Search Results (646)

This paper is concerned with the asymptotic behavior of a viscoelastic wave equation involving distributed delay, logarithmic nonlinearity and dynamic Wentzell boundary conditions. In general, when the memory kernel function $g\left(t\right)$ is monotonically decreasing, the system energy’s decay is similar to that of the kernel function. However, this work addresses the case where the kernel function does not necessarily decay; thus, at this point, whether the system energy can still decay, and especially maintain exponential decay, is a very interesting question. Assuming that the kernel function is not necessarily decreasing, which means that it may oscillate, under some proper conditions, utilizing the Lyapunov functional method and constructing auxiliary functions, an exponential decay result is attained. To some extent, the result extends and improves several earlier related results in the literature. Full article

The analysis of the time-fractional nonlinear Sharma–Tasso–Olver (STO) equation with various initial conditions has been shown in this work. Finding the appropriate approximate solution of the problems under consideration is carried out by implementing unique strategies that combine the Adomian decomposition method (ADM), and the Generalized integral transform. The proposed method computes the results as a convergent series. The main benefit of the suggested method is that it needs minimal computing effort while producing extremely accurate results. We first apply the fractional Caputo fractional derivative (CFD) and then the Atangana–Baleanu–Caputo (ABC) derivative to solve the fractional STO problem. The nonlinear wave model for harbor and coastal designs heavily relies on the wave solutions of the STO equation. Several cases of time-fractional STO equations with various initial approximations are used to illustrate the schemes under consideration. The efficiency and dependability of the methods under consideration are confirmed by executing suitable numerical simulations. We contrast our findings with those of other approaches, including the Homotopy perturbation method (HPM), and the q-Homotopy analysis Elzaki transform method (q-HAETM). Additionally, the results of using the proposed techniques at different fractional orders are analyzed, showing that their accuracy increases as the value goes from fractional order to integer order. The results gained indicate that the applied scheme is highly satisfying and investigate the complicated nonlinear problems that arise in innovation and science. Full article

Longitudinal data, characterized by repeated measurements on the same subjects over time, are ubiquitous in biomedical sciences, economics, social sciences, and engineering. Analyzing such data presents unique statistical and computational challenges, including within-subject correlation, time-varying covariates, irregular observation times, informative dropout, and high dimensionality. While traditional statistical methods, such as linear mixed-effects models and generalized estimating equations, remain foundational, they often struggle with complex nonlinear dynamics, ultra-high-dimensional feature spaces, and very large sample sizes. Over the past two decades, machine learning (ML) and artificial intelligence (AI) methods have emerged as powerful complementary approaches to address these limitations. This review provides a comprehensive survey of mathematical and computational methods for longitudinal data analysis. We cover classical statistical models, penalized regression techniques, tree-based ensemble methods, kernel machines, Bayesian hierarchical models, and modern deep learning architectures, including recurrent neural networks, temporal convolutional networks, attention-based Transformers, neural ordinary differential equations, and generative models. We propose a unified taxonomy that organizes existing methods along two primary axes: the underlying mathematical framework and the analytical objective. For each category, we present detailed mathematical formulations, discuss key theoretical properties, examine computational considerations, and summarize representative reported applications drawn from the published literature. To increase the practical value of this review, we provide a cross-cutting comparison of method families against five key challenges (within-subject correlation, irregular sampling, missing data, high dimensionality, and scalability) and offer concrete guidance on method selection according to sample size, dimensionality, and analytical objective. Finally, we critically evaluate the strengths and limitations of these approaches, with particular emphasis on interpretability, scalability, handling of missing data, robustness to covariance misspecification, and uncertainty quantification. Full article

Let $\mathcal{A}=\{p+Jq:p,q\in \mathbb{C},J^2=-1\}$ be the commutative bicomplex-type algebra in which J commutes with the scalar imaginary unit. A Cauchy–Riemann-type operator $\overline{D}$ is studied on domains in ${\mathbb{C}}^2$. In the active coordinates $\xi =z_1-i{z}_2$ and $\eta =z_1+i{z}_2$, the equation $\overline{D}f=0$ is diagonal in the idempotent basis: the $e_{+}$-component is holomorphic in $\xi$ with $\eta$ as the parameter, while the $e_{-}$-component is holomorphic in $\eta$ with $\xi$ as the parameter. The expression $e_{+}F\left(\xi \right)+e_{-}G\left(\eta \right)$ is the parameter-independent subcase. From this decomposition, one obtains a slice characterization, a criterion for separatedness, a comparison with ordinary holomorphic functions of two complex variables, active-variable Cauchy formulas and estimates, local series with parameter-dependent coefficients, reflection symmetry, and Hardy and Bergman kernel lifts on the separated Hilbert spaces. Full article

In this work, we use the Riemann–Liouville (R-L) fractional derivative of order $\alpha \in (0,1)$ to study the oscillation criteria for damped matrix fractional differential equations and determine sufficient conditions under which all prepared solutions of the system show oscillatory behaviour. The criteria are novel even for the linear undamped case and extend conventional oscillation results for integer-order matrix differential systems to the fractional setting. The goal of the current effort is to better understand the relationships between solutions and their derivatives. Using the matrix-valued Riccati transformation converts the system into a Riccati-type inequality, and the oscillation conditions are then derived by integrating against a weighted kernel via the operator L. Both results generalise the integer-order oscillation criteria to the fractional matrix setting, extending their applicability to fractional-order control systems, viscoelastic structural models, and anomalous diffusion processes. This work develops new conditions and analytical techniques that deepen insight and provide useful results for analysing oscillatory behaviour and asymptotic stability of of the considered systems. To illustrate the significance of the obtained oscillation results, we give two examples. Full article

We construct solutions of a time-harmonic Maxwell-type system within the framework of the algebra of complex quaternions. Using quaternionic analysis, we establish a connection between this system and certain first-order differential operators whose kernels consist of monogenic functions. Building on known representations of harmonic and monogenic functions, we develop a constructive procedure based on transmutation operators for generating explicit solutions of the equations $(D\pm \lambda )u=0$, and consequently of the corresponding Maxwell system. This approach provides a systematic method for reconstructing electromagnetic fields from harmonic and monogenic data, yielding an explicit link between quaternionic operator theory, transmutation methods, and the classical formulation of Maxwell equations. Full article

Hereditary damping and fractional attenuation are widely used to model wave propagation in complex media, but the variational and spectral origin of the corresponding nonlocal-in-time operators is often left implicit. In this work, we derive such operators from a minimal conservative field–reservoir model. A real scalar field is coupled locally to a continuum of harmonic reservoir modes, which are then eliminated exactly. The resulting reduced dynamics is a causal wave equation with a memory-friction term acting on the field velocity. The memory kernel is generated by the reservoir coupling spectrum through a cosine-transform relation, establishing a direct spectrum-to-kernel correspondence. This relation provides both a physical interpretation of hereditary damping and a practical admissibility criterion: macroscopic attenuation and dispersion arise from the delayed back-action of unresolved internal modes, while physically admissible kernels are constrained by the non-negativity of the underlying spectral density. The framework unifies several standard damping regimes. A broadband reservoir recovers the Markovian locally damped wave equation, reservoirs with a finite characteristic time generate finite-memory relaxation and frequency-dependent dispersion, and scale-free reservoir spectra produce power-law memory kernels. In the latter case, the hereditary damping operator reduces to a Caputo-type fractional derivative, showing that fractional wave attenuation can emerge as an effective reduced dynamics rather than being postulated phenomenologically. We further analyze dispersion, attenuation, causality, stability, and admissibility conditions in terms of the reservoir spectrum. The main contribution of the work is therefore to provide a variational and spectral derivation of hereditary and fractional wave damping, linking the structure of unresolved reservoir modes to macroscopic nonlocal wave dynamics. Full article

This paper introduces a deep learning-based methodology for reconstructing particle beam energy spectra from experimental attenuation curves. This task involves solving a classic ill-posed inverse problem for a Fredholm integral equation of the first kind. Unlike traditional Arsenin–Tikhonov regularization, the proposed framework utilizes two coupled neural networks for spectrum approximation and adaptive kernel correction. This approach explicitly accounts for measurement uncertainties in the experimental data. As a mesh-free technique, it operates directly on raw sparse experimental datasets without preprocessing. Validation using data from subnanosecond electron beams in gas-filled and vacuum diodes demonstrates that the method successfully resolves non-trivial two-peak spectral structures. In particular, it reliably identifies populations of “anomalous” high-energy electrons that are often obscured by classical regularization artifacts. Full article

We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals modelling viscoelastic memory damping. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (${\beta }_{\mathrm{NM}}=1/4$) combined with an implicit–explicit linearisation in which the linear spatial operator is treated implicitly and the nonlinear terms are treated explicitly through a second-order extrapolation. This linearisation eliminates the need for Newton–Raphson iterations at each time step. To overcome the dense memory bottleneck arising from two distinct fractional orders $\alpha \ne \beta$, the convolution memory kernels are compressed by independent sum-of-exponentials approximations obtained from a double-exponential quadrature of the kernel’s integral representation, which significantly reduces the computational complexity of the history term. A rigorous stability estimate and a global convergence bound are established using a discrete Grönwall inequality. Numerical experiments confirm the theoretical temporal and spatial convergence rates and demonstrate the practical speed-up afforded by the sum-of-exponentials acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory. Full article

In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the physical domain onto a reference square and approximates the unknown solution using a tensor-product Legendre polynomial expansion. Exact operational matrices for differentiation and lower-limit integration are constructed, allowing the original integro-differential problem to be reduced systematically to a finite-dimensional algebraic system for the spectral coefficients. The formulation provides a unified treatment of differential, Volterra, and Fredholm operators within a single spectral framework and avoids complicated discretizations of multidimensional integral terms. For a specialized linear form of the problem, rigorous convergence estimates are established in both $L^2$ and $L^{\infty }$ norms under suitable regularity assumptions on the coefficients and kernels. The analysis shows that the dominant convergence behavior is governed by the differential operator, while the integral terms contribute only higher-order consistency effects. Several benchmark examples involving both linear and nonlinear two-dimensional integro-differential equations are presented to demonstrate the performance of the proposed method. Numerical results exhibit rapid spectral-type error decay as the polynomial degree increases, with the numerical errors approaching machine precision for moderate truncation orders. These results confirm the accuracy, efficiency, and reliability of the proposed Legendre spectral operational matrix framework for solving a broad class of multidimensional integro-differential equations with nonlocal operators. Full article

This paper presents two analytic mappings defined on probability measures that extend and unify the $(a,b)$- and $(s,t)$-deformations arising in free probability for s, $b>0$ and a, $t\in \mathbb{R}$. These unified operators, denoted ${\mathbf{U}}_{(a,b,s,t)}^{\prime }$ and ${\mathbf{U}}_{(a,b,s,t)}^{″}$, are characterized by a functional equation involving the Cauchy–Stieltjes transform, providing a transform-based formulation of measure deformation. They reduce to the $(a,b)$-transformation when $s=t=1$ and to the $(s,t)$-transformation when $a=b=1$. Working in the framework of Cauchy–Stieltjes kernel families, we study the induced effect of these transformations on the associated variance functions and obtain explicit transformation formulas. These results yield a stability theorem showing that the free Meixner class is stable under both operators. In addition, we derive two properties of the semicircle law via the restricted deformations ${\mathbf{U}}_{(a,b,1/b,t)}^{\prime }$ and ${\mathbf{U}}_{(a,b,1/b,t)}^{″}$, thereby emphasizing the structural role of symmetry in measure transformations and in the preservation of canonical measures. Full article

This study develops a stochastic degradation-based reliability framework for mechanical systems subject to interacting operational stresses and imperfect maintenance. The degradation dynamics are formulated in cumulative damage space and modeled using a geometric Itô diffusion process, in which the drift term incorporates a multiplicative degradation kernel representing the combined influence of load, speed, misalignment, and environmental exposure. Imperfect maintenance is represented through a continuous attenuation functional embedded within the drift structure, allowing maintenance actions to reduce degradation growth without restoring the system to an as-good-as-new condition. Using a logarithmic transformation, the multiplicative stochastic differential equation is converted into an additive diffusion process, enabling analytical treatment via Itô’s lemma. A closed-form reliability expression is then obtained through first-passage analysis, yielding a lognormal survival function governed directly by the degradation dynamics. Numerical evaluation demonstrates physically consistent wear-out behavior and confirms the stability of the derived reliability formulation. The model further enables reliability-based maintenance optimization through preventive replacement analysis. Sensitivity results indicate that system reliability is strongly influenced by the degradation growth parameter governing the stochastic drift. The proposed framework provides a mathematically tractable connection between stochastic degradation modeling, reliability theory, and maintenance optimization. Beyond its application to conveyor belt systems, the formulation offers a general analytical structure for reliability assessment of degrading engineering systems governed by multiplicative stochastic dynamics. Full article

Classical pumping-test interpretation relies on instantaneous drawdown and its time derivative, both of which amplify measurement noise. This study introduces cumulative drawdown, or absement, as the primary state variable for confined aquifer pumping-test analysis. Time integration of the transient flow equation yields a governing relationship connecting instantaneous drawdown to the spatial curvature of cumulative drawdown. A closed-form absement kernel is derived from the Theis solution and cast in dimensionless form, yielding four diagnostic operators: absement A(t), time-averaged absement A(t)/t, windowed ΔA/Δt, and the normalized absement derivative (NAD). The four operators read different features of the same drawdown record and together cover parameter estimation, scale-resolved diagnostics, and flow-regime identification. Monte Carlo testing (N = 50) under combined Gaussian, log-time jitter, and quantization noise recovers transmissivity to within 0.1% of the true value (p > 0.5). Field tests on homogeneous and heterogeneous confined aquifers reproduce classical Theis and Cooper–Jacob estimates and identify two scale-dependent regimes in the Oude Korendijk dataset. Full article

This paper presents a meshless collocation technique for the fourth-order transient stream function formulation of the Navier–Stokes equations. The technique employs a hybrid kernel function and is augmented by the ghost point method and Picard iteration. The reduction in unknowns inherent in this stream function approach simplifies the solution process. Introducing vorticity and stream functions enables mathematical reformulation of the coupled, time-dependent Navier–Stokes system as a fourth-order partial differential equation in one variable. The Gaussian–cubic backward substitution method and time difference method are used to solve the corresponding equation, in which the nonlinear part is generally transformed into linear equations through Picard iteration methods. This paper simulates three flows to prove the feasibility of the scheme. Full article

We investigate biased and unbiased Monte Carlo algorithms for solving Fredholm integral equations of the second kind and for estimating linear functionals of their solutions. Fredholm integral equations provide a common mathematical framework in uncertainty quantification, Bayesian inference, physics, finance, engineering modeling, telecommunication systems, signal processing, and other applied problems where system responses depend on distributed, uncertain, or noise-affected inputs. The comparison covers Crude Monte Carlo and Markov Chain Monte Carlo baselines, modified Sobol quasi–Monte Carlo schemes (MSS variants), the classical Unbiased Stochastic Algorithm (USA), and a new variance-controlled unbiased estimator, the Novel Unbiased Stochastic Algorithm (NUSA). NUSA preserves unbiasedness via a randomized-trajectory representation while improving stability through two mechanisms: adaptive absorption control, governed by a parameter P_d that regulates the effective trajectory length, and kernel-weight normalization based on an auxiliary proposal density to curb heavy-tailed weight products. Extensive experiments in one- and multi-dimensional settings (including regular and discontinuous kernels and weak/strong coupling regimes) show that NUSA consistently reduces dispersion and achieves smaller errors than USA under identical sampling budgets. In representative tests, NUSA attains relative errors below 10⁻³ and improves average accuracy by approximately 30–50% compared with USA, while maintaining near-linear runtime scaling in N and competitive scaling with dimension. Although NUSA is moderately more expensive per run than USA, the variance reduction yields a superior accuracy–cost trade-off, especially near strong-coupling regimes and in higher dimensions where standard unbiased estimators become variance-limited. Full article