In this study, within the framework of recent advances in fractional-order differential and integral operators, a fractional differential equation defined by means of a generalized Caputo-type fractional derivative is investigated. The proposed approach extends the classical Caputo fractional derivative by introducing a generalized kernel structure, thereby allowing a more flexible and comprehensive description of memory and hereditary effects. From this perspective, this study contributes to the theoretical development of fractional-order operators and their analytical properties.

The considered fractional differential equation involves the Mittag–Leffler function, which plays a fundamental role in the analysis of fractional-order dynamical systems. This special function naturally arises in the solution structure of fractional differential equations and is essential for characterizing the qualitative and quantitative behavior of fractional-order models. To derive an explicit analytical solution, the Laplace transform method is employed as a powerful and systematic tool for handling generalized Caputo-type operators. By applying the Laplace transform, the fractional differential equation is transformed into an algebraic equation in the Laplace domain, significantly simplifying the analytical treatment of the problem.

Subsequently, the inverse Laplace transform is applied to obtain the exact solution in closed form, expressed in terms of the Mittag–Leffler function. This result clearly demonstrates the intrinsic connection between generalized fractional derivatives and special functions commonly used in fractional calculus. Moreover, it is shown that, under suitable parameter selections, the proposed model reduces to the classical Caputo fractional differential equation, and several well-known results available in the literature are recovered as special cases.

The methodology and results presented in this work provide a unified and effective framework for the analysis of a broader class of fractional differential equations involving generalized kernels. The proposed model and solution technique offer valuable insights into the behavior of fractional-order systems and have potential applications in mathematical physics and various applied sciences where complex memory-dependent processes are encountered.

This study investigates the existence and nonexistence of solutions for a nonlinear fourth-order viscoelastic equation with a fractional damping in a bounded domain. The model at hand describes the dynamic behavior of elastic plates, taking into account both memory effects and nonlocal dissipation. In this framework, the viscoelastic term is represented by a convolution with a suitable relaxation kernel, while the damping mechanism is defined through a fractional power of the associated linear operator.

To establish the local existence, we reformulate the governing equation as an abstract evolution system within an appropriate Hilbert space. By demonstrating that the linear operator generates a strongly continuous semigroup of contractions and ensuring the local Lipschitz continuity of the nonlinearities, we apply semigroup theory to prove the local existence and uniqueness of solutions. Furthermore, we provide conditions under which these solutions attain the regularity of weak solutions based on the initial data.

Regarding long-term behavior, we derive energy identities and a priori estimates to show that solutions exist globally for sufficiently small initial data. Here, the interplay between viscoelastic memory and fractional damping enhances the dissipative structure, which is vital for the stabilization of the system. Conversely, we explore the potential for finite-time blow-up when the system is driven by large initial energy. Using a Lyapunov functional approach and the concavity method, we establish that solutions fail to exist globally under specific conditions related to the nonlinear source. Our findings also highlight the role of the fractional damping term, noting that while it may delay the blow-up, it cannot entirely prevent it in the presence of strong nonlinearities. This work extends the existing literature by integrating fractional calculus into the analysis of fourth-order evolution equations, offering new insights into the balance between memory effects and nonlocal dissipation.

Fractional calculus, which enables the modeling of dynamic systems by generalizing the concepts of derivatives and integrals to fractional orders, is an important research topic. Fractional calculus is widely used in many scientific fields because it models real-world events more accurately than integer-order systems. This research aims to present new stability criteria for fractional-order neutral systems using a delay decomposition approach. One of the most important indicators of the qualitative behavior of fractional order differential equation systems is system stability. The Lyapunov method and the linear matrix inequality technique we used in this study are the most preferred methods. The stability criteria are obtained by constructing appropriate Lyapunov-Krasovskii functionals and using linear matrix inequalities. The difficulty of obtaining a fractional-order Lyapunov functional lies in how to design a positively defined functional V and easily determine whether the fractional derivative of V is less than zero. The Lyapunov method used in this study offers the advantage of being able to directly calculate the integer-order derivatives of the system under consideration. In conclusion, some theoretical results have been obtained for the fractional-order neutral type systems considered. A few simple examples are presented using MATLAB and Simulink to demonstrate the effectiveness and applicability of these theoretical results.

In this talk, we discuss recent advances in the theory of resolvent operators for abstract fractional differential equations. Our focus is on the construction and analysis of resolvent operators associated with abstract evolutionary fractional integro–differential equations involving the Caputo derivative of order $\alpha \in (0,1)$. Using tools from perturbation theory, we establish sufficient conditions for the existence and well-posedness of the corresponding fractional resolvent operators in Banach spaces.

By employing the concept of the fractional potential of a strongly continuous semigroup of linear operators, we derive fractional power estimates for the associated fractional resolvent operator. These estimates play a crucial role in the qualitative analysis of solutions and allow us to relax spatial regularity assumptions on the forcing terms appearing in the equations. As an application of these results, we prove the existence of mild solutions for a class of fractional neutral integro–differential equations with infinite delay, formulated in suitable phase spaces.

Finally, we illustrate the applicability of the developed theoretical framework by studying a coupled fractional partial integro–differential system involving the Caputo derivative. For this system, we establish the existence of mild solutions by combining the obtained resolvent estimates with fixed point arguments. The results presented in this talk provide a unified and robust approach to the analysis of fractional evolution equations with memory and delay effects, contributing to the ongoing development of the theory of fractional dynamical systems.

In this study, we discuss the boundary value problem involving a hybrid Caputo–Hadamard fractional sequential differential equation with variable order. The use of a variable fractional order is motivated by the fact that, in many applications, memory effects and system behavior change over time rather than remaining constant. Therefore, fractional models with constant order may be inadequate for describing the dynamics of complex systems. Our primary tool for constructing the results is the Banach space of continuous functions. Existence and uniqueness results are obtained by applying Darbo’s fixed-point theorem in combination with Kuratowski’s measure of noncompactness. This approach allows us to deal with operators that are not compact while still ensuring the existence of solutions. The hybrid concept, in which Caputo and Hadamard fractional derivatives appear sequentially, offers a flexible formulation that can describe a wider class of problems than models involving a single fractional operator. A key idea in the approach is reformulating the considered boundary value problem as an equivalent operator equation. This representation enables the direct application of fixed-point theorems and facilitates a discussion of solution uniqueness under additional suitable assumptions that are easy to verify rationally. The stability of solutions is also examined. Using the concept of Ulam–Hyers stability, it is demonstrated that small perturbations in the problem data result in bounded variations in the corresponding solutions. To illustrate the theoretical results, an example supported by numerical computations is included. The results indicate that the proposed technique is suitable for problems in which system properties evolve with time.

Fractal growth processes frequently emerge in systems characterized by anomalous transport, memory effects, and long-range correlations, features that exponential relaxations fail to capture. This motivates a central question: does fractal pattern formation give rise to power-law (fractional) kinetics, and if so, how is the fractional order linked to geometric properties such as fractal dimension? The authors explore this question through various experiments and the proposed theory of fractal elements.

Theisstudy focuses on Lichtenberg figures generated by high-voltage discharges on wood surfaces, a striking example of fractal pattern formation in a heterogeneous dielectric medium. When a wooden substrate with anisotropic conductivity and moisture variability is subjected to electric fields above its breakdown threshold, the discharge propagates through branching streamers and carbonization fronts. These fronts display long-tailed waiting-time distributions and memory-dependent evolution.

Experimental evidence reveals that the discharge does not advance with constant velocity or classical exponential behaviour. This raises a key question: Does the electrical current measured during breakdown also exhibit fractional behaviour?

To investigate this, the authors use the recent theoretical framework of Nigmatullin and Chen (2023), which provides a rigorous method for analyzing complex, self-similar signals. Their “general theory of fractal elements” shows that an experimental waveform can be decomposed into elementary self-similar modes, each associated with a power-law scaling exponent and amplitude. This decomposition reveals that the current signal is a structured combination of fractal components encoding the underlying fractional dynamics.

Essentially, the essence of this theory is decomposition of a complex self-similar signal on a combination of fractal modes governed by a set of power-law exponents that can be confirmed on many self-similar processes (in time and space). The relationship between fractal dimension and power-law exponents governed by fractal dynamics is not found due to the influence of the medium structure, where the discharge current is propagated.

The understanding of urban morphology has been enhanced by mathematical and computational approaches capable of quantifying the spatial complexity of cities. This paper investigates the hypothesis that the fractal complexity of the urban fabric varies significantly as a function of socioeconomic stratification in Alagoinhas, Bahia, Brazil. To this end, the analysis of the Fractal Dimension (FD) was employed on residential samples from three income groups (Class A: R$ 2,000; Class B: R$ 1,000 - 1,500; Class C: R$ 600), aiming to fill a gap in the understanding of the articulation between urban structure and spatial justice. The Box-Counting methodology was applied to 30 georeferenced samples (ten per class), obtained from high-resolution images (Google Earth/CNES) and standardized in square buffers of 200 meters side. The log-log regression analysis for the FD estimation demonstrated high statistical robustness, with the R-squared consistently at 1.000 for all samples. The results revealed that all classes exhibit high density, with FD values close to 2.0. Class A (mean FD 1.992) showed the highest average spatial complexity and the lowest intrinsic variation, suggesting more consolidated and detailed occupation patterns. Class C (mean FD 1.990) demonstrated the highest morphological uniformity (standard deviation of 0.002), reflecting homogeneous and dense occupation patterns, typical of peripheral areas with lower infrastructure. Class B (mean FD 1.988) exhibited the greatest internal variation, characterizing itself as a morphological transition zone. The comparative analysis (regression and K-means) highlighted a complete absence of spatial correlation between Class A and Class C, confirming the hypothesis that socioeconomic segregation manifests in structurally distinct urban patterns. We conclude that the fractal dimension is an effective metric for quantifying inequality in the production of space, demonstrating how social stratification in Alagoinhas is mapped onto its urban morphology.

Fractal Dimension (FD) is a powerful computational neuroscience method that captures local oscillations and network topology through univariate measures. A central concept in neuroscience is criticality, which describes the brain's optimal operating state at the edge of chaos, balancing stability and flexibility. Quantifying this delicate balance is a frontier in neuroscience.

The Hurst Exponent (HE) is one of the most popular FD measures and a key way to quantify the balance of criticality. The HE measures scale-free, long-term memory in temporal dynamics and has been confirmed as an essential performance indicator of brain dynamics.

In this research, we investigated the effect of healthy ageing on known resting-state functional networks (RSNs) using the HE. Using Group-Independent Component Analysis (GICA) on fMRI data from young adults (YAs) and older adults (OAs), we characterised baseline fractality in the YA group. We found that increasing HE values differentiates between subcortical, primary, and cognitive networks. This suggests that greater temporal complexity may reflect increased integrative processing in the brain.

Comparing the groups revealed widespread significant differences between young and old adults, with a general loss of scale-free long-term persistence (a decline in HE) across networks. This change may be specific to a loss of integration capability in higher cognitive functions.

Specifically, the visuospatial and dorsal default mode networks (dDMNs) were the most affected by ageing. Machine learning classifiers highlighted them as the best predictors of ageing based on the HE. Interestingly, these high-order networks exhibited a signal complexity level in OAs that resembled that of subcortical structures. We also observed increased segregation (functional decoupling) within the dDMN, as the ventral part (vDMN) showed no significant difference in HE between YA and OA.

The HE can detect early deviations from optimal function and may mark the transition from healthy ageing to pathology.

The outcomes of Transcranial Magnetic Stimulation (TMS) depend critically on the momentary intrinsic cortical state, resulting in high variability that compromises clinical efficacy. To enable reliable, personalised TMS, closed-loop systems are essential for predicting and targeting favourable brain states in real-time.

This study proposes a novel TMS-EEG approach based on a Hybrid Fuzzy Logic System (FLS) to predict single-trial brain responsiveness accurately. Inputs extracted from the pre-stimulus TEP included Power Spectral Densities across canonical bands (delta to gamma) and the non-linear measure Higuchi Fractal Dimension (HFD), which reflects network complexity. The post-stimulus response was quantified using the Area Under the Curve (AUC), with trials labelled as ‘low’ or ‘high’ responders.

The FLS is uniquely suited to model the non-linear relationships of biological data, providing transparency and interpretability absent in 'black box' models. It utilises a hybrid rule-based inference system integrating Expert-defined rules (neurophysiology-based) and Data-driven rules from a Random Forest classifier to generate understandable, linguistic mappings.

Across 1560 trials, the model achieved classification accuracy of 73% and a Cohen's Kappa score of 0.46. Rule inspection confirmed that brain states characterised by reduced HFD and reduced beta and gamma power were associated with 'high' responsiveness. This research validates the FLS as a transparent, high-performance computational engine, representing a substantial step toward practical adaptive TMS protocols guided by real-time brain-state prediction to maximise therapeutic efficacy.

Lacunarity analysis has proven effective in distinguishing between multifractal grayscale patterns that share the same correlation dimension but exhibit differing degrees of scale-dependent clustering. In our earlier studies, we demonstrated that such scale-dependent clustering significantly influences fluid flow behavior in fractal-fracture networks, even when the fractal dimensions remain constant. Building on this foundation, the present research explores whether lacunarity, a metric quantifying heterogeneity and spatial clustering, can serve as a predictor of fluid flow behavior in multifractal permeability fields. To investigate this, we considered a set of four multifractal patterns (three deterministic and one random), each characterized by identical correlation dimensions and defined by the parameters b = 7, p = 40/49, and three iterations. These patterns are subjected to lacunarity analysis using the gliding-box algorithm to quantify their scale-dependent clustering, where lacunarity (L) is related to the correlation dimension, D₂, by the equation dlog(L)/dlog(r) = D₂ - 2. We empirically tested this equation using two-dimensional multifractal grayscale patterns with known correlation dimensions. Subsequently, flow simulations are performed on these permeability fields using Trace3D: a streamline-based simulator that generates streamlines in 3D space and then solves the 1D equations analytically or numerically along the streamlines. In these simulations, the occupied regions of the multifractal fields are assumed to be highly porous and permeable, enabling all fluid flow to occur exclusively through these zones. The simulation results reveal a strong positive correlation of 95% between lacunarity and fluid recovery, indicating that increased clustering (as measured by lacunarity) enhances flow efficiency in such heterogeneous systems. These findings suggest that lacunarity is a valuable parameter for predicting fluid flow behavior in multifractal permeability fields and can potentially generate informed decisions on reservoir characterization and field development modeling strategies.