Search Results (128)

This study presents an analytical model for the long-term bending behavior of simply supported laminated glass (LG) plates with temperature-dependent viscoelastic interlayers. The glass layers are described based on three-dimensional elasticity theory, and the governing stress and displacement equations are formulated using the state-space method. The polymer interlayer is characterized by the generalized Maxwell model and the Williams–Landel–Ferry equation, while its time-dependent response is described through the Boltzmann convolution principle. By combining double Fourier series expansions with the Laplace-transform technique, analytical solutions for the stresses and displacements of multilayer LG plates are derived. The comparison shows that Kirchhoff–Love plate theory gives results close to the present solution for relatively thin LG plates, whereas the discrepancy becomes increasingly pronounced as the plate thickness increases. The finite element results agree well with those obtained from the proposed model; however, for the representative benchmark case, the present solution is approximately 1.13 × 10³ times faster than the FE simulation, and its memory usage is only about 10.88% of that required by the FE model. Parametric studies further reveal the effects of temperature, interlayer thickness, interlayer material, number of glass layers, and aspect ratio on the stress redistribution and deflection development of LG plates. Full article

Aircraft targets in SAR imagery often exhibit discrete scattering characteristics, significant variations in pose and scale, strong speckle noise in background clutter, and complex background interference, which jointly hinder stable structural feature extraction and accurate target localization. Existing detectors for SAR aircraft recognition primarily rely on bounding-box regression and classification; they do not completely exploit target structural cues, spatial attention, and frequency-domain information. To address these limitations, we propose a collaborative detection framework that integrates an uncertainty-aware keypoint-driven module (UAKM) with a fractional Fourier convolution backbone (S-FRConv). UAKM introduces a center-keypoint regression branch that jointly predicts keypoint coordinates and Laplacian scale parameters and employs a 2D Laplace negative log-likelihood loss to estimate uncertainty. The derived dense uncertainty heatmap is then used as spatial attention weights to guide distribution-based regression and multi-scale feature re-weighting, without requiring any additional annotations. S-FRConv embeds the Fractional Fourier Transform into shallow backbone layers and C2f modules, enabling joint spatial–spectral feature modeling that suppresses speckle noise and enhances edge and orientation representations. Experiments on the public SAR-AIRcraft-1.0 dataset demonstrate that the proposed method systematically improves the detection performance. For the Nano model, the overall mAP50 increases from 0.810 to 0.867, and the mAP 50:95 improves from 0.637 to 0.655 compared with the baseline, corresponding to gains of 5.7 and 1.8 percentage points, respectively. These results validate the effectiveness and generalization potential of combining uncertainty-driven spatial attention with fractional spectral feature enhancement for SAR aircraft target detection. Full article

Analytical solutions for contaminant transport in porous media are important for understanding subsurface processes and validating numerical models. However, conventional Laplace-transform-based approaches often face difficulties in handling realistic transient boundary conditions and typically result in challenging inverse formulations that require computationally intensive convolved integration. To address these limitations, this paper presents a Fourier-FFT analytical framework for solving the well-established one-dimensional advection–dispersion–reaction (ADR) equation in homogeneous and heterogeneous porous domains. The proposed Fourier-FFT approach enables straightforward mathematical formulation, rapid computation, and incorporation of realistic transient boundary conditions beyond idealized step or impulse inputs. Verification against a Laplace-based analytical solution for a homogeneous domain and a finite element solution for a dual-permeability domain show good agreement, confirming the accuracy of the method. Parametric analyses further demonstrate that the framework captures the expected physical behaviour of contaminant transport under varying hydrogeological and reaction conditions. 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

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

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

We introduce a geometric framework in which classical transforms are represented as coordinate charts on a symbolic manifold. The construction defines symbolic curvature (κ), strain (τ), compressibility (σ), and the ratio Γ = κ/τ, which together provide a diagnostic coordinate system for comparing representational stability across chart transitions. Within this setting, transforms such as Fourier, Laplace, wavelet, Jordan, and polynomial projection can be treated as charts connected by transition maps that preserve Γ on specified domains. We also introduce a symmetric positive-definite metric tensor $G_{ab}$ to quantify displacement in the invariant coordinates and to formalize minimal-effort paths (geodesics) under modeling assumptions stated in the text. The resulting framework provides a reproducible screening method for evaluating transform stability, diagnosing closure failure, and comparing transform behavior under a shared set of invariants. Full article

This study investigates the unsteady motion of a slender body in a fluid beneath an ice cover, both in the presence and absence of the ocean waves propagating through the ice–water system, using the Fourier and Laplace integral transforms. The influence of the progressive wave on ice cover deflections, specifically the Bernoulli hump and Kelvin wake angle, induced by the motion of the underwater body near the surface, is analyzed numerically. Additionally, the effect of the progressive wave on the wave resistance of the body is investigated. Conditions are derived that relate the L/D ratio to a dimensionless parameter characterizing the elastic forces of the plate, under which the presence of the ocean wave produces a minimal effect on the body’s wave resistance. Full article

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest $H_{1,1}^{1,1}\leftrightarrow H_{0,2}^{2,0}$, with H denoting the Fox H-function. Full article

The Laplace equation is an important partial differential equation, typically used to describe the properties of steady-state distributions or passive fields in physical phenomena. Its Cauchy problem is one of the classic, serious, ill-posed problems, characterized by the fact that minor disturbances in the data can lead to significant errors in the solution and lack stability. Secondly, the determination of the parameters of the classical Laplace equation is difficult to adapt to the requirements of complex applications. For this purpose, in this paper, the Laplace equation with uncertain parameters is defined, and the uncertainty is represented by fuzzy numbers. In the case of granular differentiability, it is transformed into a granular differential equation, proving its serious ill-posedness. To overcome the ill-posedness, the Fourier regularization method is used to stabilize the numerical solution, and the stability estimation and error analysis between the regularization solution and the exact solution are given. Finally, numerical examples are given to illustrate the effectiveness and practicability of this method. Full article

We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks. Full article

In this work, the advection-dispersion model (ADM) is time-fractionalized by the exploitation of Atangana-Baleanu (AB) differential operator to describe contaminant transport in a geological environment. Dispersion, adsorption, and decay, which are known as the foremost transport mechanisms, are considered. The exact solutions of the suggested Atangana-Baleanu advection-dispersion models (AB-ADMs) are acquired using Fourier sine transform and Laplace transform. The classical ADMs are demonstrated to be the special limiting cases of the suggested models. The high consistency among the suggested models and experimental data denotes that the AB-ADMs characterize contaminant transport more effectively. Additionally, the corresponding numerical and graphical results are explored to demonstrate the necessity, effectiveness, and suitability of the suggested models. Full article

Most existing multispecies transport analytical models primarily focus on inlet boundary sources, limiting their applicability in real-world contaminated sites where contaminants often arise from multiple internal sources. This study presents a novel semi-analytical model for simulating multispecies contaminant transport driven by multiple time-dependent internal sources. The model incorporates key transport mechanisms, including advection, dispersion, rate-limited sorption, and first-order degradation. In particular, the inclusion of rate-limited sorption addresses limitations in traditional equilibrium-based models, which often underestimate pollutant concentrations for degradable species. The derivation of this semi-analytical model utilizes the Laplace transform, finite cosine Fourier transform, generalized integral transform, and a sequence of inverse transformations. Results indicate that the concentrations of contaminants and their degradation products are highly sensitive to the variations in time-dependent sources. The model’s most significant contribution lies in its capability to simulate the contaminant transport from multiple internal pollution sources at a contaminated site under the influence of rate-limited sorption. By enabling the representation of multiple time-varying sources, this model fills a critical gap in analytical approaches and provides a necessary tool for accurately assessing contaminant transport in complex, realistic pollution scenarios. Full article

The paper presents an enhanced numerical framework for computing the one-dimensional fast Fractional Fourier Transform (FRFT) by integrating closed-form Composite Newton-Cotes quadrature rules. We show that a FRFT of a $QN$-length weighted sequence can be decomposed analytically into two mathematically commutative compositions: one involving the composition of a FRFT of an N-length sequence and a FRFT of a Q-length weighted sequence, and the other in reverse order. The composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, with a focus on estimating probability densities for distributions with complex-valued characteristic functions. Numerical experiments on the Variance-Gamma (VG) and Generalized Tempered Stable (GTS) models show that the proposed scheme significantly improves accuracy over standard (non-weighted) fast FRFT and classical Newton-Cotes quadrature, while preserving computational efficiency. The findings suggest that the composite FRFT framework offers a robust and mathematically sound tool for transform-based numerical approximations, particularly in applications involving oscillatory integrals and complex-valued characteristic functions. Full article

This study introduces an innovative analytical solution to the time-fractional Cattaneo heat conduction equation, which models photothermal transport in metallic thin films subjected to short laser pulse irradiation. The model integrates the Caputo fractional derivative of order 0 < p ≤ 1, addressing non-Fourier heat conduction characterized by finite wave speed and memory effects. The equation is nondimensionalized through suitable scaling, incorporating essential elements such as a newly specified laser absorption coefficient and uniform initial and boundary conditions. A hybrid approach utilizing the finite Fourier cosine transform (FFCT) in spatial dimensions and the Laplace transform in temporal dimensions produces a closed-form solution, which is analytically inverted using the two-parameter Mittag–Leffler function. This function inherently emerges from fractional-order systems and generalizes traditional exponential relaxation, providing enhanced understanding of anomalous thermal dynamics. The resultant temperature distribution reflects the spatiotemporal progression of heat from a spatially Gaussian and temporally pulsed laser source. Parametric research indicates that elevating the fractional order and relaxation time amplifies temporal damping and diminishes thermal wave velocity. Dynamic profiles demonstrate the responsiveness of heat transfer to thermal and optical variables. The innovation resides in the meticulous analytical formulation utilizing a realistic laser source, the clear significance of the absorption parameter that enhances the temperature amplitude, the incorporation of the Mittag–Leffler function, and a comprehensive investigation of fractional photothermal effects in metallic nano-systems. This method offers a comprehensive framework for examining intricate thermal dynamics that exceed experimental capabilities, pertinent to ultrafast laser processing and nanoscale heat transfer. Full article