Search Results (166)

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the $\beta$-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the $\beta$-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering. Full article

This work demonstrates that the four laws of classical thermodynamics apply to the statistics of symmetric observation distributions, and provides examples of how this can be exploited in uncertainty assessments. First, an expression for the partition function Z is derived. In contrast with general classical thermodynamics, however, this can be performed without the need for variational calculus, while Z also equals the number of observations N directly. Apart from the partition function $Z\equiv N$ as a scaling factor, three state variables m, n, and $ϵ$ fully statistically characterize the observation distribution, corresponding to its expectation value, degrees of freedom, and random error, respectively. Each term in the first law of thermodynamics is then shown to be a variation on $\delta m^2=\delta {(nϵ)}^2$ for both canonical (constant n and $ϵ$) and macro-canonical (constant $ϵ$) observation ensembles, while micro-canonical ensembles correspond to a single observation result bin having $\delta m^2=0$. This view enables the improved fitting and combining of observation distributions, capturing both measurand variability and measurement precision. Full article

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, and finally the Legendre–Clebsch condition. Our results are new in the sense that the Euler–Lagrange equation is based on duality theory, and thus build up only with left fractional operators. The Weierstrass necessary condition is a variant of strong necessary optimality condition, and it is derived from maximum condition of Pontryagin for this general analytic kernels. The Legendre–Clebsch condition is obtained under normality assumptions on data because of equality constraints. Full article

The fixation time of alleles is a fundamental concept in population genetics, traditionally studied using the Wright–Fisher model and classical coalescent theory. However, these models often assume homogeneous environments and equal reproductive success among individuals, limiting their applicability to real-world populations where environmental heterogeneity plays a significant role. In this paper, we introduce a new forward-time model for estimating fixation time that incorporates environmental heterogeneity through the use of fractional calculus. By introducing a fractional parameter α, we capture the effects of heterogeneous environments on offspring production. To solve the resulting fractional differential equations, we develop a novel spectral method based on Eta-based functions, which are well-suited for approximating solutions to complex, high-variation systems. The proposed method reduces the problem to an optimization framework via the operational matrix of fractional derivatives. We demonstrate the effectiveness and accuracy of this approach through numerical examples and show that it consistently captures fixation dynamics across various scenarios. This work offers a robust and flexible framework for modeling evolutionary processes in heterogeneous environments. Full article

Periodic solutions of high-order nonlinear differential equations are fundamental in dynamical systems, yet they remain challenging to establish with traditional methods. This paper addresses the existence of periodic solutions in general $2n$-order autonomous and nonautonomous ordinary differential equations. By extending Carathéodory’s variational technique from the calculus of variations, we reformulate the original periodic solution problem as an equivalent higher-order variational problem. The approach constructs a convex function and introduces an auxiliary transformation to enforce convexity in the highest-order term, enabling a tractable operator-theoretic analysis. Within this framework, we prove two main theorems that provide sufficient conditions for periodic solutions in both autonomous and nonautonomous cases. These results generalize the known theory for second-order equations to arbitrary higher-order systems and highlight a connection to the Hamilton–Jacobi theory, offering new insights into the underlying variational structure. Finally, numerical examples validate our theoretical results by confirming the periodic solutions predicted by the theory and demonstrating the approach’s practical applicability. Full article

This paper investigates weighted Milne-type ($\mathcal{M}-\mathfrak{t}$) inequalities within the context of Riemann–Liouville ($\mathfrak{R}-\mathfrak{L}$) fractional integrals. We establish multiple versions of these inequalities, applicable to different function categories, such as convex functions with differentiability properties, bounded functions, functions satisfying Lipschitz conditions, and those exhibiting bounded variation behavior. In particular, we present integral equalities that are essential to establish the main results, using non-negative weighted functions. The findings contribute to the extension of existing inequalities in the literature and provide a deeper understanding of their applications in fractional calculus. This work highlights the advantage of the established inequalities in extending classical results by accommodating a broader class of functions and yielding sharper bounds. It also explores potential directions for future research inspired by these findings. Full article

A geodesic or geodetic line on a sphere is called the orthodrome. Research has shown that the orthodrome can be defined in a large number of ways. This article provides an overview of various definitions of the orthodrome. We recall the definitions of the orthodrome according to the greats of geodesy, such as Bessel and Helmert. We derive the equation of the orthodrome in the geographic coordinate system and in the Cartesian spatial coordinate system. A geodesic on a surface is a curve for which the geodetic curvature is zero at every point. Equivalent expressions of this statement are that at every point of this curve, the principal normal vector is collinear with the normal to the surface, i.e., it is a curve whose binormal at every point is perpendicular to the normal to the surface, and that it is a curve whose osculation plane contains the normal to the surface at every point. In this case, the well-known Clairaut equation of the geodesic in geodesy appears naturally. It is found that this equation can be written in several different forms. Although differential equations for geodesics can be found in the literature, they are solved in this article, first, by taking the sphere as a special case of any surface, and then as a special case of a surface of rotation. At the end of this article, we apply calculus of variations to determine the equation of the orthodrome on the sphere, first in the Bessel way, and then by applying the Euler–Lagrange equation. Overall, this paper elaborates a dozen different approaches to orthodrome definitions. Full article

This paper is about a problem at the intersection of differential geometry, spectral analysis and the theory of manifolds. The study of finite-type subvarieties was initiated by Chen in the 1970s, with the aim of obtaining improved estimates for the mean total curvature of compact subvarieties in Euclidean space. The concept of a finite-type subvariety naturally extends that of a minimal subvariety or surface, the latter being closely related to variational calculus. In this work, we classify factorable surfaces in the Lorentz–Heisenberg space ${\mathbb{H}}_3$, equipped with a flat metric satisfying ${\Delta }^I{r}_i={\lambda }_i{r}_i$, which satisfies algebraic equations involving coordinate functions and the Laplacian operator with respect to the surface’s first fundamental form. Full article

This study explores the dynamics of particle motion in pseudo-Galilean $3-$space $G_3^1$ by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in $G_3^1$. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in $G_3^1$ and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications. Full article

Objectives: The purpose of the study is to evaluate the diagnostic accuracy of artificial intelligence (AI) software in detecting a common set of periodontal and restorative conditions, including marginal bone loss, dental caries, periapical lesions, calculus, endodontic treatment, crowns, restorations, and open crown margins, using intraoral periapical radiographs. Additionally, the study will assess how this AI software influences the diagnostic accuracy of dentists with varying levels of experience in identifying these conditions. Methods: A total of three hundred digital IOPARs representing 1030 teeth were selected based on predetermined selection criteria. The parameters assessed included (a) calculus, (b) periapical radiolucency, (c) caries, (d) marginal bone loss, (e) type of restorative (filling) material, (f) type of crown retainer material, and (g) detection of open crown margins. Two oral radiologists performed the initial diagnosis of the selected radiographs and independently labeled all the predefined parameters for the provided IOPARs under standardized conditions. This data served as reference data. A pre-trained AI-based computer-aided detection (“CADe”) software (Second Opinion^®, version 1.1) was used for the detection of the predefined features. The reports generated by the AI software were compared with the reference data to evaluate the diagnostic accuracy of the AI software. In the second phase of the study, thirty dental interns and thirty dental specialists were randomly selected. Each participant was randomly assigned five IOPARs and was asked to detect and diagnose the predefined conditions. Subsequently, all the participants were requested to reassess the IOPARs, this time with the assistance of the AI software. All the data was recorded using a self-designed Performa. Results: The sensitivity of the AI software in detecting caries, periapical lesions, crowns, open crown margins, restoration, endodontic treatment, calculus, and marginal bone loss was 91.0%, 86.6%, 97.1%, 82.6%, 89.3%, 93.4%, 80.2%, and 91.1%, respectively. The specificity of the AI software in detected caries, periapical lesions, crowns, open crown margins, restoration, endodontic treatment, calculus, and marginal bone loss was 87%, 98.3%, 99.6%, 91.9%, 96.4%, 99.3%, 97.8%, and 93.1%, respectively. The differences between the AI software and radiologist diagnoses of caries, periapical lesions, crowns, open crown margins, restoration, endodontic treatment, calculus, and marginal bone loss were statistically significant (all p values < 0.0001). The results showed that the diagnostic accuracy of operators (interns and specialists) with AI software revealed higher accuracy, sensitivity, and specificity in detecting caries, PA lesions, restoration, endodontic treatment, calculus, and marginal bone loss compared to that without using AI software. There were variations in the improvements in the diagnostic accuracy of interns and dental specialists. Conclusions: Within the limitations of the study, it can be concluded that the tested AI software has high accuracy in detecting the tested dental conditions in IOPARs. The use of AI software enhanced the diagnostic capabilities of dental operators. The present study used AI software to detect a clinically useful set of periodontal and restorative conditions, which can help dental operators in fast and accurate diagnosis and provide high-quality treatment to their patients. Full article

The application of fractional calculus in power electronics modeling provides an innovative method for improving inverter performance. This paper presents a three-phase inverter topology with fractional-order LC filter characteristics, analyzes its frequency response, and develops mathematical models in both stationary and rotating reference frames. Based on these models, a dual closed-loop decoupling control strategy for voltage and current is designed to enhance system stability and dynamic performance. In the power control loop, fractional-order virtual synchronous generator control (FOVSG) is employed. Observations show that increasing the fractional-order of the rotor leads to a higher transient frequency variation rate. To address this, an adaptive rotational inertia control scheme is integrated into the FOVSG structure (ADJ-FOVSG), enabling real-time adjustment of inertia to suppress transient frequency fluctuations. Experimental results demonstrate that when the reference active power changes, ADJ-FOVSG effectively suppresses power overshoot. Compared to traditional VSG, ADJ-FOVSG reduces the power regulation time by approximately 34.5% and decreases the peak frequency deviation by approximately 37.2%. Compared to the adaptive rotational inertia control in traditional VSG, ADJ-FOVSG improves regulation time by about 24% and reduces peak frequency deviation by roughly 24.4%. Full article

This research introduces an innovative fractal–fractional synergy framework for multiscale analysis of stress field dynamics in geo-energy systems. By integrating fractional calculus with multiscale fractal dimension analysis, we develop a coupled approach examining stress redistribution patterns across different geological scales. The methodology combines fractal characterization of rock mechanical parameters with fractional-order stress gradient modeling, validated through integrated analysis of core testing, well logging, and seismic inversion data. Our fractal–fractional operators enable simultaneous characterization of stress memory effects and scale-invariant fracture propagation patterns. Key insights reveal the following: (1) Non-monotonic variations in rock mechanical properties (fractal dimension D = 2.31–2.67) correlate with oil–water ratio changes, exhibiting fractional-order transitional behavior. (2) Critical stress thresholds (12.19–25 MPa) for fracture activation follow fractional power-law relationships with fracture orientation deviations. (3) Fracture network evolution demonstrates dual-scale dynamics—microscale tip propagation governed by fractional stress singularities (order α = 0.63–0.78) and macroscale expansion obeying fractal growth patterns (Hurst exponent H = 0.71 ± 0.05). (4) Multiscale modeling reveals anisotropic development with fractal dimension increasing by 18–22% during multi-well fracturing operations. The fractal–fractional formalism successfully resolves the stress-shadow paradox while quantifying water channeling risks through fractional connectivity metrics. This work establishes a novel paradigm for coupled geomechanical–fluid dynamics analysis in complex reservoir systems. Full article

This paper aims to develop a rheological model with fewer parameters that accurately describes the primary and secondary creep behavior of wood materials. The models studied are grounded in Riemann–Liouville fractional calculus theory. A comparison was conducted between the constant-order fractional Zener model and the variable-order fractional Maxwell model, with four parameters each. Using experimental creep data from four-point bending tests on two tropical wood species, along with an optimization algorithm, the variable-order fractional model demonstrated greater effectiveness. The selected fractional derivative order, modeled as a linearly increasing function of time, helped to elucidate the internal mechanisms in the wood structure during creep tests. Analyzing the parameters of this order function enabled an interpretation of their physical meanings, showing a direct link to the material’s mechanical properties. The Sobol indices have demonstrated that the slope of this function is the most influential factor in determining the model’s behavior. Furthermore, to enhance descriptive performance, this model was adjusted by incorporating stress non-linearity to account for the effects of the variation in constant loading level in wood. Consequently, this new formulation of rheological models, based on variable-order fractional derivatives, not only allows for a satisfactory simulation of the primary and secondary creep of wood but also provides deeper insights into the mechanisms driving the viscoelastic behavior of this material. Full article

The non-parametric version of Amari’s dually affine Information Geometry provides a practical calculus to perform computations of interest in statistical machine learning. The method uses the notion of a statistical bundle, a mathematical structure that includes both probability densities and random variables to capture the spirit of Fisherian statistics. We focus on computations involving a constrained minimization of the Kullback–Leibler divergence. We show how to obtain neat and principled versions of known computations in applications such as mean-field approximation, adversarial generative models, and variational Bayes. Full article

This paper extends the fractional calculus of variations to include generalized fractional derivatives with dependence on a given kernel, encompassing a wide range of fractional operators. We focus on variational problems involving the composition of functionals, deriving the Euler–Lagrange equation for this generalized case and providing optimality conditions for extremal curves. We explore problems with integral and holonomic constraints and consider higher-order derivatives, where the fractional orders are free. Full article