Search Results (557)

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

q-calculus and fractional calculus combined with geometric function theory lead to remarkable results. The fractional integral introduced by Riemann–Liouville applied to the q-multiplier transformation is used in this research to study the two dual theories of fuzzy differential subordination and fuzzy differential superordination and to develop specific fuzzy results. In the theorems examining fuzzy differential subordinations and fuzzy differential superordinations, fuzzy best dominants and fuzzy best subordinants are also provided. In addition, the demonstrated outcomes reveal corollaries by taking specific functions with established geometric features into consideration as the fuzzy best subordinant and fuzzy best dominant. The work concludes with a fuzzy differential sandwich theorem and related corollaries that combine the findings of this research on fuzzy differential subordinations and superordinations. Full article

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order $1<\delta <2$ with infinite delay. The analysis is carried out in an abstract Hilbert space framework, incorporating fractional dynamics together with stochastic perturbations. By employing techniques from fractional calculus, semigroup theory, and fixed point theory, particularly the Banach contraction principle along with compactness arguments, we establish the existence of mild solutions for the proposed system. Subsequently, sufficient conditions for approximate controllability are derived by combining operator-theoretic methods with stochastic analysis. The novelty of this work lies in extending controllability results to Sobolev-type fractional stochastic systems of order $1<\delta <2$, where both the higher-order fractional structure and stochastic effects are treated simultaneously within a unified framework. This generalizes and complements several existing results in the literature that mainly address deterministic systems or fractional differential equations of order $0<\delta \le 1$. Finally, an illustrative example is presented to demonstrate the applicability and effectiveness of the theoretical findings. Full article

The long-term prestress relaxation of soil anchors embedded in saturated clay is a critical issue affecting the safety of geotechnical structures such as slopes and foundation pits. Traditional integer-order constitutive models are often unable to accurately describe the nonlinear and time-dependent relaxation behavior observed in such anchorage systems. Based on fractional calculus theory, this study establishes a constitutive model for the coupled stress relaxation behavior of soil anchors and saturated clay. The Riemann–Liouville fractional derivative and the two-parameter Mittag-Leffler function are introduced to represent the material memory effect and continuous relaxation characteristics. To achieve reliable parameter identification, a hybrid optimization strategy combining the Adaptive Hybrid Differential Evolution (AHDE) algorithm and the Levenberg–Marquardt (L-M) method is proposed. The proposed model and identification approach are validated using field monitoring data from soil anchors in a slope engineering project at the Guangxi Friendship Pass Port. The results show that the proposed model can accurately reproduce the entire stress relaxation process, with a coefficient of determination of R² = 0.9517. Parameter sensitivity analysis further clarifies the influence of key parameters, including the fractional order and viscosity coefficient. The proposed approach provides a systematic theoretical framework and practical reference for the analysis and prediction of long-term prestress relaxation in soil anchorage systems. Full article

The large-scale rapid deployment of renewable generation and energy storage is transforming traditional power system dynamics through intermittency, reduced inertia, and pronounced long-range temporal dependence. Existing power system modeling frameworks are primarily based on short-memory assumptions and integer-order dynamics, which are unable to capture the persistence and oscillatory behavior of emerging renewable-dominant power systems. This structural mismatch leads to inaccurate system representation and degraded long-horizon prediction performance. Although fractional calculus has been applied to specific control and forecasting tasks in power systems, the joint system-level modeling of renewable generation and load demand using real-world data remains largely unexplored. In this paper, we develop a data-driven fractional-order dynamic modeling framework that explicitly incorporates long-memory effects into the governing equations through fractional differential equations based on the Caputo formulation. Using publicly available high-resolution datasets of load and renewable generation, empirical analysis reveals power-law decaying autocorrelations and dominant low-frequency spectral characteristics that motivate the use of fractional-order dynamics. Fractional orders and model parameters are jointly identified through prediction-error minimization to ensure consistency between modeled trajectories and observed persistence. The numerical results demonstrate that the proposed approach achieves a root–mean–square error of $3.12$, compared to $5.64$ and $4.98$ for integer-order and finite-memory models, respectively, and reduces the normalized root–mean–square error from $0.156$ and $0.132$ to $0.087$. Residual and spectral analyses further confirm that long-memory behavior is effectively captured by the proposed dynamics. The framework provides a scalable and physically interpretable foundation for the data-driven modeling of renewable-dominant power systems. Full article

In this paper, we introduce a new logarithmic integral operator that unifies differentiation and fractional integration within the complex domain. The present work addresses this gap by applying the proposed operator to analytic functions represented by alternating power series. The method demonstrates that the coefficients can be reorganized in a controlled manner without affecting convergence or analytic behavior. Using this framework, we derive third-order differential subordination and superordination results, which naturally lead to corresponding sandwich-type results. The findings confirm that the introduced operator offers an effective analytical tool for studying distortion, growth, and mapping properties of analytic functions, with promising potential for future applications in fluid mechanics. Full article

Recently, piecewise differential operators have been introduced to capture crossover dynamics in physical systems. In the evolution of corruption, the underlying dynamics can shift across different regimes. These crossovers occur due to policy changes, economic shocks, or shifts in social behavior. To demonstrate the crossover dynamics of a corruption mathematical system, we use a piecewise operator. The piecewise operator is divided into three pieces: a classic or integer order operator, a fractional operator, and a stochastic operator. For the fractional order case, we use the constant proportional Caputo (CPC) operator, which is a straightforward linear combination of the Riemann–Liouville (RL) integral and the Caputo derivative. Theoretical analysis such as existence and uniqueness of solutions for the fractional case under CPC derivative, is elucidated via notions of fixed point theory, specifically the implication of Perov’s fixed point result and for the stochastic model using Ito calculus. Numerical results are presented for the proposed model. Graphical analysis of the corruption model is performed using PW operators across three distinct intervals to portray the crossover dynamics of the considered system. Also, the influence of various parameters on the crossover dynamics of the corruption model is illustrated via numerical simulations. Sensitivity of parameters is demonstrated via some statistical experiments, such as scatter plots and Pearson correlation coefficients, quantifying the relationship between key parameters of the system. The validity of the result is verified by comparing the system dynamics with real data dynamics via 2D graphs. Full article

With the advent of ubiquitous healthcare and advancements in textile industry, non-invasive wearable biomedical solutions are becoming an increasingly attractive alternative to in-hospital monitoring, allowing for timely diagnostics and prediction of severe medical conditions. Non-contact biopotential monitoring is particularly promising because non-contact biopotential electrodes can be applied over clothing or embedded in the material without almost any preparation. However, due to the intricacies of capacitive coupling they rely on, the design of such electrodes and their interface with the body plays a key role in achieving measurement repeatability and their widespread utilization in clinical-grade diagnostics. Based on exhaustive investigation of several decades of the literature on non-contact and capacitive biopotential electrodes and electric potential sensors, this study is intended to serve as a state-of-the-art overview of their historical development and design challenges, a collecting point for important research theories and development milestones, a starting point for anyone seeking for a soft head start into this research area, and a remedy for occasional misnomers and conceptual errors identified in the existing papers. The ultimate goal of this comprehensive analysis is to demystify phenomena of non-contact biopotential monitoring and capacitive coupling, systematically reconciliate terminological inconsistencies, and enhance accessibility to the most important findings for future research. To accomplish this, fundamental concepts are thoroughly revisited—from fundamentals of electrochemistry and working principles of capacitors and operational amplifiers to system stability and frequency-domain analysis. With the use of various mathematical tools (Laplace transform, phasors and Fourier analysis, and time-domain differential calculus), discussions on non-contact and capacitive biopotential electrodes, collected from the 1960s onward, are for the first time compiled into a unified, abstracted, bottom-up analysis. The laid-out inspection provides analytical explanation for various aspects of measurement results available in the referenced literature, but also serves an educative purpose by devising a methodological framework that can be easily applied to other similar research fields. Firstly, the differences and similarities between wet, dry, surface-contact, non-contact, capacitive, insulated, on-body, and off-body biopotential electrodes are clarified. For this purpose, equivalent electrical models of various non-invasive biopotential electrodes are analyzed and compared. As a result, a proposal for a revised classification of biopotential electrodes is given. Secondly, instead of using the concept of a purely capacitive biopotential electrode, a test is proposed for assessing the predominant coupling mechanism achieved with an electrode over an insulating layer. Thirdly, a fundamental model of a buffer active non-contact biopotential electrode and its interface with the body is built and generalized, and the proposed test is applied for analyzing the influence of voltage attenuation and phase shifts on signal morphology. Lastly, guidelines for designing the described electrode–body interfaces are proposed, along with a discussion on practical aspects of their implementation. Full article

Accurate derivative information is central to sensitivity analysis and optimization, yet standard finite differences can lose many digits when the step size is small because of subtractive cancellation. Complex-step differentiation largely resolves this issue for first derivatives, but robust second derivatives and mixed partials remain delicate: several practical complex-step variants for $f^{″}$ still subtract nearly equal quantities, and quaternion-step rules are often presented as separate constructions. We develop a unified slice-based framework that extracts first and second derivatives from a single evaluation by projecting algebraic coefficients in commutative subalgebras of the complexified quaternions. First, we formulate a directional quaternion-steprule parameterized by an arbitrary unit pure quaternion u and provide an explicit projection operator that makes the underlying complex slice ${\mathbb{C}}_u\simeq \mathbb{C}$ transparent; the resulting first-derivative formula is rotation invariant and recovers classical j-step and planar $(j,k)$-step rules as special cases. Second, we construct a bicomplex double-step calculus in the commuting imaginary units $\mathrm{i}$ and u and show that one evaluation at $z+(\mathrm{i}+u)h$ separates derivative information into distinct coefficients, with the $\mathrm{i}u$-component equal to $h^2{f}^{″}\left(z\right)+\mathcal{O}\left(h^4\right)$, giving a subtraction-free $\mathcal{O}\left(h^2\right)$ approximation of $f^{″}$. For bivariate analytic functions we additionally derive one-shot identities for $f_x$, $f_y$, and $f_{xy}$ from $f(x+uh,y+\mathrm{i}h)$ and supply practical extraction identities, step-size guidance for $h^2$-scaled coefficients, and branch-consistency diagnostics for non-entire functions. The “cancellation-free” property here refers to avoiding the subtraction of nearly equal real quantities at the level of the differentiation formula; in floating-point arithmetic, coefficient extraction and the $1/h^2$ scaling for second-order quantities still interact with roundoff, and we quantify the resulting stable regimes numerically. Full article

This paper establishes a general parametric integral identity involving $(\mathfrak{n}+1)$-times differentiable stochastic processes, formulated entirely in terms of stochastic k-Caputo fractional derivatives. This identity serves as a unifying tool for deriving a broad class of parameter-dependent inequalities for differentiable s-convex stochastic processes. Remarkably, by assigning specific values to the underlying parameter, we have ensured our results specialize to well-known numerical integration inequalities, including those of midpoint, trapezium, Simpson, and Bullen types, in the stochastic fractional context. The findings not only enrich the theory of stochastic fractional calculus but also provide a flexible analytical apparatus for uncertainty quantification in fractional dynamical systems. Full article

Fractional calculus approach is used to analyze the model of surfactant transport by anomalous diffusion and its adsorption on an interface in a liquid-liquid system. The anomalous diffusion is modeled by time-fractional partial differential equations in the bulk phases. The adsorption of surfactant is described by the corresponding time-fractional Neumann boundary conditions at the interface. The adsorption process is considered under mixed barrier-diffusion control, described by first-order ordinary differential equation, which relates the subsurface concentration with that on the interface. A second relation between these concentrations is derived in terms of a fractional equation by application of Laplace transform technique. By combining both relations the subsurface concentration is eliminated and a single multi-term fractional ordinary differential equation for the surfactant concentration on the interface is derived. Different adsorption kinetic models are considered. In the case of Henry adsorption isotherm the model is linear and possesses analytical solution in terms of multinomial Mittag-Leffler functions. In the cases of Volmer and van der Waals adsorption isotherms nonlinear differential equations of fractional order are obtained. They are reformulated in equivalent integral form, which is used for computer simulation of the process of adsorption. Numerical results are presented and compared with analytical asymptotic predictions. Full article

The chain rule is a foundational concept in calculus, critical for differentiating composite functions, especially those appearing in modern AI techniques. Its extension to fractional calculus presents challenges due to the integral-based nature and intrinsic memory effects of these fractional operators. This survey provides a review of chain-rule formulations across major known FDs, including Riemann-Liouville (RL), Caputo, Caputo-Fabrizio (CF), Atangana-Baleanu-Riemann (ABR), Atangana-Baleanu-Caputo (ABC), and Caputo-Fabrizio with Gaussian kernel (CFG). The main contribution here is the introduction of a unified criterion, denoted as $C$, which synthesizes and extends previous classification frameworks for systematically formulating the chain rule across different operators. Each chain rule is examined in terms of its derivation, operator structure, and scope of applicability. In addition, the survey analyzes series-based approximations that appear in computing these derivatives, highlighting the minimum number of terms required to achieve acceptable mean absolute error (MAE). By consolidating theoretical developments, derivation methods, and numerical strategies, this paper provides a comprehensive resource for researchers and practitioners working with fractional-order models. Full article

Complex systems—ranging from biological organisms to turbulent fluids—exhibit multiscale heterogeneity and intermittency that traditional, differentiable calculus fails to adequately capture. Therefore, we propose a mathematical framework for analyzing complex system dynamics by assimilating the trajectories of structural units to continuous but non-differentiable multifractal curves. Utilizing the scale covariance principle, the authors recast the conservation of momentum as a geodesic equation within a multifractal space. This approach naturally separates the complex velocity field into differentiable and non-differentiable scale resolutions, where the balance of multifractal acceleration, convection, and dissipation is parametrized by a singularity spectrum f(α). We also discuss broad interdisciplinary implications, because, in our opinion, non-differentiability can enhance predictive capabilities in various fields such as oncology, pharmacology, and geophysics. Full article

This paper develops an operator-oriented framework for spectral approximation in fractional calculus by introducing a fractional inner product defined through the Riemann-Liouville integral. Instead of modifying polynomial families, the proposed approach continuously deforms the underlying Hilbert space structure, with the fractional order $\alpha$ acting as a deformation parameter. A central theoretical result shows that this fractional inner product is mathematically equivalent to a classical weighted inner product with a deformed weight $w_{\alpha }\left(x\right)={(b-x)}^{\alpha -1}w\left(x\right)$. This equivalence establishes a rigorous connection between fractional calculus and classical orthogonal polynomial theory and clarifies the structural role of the fractional parameter. For a canonical one-dimensional setting, explicit recurrence relations are derived and the limiting behavior as $\alpha \to 1$ is characterized, recovering the classical theory. The resulting orthogonal systems are naturally compatible with fractional operators and are used to construct spectral Galerkin methods for fractional differential equations. Well-posed variational formulations and optimal convergence rates are established. Numerical experiments illustrate the effectiveness of the framework, demonstrating spectral accuracy and improved performance in the approximation of fractional integrals and selected fractional differential equations when compared with standard polynomial bases. The proposed formulation provides a unifying operator-level perspective for spectral methods in fractional calculus. Full article

This paper aims to understand the teaching and learning practices and perceptions regarding the subject of Differential and Integral Calculus 1 (DIC1) based on the current French model, as implemented at Université Claude Bernard Lyon 1 (LYON 1), and the Brazilian model, as observed at the Federal University of Technology—Paraná (UTFPR). Five tutorial groups were studied at LYON 1. At UTFPR, four classes of DIC1 were analyzed. Teaching activities were observed, and teachers responded to a questionnaire regarding the frequency with which they implemented certain activities and their beliefs about which activities contribute most to student learning. Students responded to the same questionnaire, reflecting on how often their instructors employed these activities and which ones they believed were most beneficial for learning. There was general agreement between teachers and students about the instructional methodologies used in class; however, discrepancies emerged between observed practices, stated methodologies, and the activities considered essential for learning. In engineering programs, the time allocated to problem-solving—individually or on the board—emerged as a key aspect that may inspire changes and improvements in the Brazilian model. In contrast, group work and mathematical software may serve as avenues for improvement in the French model. Full article