We present an extension of the standard Poisson-Nernst-Planck model by incorporating temporal memory effects to describe the spectroscopy impedance response in electrolytic systems. This model yields a modified current-density relation in which the ionic flux depends nonlocally on the applied electric field. The resulting electrical impedance may exhibit non-Debye relaxation and fractional-like scaling at low frequencies, providing a basis for anomalous diffusion in confined electrolytes. We analyze impedance spectroscopy data from NH4Cl-glycerol solutions for various concentrations to validate the model. The comparison demonstrates how the memory kernel governs the transition between normal and anomalous diffusion regimes, enabling accurate fits to experimental data. These results evidence the relevance of memory-driven transport in complex fluids and suggest a pathway to unify standard and fractional impedance models.

Statistical testing-based framework for differentiating anomalous diffusion models with constant and random parameters

Proteins are understood to exhibit complex internal motions on multiple time scales in their rugged free energy landscapes and often show subdiffusive behavior that significantly influences their biochemical functions. In this study, we employ the fractional Fokker-Planck equation and continuous-time random walk models to investigate the anomalous diffusion of particles within rough confining potentials, drawing inspiration from protein internal dynamics. Our analysis reveals that the dynamics exhibit three distinct regimes: initial free subdiffusion, an intermediate regime where roughness markedly impacts motion, and a long-term thermal plateau due to confinement effects. We derive approximate expressions for the mean displacement and the ensemble-averaged mean squared displacement in the low-roughness limit, revealing good agreement with simulation results. Furthermore, our examination of the ergodic properties of the dynamics indicates that systems with high roughness exhibit enhanced weak ergodicity breaking. As a consequence, the time-averaged mean squared displacement does not reach a plateau but shows a power-law increase in time and individual trajectories intrinsically exhibit an amplitude scatter. In addition, we demonstrate that the mean maximal excursion effectively quantifies the extent of confinement, offering a robust measure for characterizing subdiffusive dynamics in complex systems.

Abstract This paper introduces a graph-theoretic strategy for solving nonlinear distributed-order fractional differential equations (NDOFDEs) by utilizing the Caputo fractional derivative and the clique polynomial of Star–Sun graphs. By encoding weight functions via clique polynomials, we achieve 85–92% error reduction versus Legendre-wavelet methods and 40% faster computation by transforming complex integration into summations. This novel weighting scheme offers a new way to represent distributed-order kernels. The Caputo derivative leverages our method's compatibility with initial conditions, while python-implemented clique computations validate scalability for massive graphs. Error behavior and convergence analysis confirm that the clique polynomial collocation method (CCM) remains stable compared to classical collocation techniques. Theoretical novelty lies in bridging clique polynomials with distributed-order calculus—a paradigm shift for multiscale modeling in viscoelasticity, signal processing, and anomalous diffusion. Future directions include orthogonalized bases and broader applications to complex solution profiles. Some of the highlights are as follows: (1) Developed a clique polynomial-based method for solving nonlinear distributed-order fractional differential equations. (2) Utilized the Caputo derivative to effectively manage initial conditions. (3) Demonstrated computational advantages by transforming integral equations into summation expressions. (4) Validated the proposed method through numerical comparison with existing schemes. (5) Implemented a python program for computing clique polynomials, facilitating further research in graph-theoretic applications. (6) Bridged graph theory and fractional calculus, fostering interdisciplinary applications in diffusion modeling, viscoelasticity, and signal analysis.

Anomalous diffusion of a two-state random process in a constant bias field

Anomalous diffusion in coupled viscoelastic media: A fractional Langevin equation approach

Diffusion on curved surfaces deviates from the flat case due to geometrical corrections in the evolution of its moments, such as the geodesic mean square displacement. Moreover, anomalous diffusion is widely used to model transport in disordered, confined, or crowded environments and can be described by a temporal subordination scheme, leading to a time-fractional diffusion equation. In this work, we analyze the dynamics of time subordinated anomalous diffusion on curved surfaces. By using a generalized Taylor expansion with fractional derivatives in the Caputo sense, we express the moments as a temporal power series and show that the anomalous exponent couples with curvature terms, leading to a competition between geometric and anomalous effects. This coupling indicates a mechanism through which curvature modulates anomalous transport.

ABSTRACT This paper introduces a novel analytical framework for deriving multiple soliton and singular soliton solutions to M-coupled fractional evolution equations. By integrating conformable fractional derivatives with an extended Hirota direct method, we systematically solve fractional versions of the KdV, mKdV, KP, and modified KP equations. The conformable derivative permits effective bilinearization, facilitating the construction of explicit solutions. We further provide a geometric interpretation through curvature analysis of soliton surfaces in fractional space. Theoretical results are validated against classical cases (α = 1), demonstrating consistency and enhancing the analytical toolkit for modeling wave propagation in nonlinear optics, plasma physics, and anomalous diffusion.

Physical mechanism analysis of anomalous diffusion characterized by scaling law

YaxA prepore formation underlies bipartite YaxAB toxin assembly on living membranes.

Understanding protein motion within the cell is crucial for predicting reaction rates and macromolecular transport in the cytoplasm. A key question is how crowded environments affect protein dynamics through hydrodynamic and direct interactions at molecular length scales. Using megahertz X-ray Photon Correlation Spectroscopy (MHz-XPCS) at the European X-ray Free Electron Laser (EuXFEL), we investigate ferritin diffusion at microsecond time scales. Our results reveal anomalous diffusion, indicated by the non-exponential decay of the intensity autocorrelation function g2(q, t) at high concentrations. This behavior is consistent with the presence of cage-trapping between the short- and long-time protein diffusion regimes. Modeling with the δγ-theory of hydrodynamically interacting colloidal spheres successfully reproduces the experimental data by including a scaling factor linked to the protein direct interactions. These findings offer insights into the complex molecular motion in crowded protein solutions, with potential applications for optimizing ferritin-based drug delivery, where protein diffusion is the rate-limiting step.

In this paper, we investigate the dynamics of travelling waves and wrinkle formation in graphene sheets using the thermophoretic motion equation and generalized Mittag-Leffler function (GMLF) with Caputo sense. The practical importance of understanding wrinkles in graphene to crucial for understanding the wave propagation in nanomaterials, as well as nanomechanical resonators and energy storage devices, and thermal management systems, where wrinkle dynamics can affect heat transfer. The limitations of prior models are that integer models may not fully capture the complex, memory-dependent, viscoelastic, and anomalous diffusion behaviors in graphene sheets. This limitation leads us to present this work, where a fractional-order model provides a more effective framework for accurately describing the intricate dynamics of wrinkle formation and diffusion. To bridge this gap, we introduce a new fractional form of the wrinkle graphene sheet model and transform the thermophoretic motion equation into a fractional form using Caputo fractional derivative (CFD) and GMLF. We obtain novel wrinkle-like soliton solutions for the proposed model. Our outcome is in good agreement with the exact solution of the original equation when the fractional order is equal to 1. We compare the exact and approximate solutions and present the results through 2D and 3D figures. We discuss the impact of system parameters on the dynamics of the obtained solutions. The findings reveal that the fractional order and model parameters significantly impact the solution dynamics. This implies that the fractional order and system parameters can serve as controllers for the dynamics of graphene wrinkles. The outcomes reveal the advantages and efficacy of the MGMLFM, which include that it does not require any transformation, perturbation, or linearization, unlike other techniques in the published papers. It is easily computable components, implemented directly on the problems, and produces approximate solutions of high accuracy with a small absolute error.

Tempered anomalous diffusion with stochastic resetting: a framework for adsorption kinetics

One-stop early noninvasive evaluation of renal allograft rejection and fibrosis: microstructural mapping via time-dependent diffusion MRI

Cutaneous candidiasis, mainly caused by Candida albicans, is a growing global health concern and is listed by WHO as a high-priority fungal threat. Suboptimal penetration of conventional vehicles limits the efficacy of current topical antifungals, increasing the risk of severe and invasive infections. Therefore, there is an innovative research field in advanced topical delivery systems to improve drug deposition, retention and antifungal efficacy. The main objective of this work was to develop nanocarriers based on hyalurosomes for the delivery of voriconazole (VCZ) and evaluate their potential to enhance the drug's cutaneous penetration and antifungal activity. Four VCZ-loaded hyalurosomal formulations were prepared (H1-H4) by modulating the proportions of phospholipid and polyols. Although changes in some physicochemical properties were observed, all the VCZ-loaded nanosystems were nanosized (< 140nm), spherical, multilamellar and exhibited high entrapments efficiencies (> 72%), excellent biocompatibility with human keratinocytes and potent antifungal activity against C. albicans. VCZ release from formulation H1 (1% phospholipid, 10% ethanol) followed a Fickian mechanism, while H2-H4 (4-10% phospholipid, 2.5-10% ethanol) exhibited anomalous diffusion involving both diffusion and matrix relaxation or erosion. Additionally, H1 and H2 (1-4% of phospholipid, 10% ethanol) achieved significantly enhanced drug penetration into deeper skin layers and superior in vivo antifungal efficacy compared to VCZ dispersion. The results highlight the potential of hyalurosomes as a next-generation topical antifungal delivery system, effective against both superficial and invasive candidiasis, with formulations H1 and H2 emerging as the most promising candidates for the treatment of the more invasive forms.

Numerical simulation of two-dimensional fractional space–time partial differential equation arising from anomalous diffusion with zero magnetic field gradient

This paper introduces the novel concept of Fractional-Order Neutrosophic MR-Metric Spaces (FoNMR-MS), which synergistically combines three powerful mathematical frameworks: MR-metric spaces, neutrosophic logic, and fractional calculus. We begin by extending the classical MR-metric structure through the incorporation of fractional integrals, defining a comprehensive fractional-order metric Mα and corresponding neutrosophic functions Tα, Iα, Fα. Fundamental properties including non negativity, identity, symmetry, and a generalized fractional triangle inequality are rigorously established. The core theoretical contribution is a comprehensive fixed point theorem for contraction mappings in complete FoNMR-MS, accompanied by detailed convergence analysis and neutrosophic consistency conditions. We further provide extensive examples and applications demonstrating the utility of our framework in modeling anomalous diffusion processes, image denoising, and machine learning under uncertainty. This work significantly generalizes existing results in fixed point theory and offers a robust mathematical foundation for handling complex systems characterized by fractional dynamics and neutrosophic uncertainty.

Nonlocal models offer a unified framework for describing long-range spatial interactions and temporal memory effects. The review briefly outlines several representative physical problems, including anomalous diffusion, material fracture, viscoelastic wave propagation, and electromagnetic scattering, to illustrate the broad applicability of nonlocal systems. However, the intrinsic global coupling and historical dependence of these models introduce significant computational challenges, particularly in high-dimensional settings. From the perspective of algorithmic strategies, the review systematically summarizes high-dimensional numerical methods applicable to nonlocal equations, emphasizing core approaches for overcoming the curse of dimensionality, such as structured solution frameworks based on FFT, spectral methods, probabilistic sampling, physics-informed neural networks, and asymptotically compatible schemes. By integrating recent advances and common computational principles, the review establishes a dual “problem review + method review” structure that provides a systematic perspective and valuable reference for the modeling and high-dimensional numerical simulation of nonlocal systems.

Anomalous diffusion is an important physical property of active Brownian particles and results mainly from their rich dynamic behavior. We use nonequilibrium molecular dynamics methods to study the mass diffusion of two-dimensional active particles under concentration gradients. It is found that active particles exhibit anomalous collective diffusion behavior that violates Fick's law, whereby their diffusion coefficient diverges as the system size increases. The particles exhibit a flow pattern characterized by large-scale structures, where their movement trajectories and directions are correlated and synchronous. This correlation gives rise to the phenomenon of superdiffusion. This work provides a detailed interpretation of the divergent characteristics and physical mechanisms of active particles based on long-time correlations, velocity spectrum analysis, and local synchronization rates. Our results offer a path for deeply understanding the nonequilibrium dynamic properties of active particles under complex conditions.

A novel fractional-order susceptible-infected-susceptible (SIS) epidemic model incorporating anomalous diffusion in heterogeneous networks is proposed, derived from the continuous-time random walk (CTRW) framework, to capture the significant effects of individual residence times and network topology on epidemic spreading. We begin by deriving the fractional-order reaction-diffusion model on complex networks from the CTRW framework. The existence and uniqueness of solutions on [0,∞), as well as those of the disease-free and endemic equilibria, are then established. Subsequently, we analyze the global asymptotic stability of the disease-free equilibrium when R0<1 and the local asymptotic stability of the endemic equilibrium when R0>1. Finally, the Ulam-Hyers stability of the SIS epidemic model is investigated. Theoretical analysis and numerical simulations demonstrate that the memory effects induced by power-law waiting times influence the convergence rate of the solution toward the steady state. When R0>1, the density of infected individuals converges to 1-1R0. Moreover, the steady-state number of infected individuals on each network node is approximately linearly related to the degree of node, and the number of individuals on each node converges to a constant multiple of the eigenvector component corresponding to the 0 eigenvalue of matrix A, clearly reflecting the effect of network topology on disease transmission.