Introduction: The repair of large segmental bone defects has always been a major challenge in clinical practice, and "stress shielding" caused by the mismatch in elastic modulus between the bone implant and the native bone is one of the key factors leading to implant failure.

Methods: Inspired by the branching structure of tree branches and trabecular bones, this study constructs a multi-level fractal scaffold geometric model with controllable fractal dimension, porosity, and pore size distribution. The influence of key structural parameters such as fractal iteration times, pore size, and trabecular diameter on the overall elastic modulus, compressive strength, and yield behavior of the scaffold was systematically studied using a combination of finite element calculations and mechanical experiments. Furthermore, scanning electron microscopy (SEM) was used to characterize the microstructure of the scaffolds fabricated by 3D printing.

Results: The porosity distribution of the 2-iteration and 3-iteration models is consistent with that of natural bone, featuring a radial gradient distribution with larger internal porosity and smaller external porosity. The elastic modulus of the 3-iteration model is 21% lower than that of the 2-iteration model, while its compressive strength is 17.3% higher than that of the 2-iteration model. The pore size of the fabricated scaffolds is consistent with that of the designed model, with a pore size error within 10%, indicating that the multi-level fractal scaffold designed in this study has good processability.

Conclusions: The 3-iteration fractal scaffold has higher strength and lower elastic modulus, which is similar to the mechanical properties of natural bone and can effectively alleviate stress shielding. The bionic scaffold based on fractal theory and the research on its structure-modulus matching proposed in this study can provide a new idea for alleviating the stress shielding of implants.

In this paper, we present a numerical method for solving inverse problems associated with nonlinear fractional-order differential equations, with the goal of identifying an unknown right-hand side function from over-measured data. The proposed approach is based on a newly introduced hybrid basis, referred to as the fractional-order hybrid Fibonacci function, constructed by combining block-pulse functions with Fibonacci polynomials. This hybrid representation exploits the strong approximation capabilities of Fibonacci polynomials together with the effectiveness of block-pulse functions in modeling discontinuous behavior, allowing for accurate approximation of both continuous and discontinuous solutions. The fractional-order feature is incorporated through the transformation x ->x^α applied to the Fibonacci polynomials, where α is a real parameter. To the best of our knowledge, this hybrid basis is employed for the first time in the context of inverse problems for fractional differential equations. An exact Riemann-Liouville fractional integral operator is derived in closed form using the regularized beta function. By expanding the solution in terms of the proposed hybrid functions, the inverse problem is reduced to a system of algebraic equations with unknown coefficients corresponding to the right-hand side function. Discretization using Newton-Cotes quadrature nodes leads to a homogeneous system of algebraic equations, from which the coefficients of the hybrid expansion are determined. Substituting these coefficients into the solution representation allows for the recovery of the unknown nonlinear term. Several numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method. The results indicate that the use of an exact fractional integral operator significantly improves the quality of the approximation when compared with existing numerical approaches.

Multifractality has become a central concept for characterizing complexity across scientific domains. Yet, its reliable detection in empirical time series remains challenging due to the interplay of temporal correlations and heavy-tailed fluctuations. Recent analytical and numerical results demonstrate that genuine multifractality cannot arise in the absence of long-range temporal correlations. In contrast, apparent multifractal spectra observed in shuffled or short uncorrelated data are finite-size artifacts or manifestations of bifractality in Lévy-stable regimes. Motivated by these findings, we revisit multifractal detrended fluctuation analysis (MFDFA) as the most stable and widely used tool for testing multiscaling properties in complex systems, from financial markets to natural language. We show, using controlled multiplicative cascades, how the distribution of fluctuations—modulated via q-Gaussian transformations—can broaden the singularity spectrum only when correlations are already present. This provides a principled way to separate true multifractal effects from those induced by fatter tails. We further demonstrate that empirical systems often exhibit multiscale organization consistent with correlation-driven multifractality, including financial markets, where nonlinear dependencies and volatility clustering contribute to the observed spectra. Additional analogies arise in linguistic time series, where long-range dependencies and hierarchical structure produce fractal or multifractal patterns even in symbolic data. Building on these theoretical and empirical insights, we propose a robust protocol for testing multifractality that incorporates surrogate data, tail-stabilized reference distributions, and scale-range diagnostics. This procedure allows researchers to quantify how much of the spectrum width originates from temporal organization versus the heaviness of fluctuations. The talk concludes by emphasizing that properly disentangling these components is essential for interpreting multifractality as a genuine marker of complexity in natural and socio-economic systems.

Earthquakes in the Himalayan region primarily result from the ongoing convergence of the Eurasian and Indian tectonic plates, with the Indian plate being underthrust beneath the Eurasian plate, leading to complex seismic activity. The Nepal region, located within a seismic gap, is especially significant due to its potential for large earthquakes. In this study, we investigate the temporal evolution of seismicity in the Himalayan region to identify precursor signals preceding the 2015 Mw 7.8 earthquake in Nepal Himalaya, with a focus on the fractal characteristics of seismic events. The magnitude of completeness (Mc) for the dataset was found to be 3.9 and above. The sliding window analysis using 100-event windows showed variation in fractal correlation dimension (Dc) values over time. During the first nine years, the trend increases; later, it significantly decreases. Increasing and decreasing trends demarcate the two time periods during which the fractal behaviour of earthquake occurrences differs. We detected unusual patterns in the fractal correlation dimension (Dc) of seismic events, with values ranging from 0.94 to 1.66. A sinusoidal-like pattern emerged in the initial years of the dataset (within the first 9 years), followed by a notable change in the fractal dimension (after 9 years), just prior to the main earthquake. This indicates a shift towards stronger clustering of earthquakes as stress built up in the region. The patterns in the fractal dimension suggest significant changes in seismic behaviour before 11 years leading up to the 2015 earthquake, indicating stress accumulation and the likelihood of a large seismic event. Notably, the time period of the decreasing trend exhibits a comparatively higher crustal stress than the period of the increasing trend. This study provides insights into long-term stress accumulation patterns and their application in seismic hazard assessments.

This work develops a rigorous analytical framework for the study of nonlinear fractional pantograph equations involving sequential Riemann–Liouville derivatives of different fractional orders together with nonlocal integral boundary conditions. These models arise naturally in systems that exhibit memory, scaling effects, and nonlocal interactions, making their qualitative analysis both challenging and essential. The methodological approach relies on a combination of Schauder’s fixed-point theorem and the Banach contraction principle to obtain fundamental results on the solvability of the problem. Within the Banach space $L^{1}(J)$, we establish the existence of at least one solution by demonstrating the compactness of the associated integral operator through the Kolmogorov compactness criterion. Under suitable Lipschitz-type assumptions, uniqueness is further guaranteed by showing that the corresponding operator is a contraction.
In addition, we investigate the continuous dependence of solutions on initial data and nonlinear terms, confirming the stability and robustness of the proposed model under small perturbations. An illustrative example is included to verify the theoretical framework and to demonstrate how the analytical results can be applied in practice. Overall, the findings of this study contribute to the qualitative theory of fractional pantograph systems and provide a unified basis for modeling complex nonlocal phenomena with memory effects in applied mathematics, physics, and engineering.

The Gamma function Γ(z) is fundamental in analysis and fractional calculus; however, its poles at the negative integers prevent its direct use as an integral transform kernel. In this paper, we introduce the Gamma transform, a novel analytic operator derived from a distributional decomposition of Γ(z). By representing the Gamma function as a convergent series of complex delta functionals, we obtain a regularized formulation that is free of singularities and well defined for real, fractional, and complex orders. The proposed transform exhibits structural properties analogous to classical integral transforms, including the Laplace and Mellin transforms, while incorporating discrete factorial-type weighting inherent to the Gamma function. Fundamental properties of the Gamma transform are established, including linearity, boundedness, inversion, and convolution theorems, ensuring consistency with standard operational calculus. This framework naturally extends classical transform theory to fractional settings without loss of analytic rigor. Applications to ordinary and fractional differential equations are presented to demonstrate the effectiveness of the Gamma transform in solving initial- and boundary-value problems, where it provides compact representations and inherent regularization. The results illustrate how the transform unifies classical analytic methods with modern fractional analysis. The Gamma transform offers a new, singularity-free tool for fractional calculus and related fields, contributing to the development of analytic techniques within the context of fractal and fractional systems.

The definition and properties of the multiplicative derivative for the function $f: I^{\circ}\subseteq \mathbb{R}\to \mathbb{R^+}$ are given at the beginning of this abstract. Additionally, this paper gives a comprehensive description of the multiplicative integral and clarifies the characteristics that differentiate it from other types of integrals. In the realm of non-Newtonian calculus, the technique of multiplicative integration by parts is considered an important instrument. Secondly, this constitutes the first definition of $h$-convexity that is multiplicative by the use of the $B$-function, in addition to a number of cases in which convexity is multiplicative. This research provides evidence of a new form that utilizes the multiplicative absolute value, which is represented as $| \cdot |^{*}$, as well as other distinctive traits that have not been seen before. A presentation is offered that discusses the multiplicative Riemann–Liouville fractional integral operators of order $\alpha > 0$. These operators are formed based on a function $f : [a,b] \to \mathbb{R^{+}}$, where $\mathbb{R^{+}}$ represents the set of positive real numbers. This strategy framework proposes a new technique for dealing with the idea of multiplicative Weddle inequalities, and it is outlined in this publication. This is based on the ideas of fractional integrals as described by Riemann–Liouville. We put forward evidence that suggests the existence of yet another level of inequality. This conclusion is reached on the premise that the function in question is a multiplicative h-convex and that a multiplicative absolute value with unique properties exists. The findings that are provided are calculated by using multiplicatively $P$-functions in conjunction with multiplicatively $s$-convex functions.

In this talk, contributions by Abel and Sonin to the origins of the general fractional integrals and derivatives with the Sonin kernels are first discussed. While Abel introduced and employed the integral and integro-differential operators with the power law kernels that are nowadays referred to as the Riemann–Liouville fractional integral and the Caputo fractional derivative, Sonin extended his method to the case of pairs of arbitrary kernels whose Laplace convolutions are identically equal to one.

Following this, we mention some recent results regarding the properties of general fractional integrals and derivatives with the Sonin kernels. In particular, the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives, the regularized general fractional derivatives, and the sequential general fractional derivatives are formulated in the appropriate spaces of functions.

As an application of this theory, we discuss the convolution series that are a far-reaching generalization of the power law series as well as a representation of functions in form of the generalized convolution Taylor series. Another application is the generalized convolution Taylor formula that contains convolution polynomials and remainders given in terms of the general fractional integrals and the general fractional sequential derivatives. A known case of this formula is the fractional Taylor formula with polynomials involving the power law functions with the fractional exponents and a remainder in terms of the Riemann–Liouville fractional integrals and the sequential Riemann–Liouville fractional derivatives.

We study non-autonomous semilinear evolution equations

$\partial_t^\alpha u=A(t)u(t)+J(u(t))$ $t\in (0,T)$

$u(0)=u_0$

with fractional-in-time derivatives, governed by sectorial operators A(t) that satisfy the classical Acquistapace–Terreni conditions.

These conditions ensure the well-posedness of the associated linear evolution families despite the lack of time invariance. Our analysis introduces the fractional solution operators S_alpha(t, τ) and P_alpha(t, τ), for which we establish ultracontractivity estimates that generalize classical heat-kernel bounds to the fractional and non-autonomous setting. These estimates provide a crucial tool for controlling nonlinearities.

Building on this linear foundation, we address the semilinear equation through fixed-point arguments formulated in weighted function spaces adapted to fractional temporal behavior. We prove local well-posedness for a broad class of nonlinearities, requiring only localized Lipschitz continuity and suitable growth conditions. Furthermore, under additional smallness assumptions on the initial data, we obtain global-in-time existence results. These findings extend and refine existing theories for both autonomous fractional equations and classical parabolic problems.

To illustrate the applicability of our abstract theory, we discuss a fractional heat equation with time-dependent, uniformly elliptic operators in non-divergence form. This example highlights how the developed framework accommodates PDEs with variable coefficients, nonlinear effects, and fractional temporal dynamics.

Reference: S. Creo and M. R. Lancia, Non-autonomous semilinear fractional evolution equations: well-posedness and ultracontractivity results, 2025.

We study uniform weak convergence rates for probabilistic numerical schemes that approximate solutions of time-fractional diffusion equations with unbounded coefficients. The spatial part of the model is generated by a diffusion process with linearly growing drift and diffusion coefficients, which includes, in particular, geometric Brownian motion. The time-fractional structure is described by the classical Caputo–Dzherbashian derivative. In probabilistic terms, the solution can be represented as a subordinated Markov process obtained by time-changing a diffusion with the inverse of a stable subordinator.

We investigate a solution method based on continuous-time random walk (CTRW) approximations. The construction combines discrete-time Markov chain schemes for the diffusion component with heavy-tailed random walk approximations of the stable subordinator.

Our analysis develops a high-order sensitivity framework for the associated diffusion semigroup, relying on Kunita’s stochastic flow theory and the chain rule for tensor fields. This allows us to control derivatives of the flow with respect to the initial condition as random tensor fields, obtain uniform (in space) and exponentially growing (in time) bounds for all orders of sensitivities, and derive corresponding estimates for the derivatives of the Markov semigroup. We also establish a quasi-contraction property of the semigroup in suitable weighted function spaces.

Using these ingredients, we prove explicit uniform weak convergence rates for the CTRW approximation applied to smooth test functions of linear growth. Depending on the spatial part of the equation, the convergence rate is either logarithmic in the time step or follows a power law. Our results extend previous CTRW convergence theory from bounded to linearly growing coefficients, including applications to fractional Black–Scholes models.