Search Results (1,014)

Oscillatory motions in stratified atmospheric fluid layers significantly influence weather and climate dynamics. Shallow-water equations effectively describe these motions. This study extends the shallow-water model to the time-fractional domain using the conformable fractional derivative. This derivative preserves the local differential structure while introducing tunable time scaling in the dynamics. Approximate analytical solutions were obtained using the conformable Laplace Adomian Decomposition Method (CLADM). This method combines the conformable Laplace transform with Adomian decomposition. Numerical results for fractional orders $\vartheta \in (0, 1]$ demonstrate that the fractional parameter systematically modulates the system dynamics. The solutions at $\vartheta =1$ align well with the established Elzaki Adomian Decomposition Method (EADM), Homotopy Analysis Method (HAM), Fractional Reduced Differential Transform Method (FRDTM), and reference numerical solutions (NUM). This fractional framework offers a flexible approach to modeling atmospheric fluid-layer dynamics. Full article

This paper proposes a machine-learning-assisted framework for modal sensitivity analysis of systems with viscoelastic damping elements, including both classical and fractional rheological models. Surrogate models are trained to approximate natural frequencies over a prescribed parameter space using two sampling strategies (Grid and Latin Hypercube) and two regression approaches: multi-layer perceptron (MLP) and Gaussian process regression (GPR). Sensitivities are obtained from the surrogates by finite differences and complemented by model-interpretability measures, namely permutation feature importance (PFI) and Shapley Additive Explanations (SHAP). The surrogate-based results are compared with analytically obtained sensitivities. Local first- and second-order sensitivities of natural frequencies are derived analytically using the direct differentiation method (DDM) for a nonlinear eigenvalue problem formulated in the Laplace domain and further transformed into dimensionless sensitivity measures. The methodology is demonstrated for a single-degree-of-freedom oscillator with classical and fractional Kelvin damper models and a two-story frame equipped with a fractional Kelvin damper. The results show very good agreement between analytical and surrogate-based sensitivities. Feature-importance rankings obtained by PFI and SHAP are consistent with the dimensionless sensitivities and capture changes in parameter influence under varying damping levels. Dispersion studies indicate only minor ranking variations. Full article

Fractional calculus and distribution theory share a common conceptual origin in the symbolic interpretation of differentiation and integration. Despite this connection, most developments in fractional calculus have traditionally been formulated within the framework of ordinary functions, while the systematic use of distributions remains limited. In this work, a novel distributional framework is developed by constructing a fractional Taylor representation of the product of Euler gamma and Riemann zeta functions in terms of fractional derivatives of the Dirac delta distribution. The proposed formulation enables the derivation of new fractional identities via Laplace transformation and facilitates the analytical solution of fractional differential equations containing such functions. Closed-form solutions are obtained in both classical and generalized distributional senses, allowing the extension of solutions from the positive real axis to the entire real line. Furthermore, the framework is applied to fractional operators of Erdélyi–Kober type, yielding new integral and derivative transforms. Fractional differential and integral equations with singular terms arise naturally in several engineering models involving memory effects, impulsive responses, and anomalous transport phenomena. However, the presence of nonremovable singularities—such as those associated with Euler gamma and Riemann zeta functions—significantly restricts the applicability of classical analytical methods. Overall, the proposed distributional framework bridges the gap between abstract fractional calculus and practical engineering models by enabling analytical solutions of fractional systems with singular memory kernels that were previously inaccessible using classical methods. Full article

The double natural generalized Laplace transform decomposition method (DNGLTDM) is proposed as a new approach for solving singular linear and non-linear two-dimensional Boussinesq equations involving fractional partial derivatives. This method combines the decomposition technique with the double natural generalized Laplace transform to construct solutions in the form of rapidly convergent infinite series that approximate the exact solutions. The paper presents a detailed study of the fundamental properties of the transform, including the convolution theorem, the periodicity theorem, the treatment of partial derivatives with non-constant coefficients, and partial fractional derivatives. In addition, the convergence of the obtained series solutions and the associated error analysis are thoroughly investigated. Finally, two illustrative examples are provided to demonstrate the accuracy and effectiveness of the DNGLTDM, one of which considers a problem of partial fractional order. Full article

In this paper, a cold standby repairable system comprising two heterogeneous components, each characterized by multiple types of mutually independent failure modes, is investigated. The operational lifetimes of the components follow exponential distributions, while their repair times after failure are governed by general distributions. By applying the theory of the Markov renewal process together with the Laplace and the Laplace–Stieltjes transform techniques, we derive analytical expressions for the time to the first system failure, system availability, and the rate of occurrence of system failures. Some results for these reliability measures under several special cases are also presented. Finally, numerical examples are provided under different repair time distributions to analyze the influence of model parameters on the system’s reliability performance. Full article

An analytical model of frictional heat transfer during the uniform sliding of two layers is proposed. One layer is composed of a functionally graded material (FGM) with a thermal conductivity coefficient that varies exponentially across its thickness, while the second layer is homogeneous, with constant thermophysical properties. The thermal problem of friction is formulated as an initial boundary value problem of heat conduction, accounting for the thermal contact conductance and convective heat exchange with the environment. An exact solution for constant friction power was obtained using the Laplace integral transform, supplemented by an asymptotic form for the initial stage of heating. Based on these analytical solutions, a comprehensive study was carried out for a frictional system comprising a ceramic–metal FGM composite in contact with a homogeneous friction material. A dimensional analysis allowed for both a qualitative and quantitative investigation into the influence of contact conductance, convective heat exchange, layer thickness and the FGM gradient parameter on the temperature evolution and distribution, as well as the time to reach the steady state. It was demonstrated that the implementation of an appropriately graded material can substantially improve thermal operating conditions by enhancing heat dissipation into the material bulk and intensifying convective cooling. Full article

Umbral theory, formulated in its modern version by S. Roman and G. C. Rota, has been reconsidered in more recent times by G. Dattoli and collaborators with the aim of devising a working computational tool in the framework of special function theory. Concepts like the umbral image and umbral vacuum have been introduced as pivotal elements of the discussion which, albeit effective, lack generality. This article is directed towards endowing the formalism with a rigorous formulation within the context of formal power series with complex coefficients $(\mathbb{C}⟦t⟧,\partial )$. The new formulation is founded on the definition of the umbral operator $\mathfrak{u}$ as a functional in the “umbral ground state” subalgebra of analytically convergent formal series $\phi \in \mathbb{C}\left\{t\right\}$. We consider in detail some specific classes of umbral ground states $\phi$ and analyse the conditions for analytic convergence of the corresponding umbral identities, defined as formal series resulting from the action on $\phi$ of operators of the form $f\left(\zeta {\mathfrak{u}}^{\mu }\right)$ with $f\in \mathbb{C}\left\{t\right\}$ and $\mu ,\zeta \in \mathbb{C}$. For these umbral states, we exploit the Gevrey classification of formal power series to establish a connection with the theory of Borel–Laplace resummation, allowing us to make rigorous sense of a large class of—even divergent—umbral identities. As an application of the proposed theoretical framework, we introduce and investigate the properties of new umbral images for the Gaussian trigonometric functions, which emphasise the trigonometric-like nature of these functions and enable defining the concept of a “Gaussian Fourier transform”, a potentially powerful tool for applications. Full article

In this work, we address the nonhomogeneous initial-boundary value problem for the coupled Hirota equation posed on the finite interval $[0,L]$. To investigate the well-posedness of this problem, we first adopt an appropriate transformation, namely the Laplace transform, which is tailored to the specific characteristics of the problem, and further obtain an explicit solution formula for the linear inhomogeneous coupled system. Subsequently, the local well-posedness of the original nonhomogeneous initial-boundary value problem in $X_{s,T}\times X_{s,T}\left(X_{s,T}=C(0,T;H^s(0,1))\cap L^2(0,T;H^{s+1}(0,1))\right)$ is rigorously established through the combination of this explicit formula, the contraction mapping principle and energy estimates. Full article

The primary objective of this study is to develop a specialized deep learning framework specifically adapted for the unique physical characteristics of neurovascular Optical Coherence Tomography (OCT) imaging. Although Polyp-PVT, originally designed for polyp segmentation, shows promise for OCT analysis, it faces limitations in neurovascular applications. The default RGB input wastes resources on duplicated grayscale data, while its fixed-scale fusion struggles with vascular curvature variations. Furthermore, the attention mechanism fails to capture radial vessel patterns, and geometric constraints limit thin boundary detection. To address these challenges, we propose Brain-OCT-PVT with key innovations: a single-channel input stem reducing parameters by two-thirds; a Radial Intensity Module (RIM) using polar transforms and angular convolution to model annular structures; and a Deformable Cross-scale Fusion Module (D-CFM) with learnable offsets. The Boundary-aware Attention Module (BAM) combines Laplace edge detection with Swin-Transformer for sub-pixel consistency. A specialized loss function combines Dice Similarity Coefficient (Dice), BoundaryIoU on 2-pixel dilated edges, and Focal Tversky to handle extreme class imbalance. Evaluation on 13 clinical cases achieves a Dice score of 95.06% and an 95% Hausdorff Distance (HD95) of 0.269 mm, demonstrating superior performance compared to existing approaches. Full article

This paper investigates the finite-horizon survival probability for a system of correlated arithmetic Brownian motions with heterogeneous drifts and volatilities, focusing on the event in which one component remains strictly below all others. Using a whitening transformation of the covariance structure, we reduce the problem to the survival of a standard Brownian motion in a simplicial cone, characterized by its spherical cross-section. While explicit solutions are available in low dimensions, we address the computationally challenging tetrahedral angular case. We derive a semi-analytic formula for the survival probability via an eigenfunction expansion of the Dirichlet Laplace–Beltrami operator on this curved domain. For efficient implementation, we construct a diffeomorphism from the spherical tetrahedron to a fixed Euclidean tetrahedron, enabling the computation of angular eigenpairs through a stable finite-element scheme. For higher-dimensional regimes, we also introduce a covariance-based difficulty index and geometric bounds based on an inscribed spherical cap to assess spectral convergence and estimate long-time decay rates. Numerical experiments show that this offline–online approach achieves high accuracy and substantial speedups relative to Monte Carlo benchmarks. Full article

In this study, we investigate the upper- and lower-bound approximations of numerical eigenvalues derived by weak Galerkin spectral element methods on arbitrary convex quadrilateral meshes for the Laplace eigenvalue problem. Firstly, the Piola transformation is employed to construct the approximation space for weak gradients on each convex quadrilateral element, while a one-to-one mapping is used to establish the approximation space for weak functions. Subsequently, based on the weak Galerkin spectral element approximation space defined on convex quadrilateral meshes, a Galerkin approximation scheme is formulated, and its well-posedness is then analyzed. Furthermore, numerical experiments are performed on arbitrary convex quadrilateral meshes of the square and L-shaped domains to explore the upper- and lower-bound approximations of numerical eigenvalues. Numerical findings indicate that the presented method not only obtains optimal orders of convergence with respect to both the mesh size and the polynomial degree, but also provides upper- and lower-bound approximations for the reference eigenvalues by proper choices of polynomial degrees in approximation spaces and parameters of the approximation scheme in both h-version and p-version weak Galerkin spectral element methods. This study offers new perspectives and methodologies for the high-precision numerical solution of eigenvalue problems in elliptic equations. Full article

The proliferation of Location-Based Services (LBSs) has generated vast trajectory datasets that offer immense analytical value but pose critical privacy risks. Achieving an optimal balance between data utility and privacy preservation remains a challenge, a difficulty compounded by the limitations of existing methods in modeling complex, long-term spatiotemporal dependencies. To address this, this paper proposes a trajectory data publishing scheme combining a Transformer decoder with differential privacy. Unlike traditional single-layer approaches, the proposed method establishes a systematic generation–generalization framework. First, a Transformer decoder is integrated into a Generative Adversarial Network (GAN). This architecture mitigates the gradient vanishing issues common in RNN-based models, generating high-fidelity synthetic trajectories that capture long-range correlations while decoupling them from sensitive source data. Second, to provide rigorous privacy guarantees, a clustering-based generalization strategy is implemented, utilizing Exponential and Laplace mechanisms to ensure $ϵ$-differential privacy. Experiments on the Geolife and Foursquare NYC datasets demonstrate that the scheme significantly outperforms leading baselines, achieving a superior trade-off between privacy protection and data utility. Full article

To meet the requirements of effectiveness and strength in actual engineering, based on the dynamic fracture characteristics, the dynamic propagation of orthogonal anisotropic interface cracks in piezoelectric bimaterials was analyzed. By performing Laplace transformation and Fourier transformation on the governing equations, the problem was transformed into a singular integral equation. Using the Chebyshev point method and Laplace inversion, the stress and electric displacement intensity factors at the crack tip of the orthogonal anisotropic interface were obtained. The results show that the crack length affects the dimensionless function. The longer the crack, the larger the dimensionless function. Under certain conditions, the smaller the elastic parameters, the smaller the dimensionless dynamic stress intensity factor. At the same time, the impact time also affects the dynamic crack propagation. With the passage of time, the dimensionless function first increases, then reaches a peak, and finally oscillates and converges to the static value. On this basis, the response surface method was used for analysis and prediction. The R² value of the random forest model is 0.9886, which indicates that the model has high predictive accuracy. When the optimal values of A ($d_1/a$), B ($c_{p}t/a$) and C ($c_{44}^{(2)}/c_{44}^{(1)}$) are 0.4045, 1.6797 and 1.9035 respectively, the stress intensity reaches its maximum value of 1.3375. Full article

Analytical models of transdermal drug delivery (TDD) often represent deeper skin layers using ideal sink assumptions or phenomenological interfacial resistances. While mathematically convenient, these approaches obscure the physical role of the dermis and hypodermis in controlling molecular transport. Here, we develop an impedance-based analytical model for diffusion across multilayer skin membranes, in which the epidermal barrier is dynamically coupled to a finite diffusive backing layer representing the dermis–hypodermis composite. Diffusion impedance links transport conductivity, storage capacity, and layer thickness, while preserving continuity of concentration and flux at all interfaces. Closed-form expressions in the Laplace domain describe concentration fields and interfacial fluxes, and cumulative drug uptake is computed in the time domain via inverse Laplace transformation. The model identifies distinct short- and long-time transport regimes. Commonly used Dirichlet and Robin boundary conditions emerge as limiting cases but cannot reproduce the regime-dependent behavior of a backing layer. In particular, Robin formulations reduce the backing layer to a constant effective resistance, neglecting its storage capacity and time-dependent impedance. By replacing ad hoc boundary conditions with a physically grounded impedance framework, this approach provides a unified and extensible method for analyzing multilayer transport systems, including extensions to anomalous or memory-dependent diffusion. 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