended rational sine–cosine method is utilized to construct new optical soliton solutions of the model. The fmZKe is reduced to a non-differentiable ordinary differential equation by applying a non-differentiable wave transformation defined on Cantor sets. This reduction leads to a system of linear algebraic equations, which upon solving yields several exact solutions of the model. Furthermore, to establish a clear understanding of the model’s behavior, non-smooth graphical representations of the solutions are presented for various parameter values. The stability analysis of the newly obtained solutions in a classical sense is examined using stability theory, and the real-life applications of the results are highlighted. Full article

The Yang local fractional calculus was developed to analyze discontinuous mappings. The results obtained within this framework are equally useful in the classical sense. The study of integrals and their approximation rules is an interesting field of research. Among them, the Gaussian quadrature rules are more useful due to their accuracy. In this manuscript, we will explore the error inequalities of the four-point Lobatto–Legendre Quadrature rule incorporating fractal calculus. Our approach is based on the generation of inequalities through a generalized fractal identity. First, we develop an auxiliary result. Then, the applications of various classes of mappings are defined over ${\mathbb{R}}^{\mathsf{t}}$ and auxiliary results, and we develop several new estimates. Additionally, an Artificial Neural Network ($\mathit{A}\mathcal{N}\mathcal{N}$) framework is used to analyse the profile and computational stability of the derived inequalities. Lastly, we have focused on the applicable analysis of the proposed results. This is the first study carried out on Lobatto-type inequalities within fractal space. Full article

Population-based metaheuristic algorithms are widely used for multi-objective city evacuation planning, yet their opaque internal dynamics limit practitioner trust in safety-critical contexts. This study introduces, to the best of our knowledge, the first unified coupling of fractional calculus and fractal analysis with the EvoMapX process-level explainability framework in the context of evacuation optimization. In contrast with classical integer-order EvoMapX paired with exponential moving averages of operator credit, the proposed formulation embeds long-range memory directly into the explainability pipeline through Caputo and Grünwald–Letnikov derivatives. The Operator Attribution Matrix (OAM), Population Evolution Graph (PEG), and Convergence Driver Score (CDS) are extended with fractional-order formulations employing Caputo and Grünwald-Letnikov fractional derivatives with adaptive memory parameters, alongside Mittag–Leffler urgency escalation dynamics. A Fractional-Order PSO variant (FO-EPSO) with segment-specific fractional velocity updates and a fractal fitness landscape analysis module for adaptive parameter tuning are introduced. The framework incorporates nine evacuation-specific operators, a spatial OAM for zone-level attribution, and a multi-stakeholder explanation pipeline. Experiments across 520 disaster scenarios demonstrate that explainability and optimization performance are not mutually exclusive: the EvoMapX-integrated NSGA-II achieved a mean hypervolume of 0.731 versus 0.728 for the standard variant, with less than 5% computational overhead. The OAM revealed disaster-type-specific operator patterns invisible to conventional analysis. Real-world validations on Beijing Chaoyang District and Kigali, Rwanda, confirmed these findings. From an operational standpoint, the most consequential outcome of this work concerns its impact on human decision-makers: a controlled study with 45 emergency-management professionals showed that incorporating EvoMapX explanations cut the time required to commit to an evacuation plan by 24.9%, raised reported decision confidence by 20.3%, and lifted self-assessed algorithm understanding from 18.1% to 78.9% (all p < 0.001). Equally important for real-time disaster response, this entire layer of process-level transparency is delivered with a runtime penalty of under 5% relative to the non-explainable baselines, which we view as a key practical advantage for field deployment. This work establishes fractional-order process-level transparency as a feasible and beneficial paradigm for interpretable optimization in safety-critical domains. Full article

Fractional quantum calculus provides a powerful mathematical framework for incorporating memory and scaling effects into numerical models. However, classical iterative methods for nonlinear equations often suffer from limited stability, sensitivity to initial guesses, and restricted convergence domains, particularly in highly nonlinear settings. In this work, we introduce a new Caputo fractional–quantum iterative scheme, denoted by ${\mathrm{MSB}}_{\mathfrak{q}:\alpha }$, formulated as a parameterized two-step method based on a Caputo-type fractional quantum derivative. The proposed framework incorporates additional structural parameters that regulate the iterative dynamics and enable enhanced control over convergence behavior and stability properties. To assess the performance of the method, we employ tools from complex dynamical systems, including stability analysis and fractal basin investigations in the complex plane. These analyses provide insight into how the fractional and quantum parameters influence the geometry of attraction domains and the global convergence behavior of the scheme. Numerical experiments on representative nonlinear problems arising in engineering and biomedical applications demonstrate improved robustness with respect to initial guesses, reduced residual errors, and competitive computational efficiency compared with existing iterative methods. Overall, the results indicate that the proposed fractional–quantum framework offers an effective and versatile approach for the numerical solution of challenging nonlinear equations. Full article

In this article, the $\alpha$-fractal interpolation function $f^{\alpha }$ corresponding to any function f belonging to the weighted Sobolev space ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is defined. The convergence of sequences of $\alpha$-fractal interpolation functions corresponding to mappings in ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ with respect to the uniform norm as well as the weighted Sobolev norm is discussed. It is proved that the Riemann–Liouville fractional order integral of an $\alpha$-fractal interpolation function of any map $f\in {\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is also a self-referential function interpolating a specific data set. Some aspects of the convergence of the Riemann–Liouville integral of $\alpha$-fractal functions when the original mappings converge are also analyzed. In short, by imposing certain conditions on the base function and the scale vector of a specific iterated function system, fractal perturbations of functions from weighted Sobolev spaces are defined. It is also proved that, under suitable hypotheses, the Riemann–Liouville fractional integral of these fractal mappings on Sobolev spaces is a fractal function of the same kind. Full article

Rock pore–fracture systems exhibit inherent fractal characteristics, which exert a significant influence on fluid transport. In this study, coal rock is selected as the representative medium. Based on fractional calculus in physical fractal space, and by integrating operator algebra with the force–electric analogy method, a fractional order control equation is derived. To validate the proposed model, porosity measurements of coal and limestone were performed using the two-compartment Boyle’s law method and compared with conventional porosity calculation approaches. The results demonstrate that the fractional order model achieves a coefficient of determination (R²) of up to 0.99 for porosity and 0.98 for pressure, representing an improvement of approximately 0.07 over the exponential model. Moreover, the root mean square error (RMSE) of porosity is as low as 0.0008, while the RMSE of pressure is 0.0715, both significantly lower than those obtained using the exponential model. These results indicate that the fractional order model more effectively captures the non-Darcy flow behavior and the temporal evolution of porosity, providing substantially improved fitting accuracy. Further analysis reveals that the porosity–time relationship is jointly governed by fluid compressibility and pore compressibility under effective stress conditions. Comparative results across different lithologies reveal that the pore compressibility coefficient increases with porosity; for the same rock type, a higher coefficient implies a more complex pore structure and a longer equilibration time. Overall, the proposed fractional order framework provides a more accurate description of the fractal pore structures in rocks, establishing a clear link between microscale fractal geometry and macroscale fractional order response. Full article

A fractal approach for the Timoshenko beam theory by applying differential vector calculus in a three-dimensional continuum with a fractal metric is developed. First, a summary of the tools needed, mathematical relationships, and background of fractal continuum mechanics is presented. Then, the static and dynamical parts of the Timoshenko beam equation are extended to fractal manifolds. Afterwards, an intrafractal beam constructed as a Cartesian product is suggested and the fractal dimensionalities of the Balankin beam are scrutinized. This allows comparing both intrafractal beams when they have the same Hausdorff dimension but different connectivity. Finally, the effects of fractal attributes on the mechanical properties of the deformable fractal medium are highlighted. Some applications of the developed tools are briefly outlined. Full article

This paper introduces new generalized forms of contractive mappings in the framework of complete metric spaces. By extending the classical Reich and Sehgal contractions to their iterated counterparts in Singh’s sense, we establish unified fixed-point theorems that ensure both existence and uniqueness under constant and variable contractive parameters. The proposed p-Reich and p-Sehgal contractions encompass several well-known results, including those of Banach, Kannan, Chatterjea, Reich, and Sehgal, as special cases. Convergence of the associated Picard iterative process is rigorously analyzed, revealing deeper insights into the iterative stability and asymptotic behavior of nonlinear mappings in metric spaces. The practical utility of our unified fixed-point theorems is illustrated through concrete applications in fractal and fractional calculus. Full article

We explore the physical quantum properties of atoms in fractal spaces, both as a theoretical generalization of normal integer-dimensional Euclidean spaces and as an experimentally realizable setting. We identify the threshold of fractality at which Ehrenfest atomic instability emerges, where the Schrödinger equation describing the wavefunction of a single electron orbiting around an atom becomes scale-free, and discuss the potential of observing this phenomena in laboratory settings. We then study the Rydberg states of stable atoms using the Wentzel–Kramers–Brillouin approximation, along with a proposed extension for the Langer modification, in general fractal dimensionalities. We show that fractal space atoms near instability explode in size even at low-number excited state, making them highly suitable to induce strong entanglements and foster long-range many-body interactions. We argue that atomic physics in fractal spaces—“fractatomic physics”—is a rich research avenue deserving of further theoretical and experimental investigations. Full article

This paper uses the theory of self-affine fractal functions to model the dynamic flight graphs of starling flocks, integrating the fractional calculus of self-affine fractal functions to quantitatively characterize the intrinsic nonlinear dynamics and memory effects within the system, employing statistical inference methods to find the fractal fit for the images. The changes in box dimensions over time could characterize the phase transition process of the starling flight flocks. By analyzing the rate of change of fractal dimensions, we identify critical points corresponding to phase transitions during collective flight behavior. During the flight of the starling flocks, a real-time phase transition process for evading attacks and effective advancement has been identified. Experimental data confirms the effectiveness of controlling the phase transition. Full article

Integration in non-integer-dimensional spaces (NIDS) is actively used in quantum field theory, statistical physics, and fractal media physics. The integration over the entire momentum space with non-integer dimensions was first proposed by Wilson in 1973 for dimensional regularization in quantum field theory. However, self-consistent calculus of integrals and derivatives in NIDS and the vector calculus in NIDS, including the fundamental theorems of these calculi, have not yet been explicitly formulated. The construction of precisely such self-consistent calculus is the purpose of this article. The integral and differential operators in NIDS are defined by using the generalization of the Wilson approach, product measure, and metric approaches. To derive the self-consistent formulation of the NIDS calculus, we proposed some principles of correspondence and self-consistency of NIDS integration and differentiation. In this paper, the basic properties of these operators are described and proved. It is proved that the proposed operators satisfy the NIDS generalizations of the first and second fundamental theorems of standard calculus; therefore, these NIDS operators form a calculus. The NIDS derivative satisfies the standard Leibniz rule; therefore, these derivatives are integer-order operators. The calculation of the NIDS integral over the ball region in NIDS gives the well-known equation of the volume of a non-integer dimension ball with arbitrary positive dimension. The volume, surface, and line integrals in D-dimensional spaces are defined, and basic properties are described. The NIDS generalization of the standard vector differential operators (gradient, divergence, and curl) and integral operators (the line and surface integrals of vector fields) are proposed. The NIDS generalizations of the standard gradient theorem, the divergence theorem (the Gauss–Ostrogradsky theorem), and the Stokes theorem are proved. Some basic elements of the calculus of differential forms in NIDS are also proposed. The proposed NIDS calculus can be used, for example, to describe fractal media and the fractal distribution of matter in the framework of continuum models by using the concept of the density of states. Full article

Background: This paper develops a novel, interdisciplinary framework for optimizing high-speed rail (HSR) freight logistics hubs in the Ottawa–Quebec City corridor, addressing critical gaps in geospatial mismatches, static optimization limitations, and narrow sustainability scopes found in the existing literature. Methods: The research methodology integrates a hybrid graph neural network-reinforcement learning (GNN-RL) architecture that encodes 412 nodes into a dynamic graph with adaptive edge weights, fractal accessibility (α = 1.78) derived from fractional calculus (α = 0.75) to model non-linear urban growth patterns, and a multi-criteria sustainability evaluation framework embedding shadow pricing for externalities. Methodologically, the framework is validated through global sensitivity analysis and comparative testing against classical optimization models using real-world geospatial, operational, and economic datasets from the corridor. Results: Key findings demonstrate the framework’s superiority. Empirical results show an obvious reduction in emissions and lower logistics costs compared to classical models, with Pareto-optimal hubs identified. These hubs achieve the most GDP coverage of the corridor, reconciling economic efficiency with environmental resilience and social equity. Conclusions: This research establishes a replicable methodology for mid-latitude freight corridors, advancing low-carbon logistics through the integration of GNN-RL optimization, fractal spatial analysis, and sustainability assessment—bridging economic viability, environmental decarbonization, and social equity in HSR freight network design. Full article

The Riemann-Liuoville fractional integrals are the simplest and most popular operators of the classical fractional calculus. But their variants, the Erdélyi-Kober operators of fractional integration, have many more applications due to the freedom to choose the additional (three) parameters. We introduce and study a generalization of the Erdélyi-Kober and Riemann-Liuoville fractional integrals, where the elementary kernel function is replaced by a suitably chosen $I_{1,1}^{1,0}$-function. The I-functions introduced by Rathie in 1997 are generalized hypergeometric functions extending the Fox H-functions and the Meijer G-functions. Note that till recently this new class of special functions has not been popular because of their too complicated structure involving fractional powers of the Gamma functions and their multi-valued behavior. However, the I-functions happened to arise not only for the needs of statistical physics, but also since they included important special functions in mathematics that were not covered by the H- and G-functions. In our previous works, as Kiryakova and Paneva-Konovska, we have shown the relations of such functions, among which are the Mittag-Leffler and Le Roy type, their multi-index variants, and others related to fractional calculus, to the I-functions. Here, we propose a new theory of generalization of the Erdélyi-Kober fractional integrals, based on the use of an I-function as a kernel. This will serve next as a base to extend our generalized multi-order fractional calculus with operators involving $I_{m,m}^{m,0}$. In this paper, we also evaluate the images under these new generalized fractional integrals of special functions of very general form. Finally, in the Conclusion section, we comment on some earlier discussions on the relations between fractal geometry and fractional calculus, nowadays already without any doubts. Full article

This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ($LF{L}_{\zeta }^2$) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the $LF{L}_{\zeta }^2$ transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries. Full article

Fractional calculus plays a central role in modeling memory-dependent processes and complex dynamics across various fields, including control theory, fluid mechanics, and bioengineering. This study introduces an efficient and stable fractional-order iterative method based on the Caputo derivative for solving nonlinear equations. By employing a Taylor series expansion, a local convergence analysis shows that for $\gamma \in (0,1]$, the method achieves a convergence order of $2\gamma +1$. To address challenges related to memory effects and instability in existing approaches, the proposed scheme incorporates parameter optimization through chaos and bifurcation analysis. Dynamical plane analysis reveals that parameter values within chaotic regimes lead to divergence, while those in stable regions converge uniformly. The method’s performance is evaluated using a set of nonlinear models drawn from biomedical engineering, including enzyme kinetics with inhibition, extended glucose–insulin regulation, drug dose–responses, and lung volume–pressure dynamics. Comparative results demonstrate that the proposed approach outperforms existing methods in terms of iteration count, residual error, CPU time, convergence order, fractal behavior, and memory efficiency. These findings underscore the method’s applicability to complex systems characterized by nonlinearity and memory effects in scientific and engineering contexts. Full article