Search Results (157)

This manuscript presents an efficient hybrid numerical method for solving one- and two-dimensional time-fractional Cattaneo models that incorporate the Riesz space-fractional derivative. The proposed approach combines the Crank–Nicolson scheme for temporal discretization with the classical Galerkin finite element method for spatial discretization. The stability of the semi-discrete scheme is established in the $L^2$ norm, and a convergence analysis of the fully discrete scheme is carried out, yielding a temporal convergence order of $\mathcal{O}\left({(\Delta t)}^{3-\mu }\right)$ and a spatial convergence order of $\mathcal{O}\left({\mathit{R}}^2\right)$. Finally, two numerical examples, in one and two dimensions, are provided to demonstrate the accuracy and efficiency of the proposed method and to verify the theoretical findings. Full article

Moving boundaries (often called free boundaries) in shallow-water equations mark the time-dependent water line between ‘(flooded) wet’ and ‘dry’ regions and this feature is of utmost practical importance and should be adequately incorporated into the simulation. An efficient and conceptually simple implicit scheme is proposed for the one- and two-dimensional shallow-water equation systems with moving boundary treatment (i.e., wet–dry scheme). For the implementation of a wet–dry scheme in which very low water depth is present, a fully implicit backward-Euler scheme was developed for the fixed Eulerian mesh domain and a positivity-preserving limiter was employed. For the approximate Riemann problem, the local Lax–Friedrichs flux is used. The Jacobians are estimated by centered finite differences in an approximate way rather than exact formulas, which, in principle, greatly reduce the coding efforts compared to the analytical formulation and will make extensions to higher dimensional schemes. Some benchmark problems with exact solutions and/or reference experimental data are presented for the domain where the initially dry bed presents. Overall, satisfactory results are obtained. Full article

This paper proposes a finite element approach to the numerical solution of boundary value problems of linear elasticity theory formulated directly in terms of the stress tensor components. In contrast to the classical finite element formulation, in which displacements are the primary unknowns, the approach considered here treats the stress components as the sought quantities. Two forms of the boundary value problem are investigated. The first is based on the joint use of the equilibrium equations and the Beltrami–Michell equations, while the second is a transformed system of Poisson-type equations for the stress tensor components. Variational relations based on the Galerkin method are obtained for both formulations. Using linear basis functions on triangular finite elements, local matrices are constructed, and global systems of algebraic equations are formed, whose unknowns are the nodal values of the stresses. The numerical implementation of the developed schemes is carried out in an in-house C++ program and in the FreeFEM++ software environment. To verify the reliability of the proposed mathematical and numerical models, the classical Kirsch problem of a stretched elastic plate with a circular hole—characterized by a pronounced stress concentration near the edge of the hole—is solved. The numerical values of the stress components are compared with the analytical solution obtained using the Airy stress function method, as well as with the results of calculations performed with the FreeFEM++ software package. The comparison shows good agreement between the obtained solutions and confirms the possibility of determining stresses directly, without first computing the displacement field. A mesh-refinement study further showed a systematic reduction in the numerical error: on the finest mesh considered, the relative error decreased to 3.189% for formulation A and to 7.415% for formulation B. The proposed approach extends the applicability of the finite element method to boundary value problems of elasticity theory formulated in terms of stresses and can be used to study problems with complex geometry and local stress concentration. Full article

This study establishes the class of Sobolev-type neutral stochastic differential equations with Hilfer derivative and finite delay. Existence and uniqueness of the mild solution are then established in the mean-square sense via the Banach contraction principle, and a Faedo–Galerkin scheme is constructed whose approximate solutions are shown to converge, in mean square, to the unique mild solution. An application to a stochastically perturbed Sobolev-type filtering/diffusion model closes the paper. Full article

This article studies a time-dependent latent sentiment representation governed by a semilinear stochastic evolution equation on a separable Hilbert space. The objective is to derive uncertainty, interpretability, and robustness from one stochastic law. The mathematical analysis establishes covariance propagation and spectral uncertainty attribution, Fréchet differentiability of the forcing-to-state and forcing-to-readout maps with quadratic remainders, trace-norm differentiation of the covariance, and an adjoint influence kernel whose norm equals the exact worst-case displacement over an energy-bounded forcing ball in the linear regime. The dependence of the certificates on dissipativity, forecast horizon, noise geometry, and spectral truncation is made explicit. A Galerkin state-space reduction yields exact linear transitions, a likelihood-based calibration scheme, and structural identifiability conditions. A reproducible three-mode study verifies the covariance and duality identities, evaluates the sensitivity constants, and compares Gaussian with split-conformal predictive intervals under controlled synthetic conditions. The results provide a rigorous operator calculus and a tractable finite approximation. They establish internal mathematical and numerical validity without asserting a generalised superiority. Full article

Accurate and computationally efficient numerical modelling of shallow-water flows is essential for flood prediction and hydrodynamic risk assessment. This study develops an implicit finite-difference scheme for the numerical solution of the two-dimensional shallow-water equations. First, the main numerical approaches used for shallow-water modelling, including finite-difference, finite-volume, finite-element, and discontinuous Galerkin methods, are analysed in terms of accuracy, stability, treatment of discontinuities, and computational requirements. Based on this analysis, an implicit finite-difference formulation is developed that uses central approximations for spatial derivatives and averages flow variables at cell boundaries. The nonlinear terms are treated using Newton linearization, resulting in an iterative scheme that allows larger time steps than explicit formulations constrained by the Courant–Friedrichs–Lewy condition. The proposed method is implemented in MATLAB as a computational module for two-dimensional hydrodynamic simulations. Its performance is demonstrated on a test problem that describes the propagation of an initially localised disturbance in a rectangular computational domain with rigid boundaries. The numerical results demonstrate stable wave propagation, conservation of the modelled flow dynamics, and physically consistent boundary reflections. The developed approach provides a computational basis for further integration of shallow-water hydrodynamic models with spatial data and geographic information systems for flood forecasting and risk assessment. The implicit scheme allowed the release of time steps $\Delta t = 0.1, 0.5$ and 0.9, which significantly exceeds the limit stability of the explicit scheme, which, due to the Courant–Friedrichs–Lévy conditions, was limited to the value Δt ≤ 0.01. The simulation results show that the developed scheme provides stable wave growth and physically correct separation from impermeable boundaries for all investigated time step indicators. The obtained water depth profiles at times $t = 10, 15, 20$ and 25 s illustrate the correct evolution of the initial combustion: the wave front expands symmetrically while preserving the conservative properties of the model hydrodynamics. The numerical solution demonstrates the accuracy and robustness of the proposed implicit finite-difference scheme, even when using time steps that are almost two orders of magnitude larger than those allowed by explicit methods. The results confirm that the developed approach is a robust and computationally efficient tool for hydrodynamic modelling, intended for further integration with geographic information systems in behaviour prediction and risk assessment tasks. Full article

This paper develops and rigorously analyzes a fully discrete upwind interior penalty discontinuous Galerkin (IPDG) scheme for simulating solute transport in two-dimensional variable-order fractional mobile–immobile media. The temporal variable-order Caputo derivative is discretized via a Grünwald–Letnikov approximation in conjunction with a first-order backward difference, while the spatial discretization employs an IPDG method featuring an upwind numerical flux for the convection term and a penalty formulation for the diffusion operator. Under the physically relevant assumption of a divergence-free velocity field, we establish the unconditional stability of the proposed scheme. A comprehensive error analysis in the $L^2$ norm yields a convergence rate of $O(\Delta t+h^{\min (k+1,s)-\chi -1/2})$, explicitly linking the polynomial degree k, solution regularity s, and the penalty variant $\chi$. Numerical experiments in two dimensions are conducted to verify the accuracy and robustness of the proposed scheme in simulating anomalous transport phenomena in subsurface environments. Full article

In this paper, we consider a nonlinear fractional superdiffusion equation involving a Caputo derivative of order $\alpha \in (1,2)$. Solutions to this class of problems typically exhibit weak singularities near the initial time. To handle this singularity, we introduce an auxiliary variable $p:=D_t^{\alpha /2}(u-t{u}_1)$ and reformulate the original problem as an equivalent coupled system. This system is then discretized using the nonuniform Alikhanov scheme in time, the conforming Galerkin finite element method in space, and Newton linearization for the nonlinear term. We prove that the numerical solutions remain uniformly bounded, independent of the temporal and spatial mesh parameters. Combining this result with a discrete fractional Grönwall inequality and a temporal-spatial splitting argument, we establish unconditional optimal error estimates of order $O(N^{-min\{2,r\alpha /2\}}+h^k)$. Finally, numerical experiments are presented to verify the sharpness of the theoretical convergence rates. Full article

We present and analyze a time-fractional, partly diffusive model of two electrically coupled Hindmarsh–Rose neurons in which only the membrane potentials undergo spatial transport, through a nonlinear Neumann m-Laplacian, while the temporal evolution is governed by a Caputo derivative of order $\rho \in (0,1]$. The fractional operator incorporates hereditary relaxation, whereas the m-Laplacian represents gradient-dependent degenerate transport and recovers ordinary diffusion when $m=2$. In a Gelfand triple adapted to the no-flux boundary condition, we derive a fractional energy inequality, a uniform dissipative estimate, and the existence of a global weak solution by a Faedo–Galerkin approximation, fractional compactness, and Minty’s method. Uniqueness and continuous dependence are obtained in the stated bounded solution class. We prove global Mittag–Leffler synchronization above an explicit coupling threshold and establish a practical synchronization bound under parameter mismatch. A fully implicit L1 finite-volume method is then constructed; every time step is solvable, uniqueness follows under an explicit monotonicity condition, and the scheme is unconditionally energy dissipative and convergent. Manufactured-solution tests recover the expected $2-\rho$ temporal and second-order spatial rates. In the neuronal simulations, reducing $\rho$ from 1 to $0.90$ lengthens the mean bursting period from about 379 to 565 time units, an increase of roughly one half, and raises the number of spikes per burst from about $31.7$ to $38.8$. Over $2\le m\le 4$ the temporal rhythm is essentially unchanged, the burst period staying near 362 time units, while the diffusion exponent reshapes the peak amplitude and the spatial gradient profiles of the traveling fronts. The empirical synchronization threshold for the canonical parameter set is approximately $11.0$, far below the global sufficient bound $2.3917\times {10}^4$, which quantifies the conservatism of the analytical certificates. Full article

This paper presents a Galerkin finite element method based on cubic $\beta$-spline basis functions for the numerical solution of multi-term time-fractional differential equations. The spatial discretization is constructed using a parametric family of cubic $\beta$-splines, which provide enhanced flexibility and smoothness in the approximation space. For the temporal discretization, the Caputo fractional derivatives are approximated using the $L1$ scheme combined with a $\theta$-method to improve stability and numerical accuracy for multi-term fractional models. The resulting fully discrete formulation is solved efficiently and the source terms are evaluated using Gauss–Legendre quadrature to obtain accurate load vector approximations. The convergence behavior of the proposed scheme is investigated through mesh refinement analysis and numerical accuracy is assessed using standard error norms. Several benchmark test problems are considered to validate the proposed method. The numerical results demonstrate that the cubic $\beta$-spline Galerkin approach provides accurate and stable approximations, with error norms decreasing consistently as the mesh is refined. Comparisons with available exact solutions further confirm the efficiency and reliability of the proposed numerical scheme for solving complex multi-term time-fractional differential equations. Full article

A spectral Galerkin algorithm is developed for the numerical treatment of modified anomalous time-fractional sub-diffusion equations (MATFSDEs). The algorithm is constructed using Fibonacci coefficient polynomials, from which two space–time trial families are formed. These families are selected so that the homogeneous initial and boundary conditions are automatically incorporated after a suitable transformation of the original problem. The Galerkin formulation then reduces the model to a finite algebraic matrix system whose entries can be explicitly evaluated. To support the construction, inversion, moment, and linearization identities for the selected polynomials are obtained and then used to express the required matrices in closed form. The convergence of the expansion is investigated, and explicit error bounds are derived. Numerical tests are reported to demonstrate the accuracy, applicability, and competitiveness of the proposed scheme in comparison with some existing methods in the literature. Full article

In this paper, the nonlinear vibration (NV) response of carbon nanotube-reinforced composite (CNTRC) marine panels (MPs) fitted with symmetric multi-vibration absorbers (MVAs) subjected to steady, velocity-dependent hydrodynamic loads is investigated. To model actual marine conditions more realistically, the lift and drag forces varying with flow velocity were taken into account using experimentally supported Matveev-based formulations for a specific ship. Within the shell, three carbon nanotube (CNT) distribution schemes are considered: one uniformly distributed (UD) CNT configuration and two functionally graded (FG) CNT patterns, namely FG-V and FG-X. The analytical framework is further constructed using classical shell theory (CST) by incorporating geometric nonlinear terms, and the Galerkin technique is employed to obtain a reduced-order model. Thereafter, the NV response of the CNTRC-MPs is predicted through the P-T method, which relies on the joint application of the piecewise constant argument and Taylor series expansion. The results indicate that symmetric MVAs can effectively suppress NV behavior and significantly decrease the maximum NV amplitude of the panel. Moreover, the effectiveness of the proposed configuration is shown to depend on both the absorber characteristics and the reinforcement pattern of CNTs. The study demonstrates that the use of symmetric absorber systems offers a practical and efficient passive vibration-control solution for advanced marine composite panels. Full article

Wave propagation modeling in heterogeneous media requires numerical methods that can simultaneously handle complex geometries, artificial boundary reflections, and spatially varying resolution demands. In this study, we present a Discontinuous Galerkin (DG) forward modeling method for wave propagation with split-field absorbing boundary conditions and gradient-based adaptive meshes. The wave equation is formulated as a first-order hyperbolic system and discretized by the DG method, which preserves local conservation and is well suited for explicit Runge–Kutta time integration on unstructured meshes. To reduce spurious reflections from truncated computational boundaries, a split-field absorbing boundary treatment is introduced in the absorbing layer through directional damping terms, maintaining the first-order structure and local update form of the DG scheme. In addition, a physics-based mesh metric is constructed from the local velocity-gradient length scale, allowing the mesh to be automatically coarsened in smooth regions and refined near strong velocity contrasts, interfaces, and discontinuities. Numerical convergence tests show that the quadratic DG scheme achieves the expected third-order accuracy in the L2 norm. Quantitative PML evaluation gives a reflection coefficient of approximately $1.65\times {10}^{-5}$, indicating effective suppression of artificial boundary reflections. For the three-dimensional Marmousi model, the proposed adaptive mesh reduces the number of tetrahedral elements from 158,492 to 88,681, decreases the CPU time from 241.4453 s to 139.1328 s, and reduces memory consumption from 3788.67 MB to 2109.55 MB compared with the uniform mesh. These results demonstrate that the proposed method can improve the balance between computational efficiency and solution accuracy while maintaining stable and physically interpretable wavefield modeling in heterogeneous media. Full article

In this study, a high-order Galerkin–Numerov approach is presented to solve the transient vibration problem of anisotropic Kirchhoff plates supported by a uniform Vlasov foundation. A discretization of the governing fourth-order plate equation is derived based on a mixed boundary value problem and a hybrid Hermite–sine Galerkin formulation, which maintains the C1-continuity properties of classical plate theory. The resulting reduced-order modal system is integrated in time with the Numerov scheme, which is fourth-order accurate, and has a small numerical dispersion and good phase-preserving properties for oscillatory dynamics. The proposed methodology is evaluated using stability and convergence tests and parametric investigations. The fourth-order temporal convergence and rapid spectral-like spatial convergence of the numerical results are validated, and the long-time accuracy and robustness of the formulation is confirmed by the negligible phase error and bounded energy drift. The results from the parametric study indicate that the thickness of the plates and the stiffness of the Winkler foundation are the two most important mechanisms for vibration suppression, while the orthotropic coupling and the Vlasov shear interaction have substantial effects on the modal redistribution and transient deformation properties. The proposed method is compared with the conventional lower-order integration schemes, and it is observed that the method gives better phase fidelity and computational efficiency, and it is possible to predict the vibration amplitude and vibration timing accurately. In addition to the numerical benefits, the framework also provided physical insights on the coupled effect of anisotropy, foundation interaction and boundary restraint. The suggested model is directly applicable for composite floor systems, aerospace panels, foundation supported slabs, biomechanical plate analogs, etc., and smart vibration control platforms. This work thus lays the groundwork for future studies of nonlinear behavior, adaptive foundations and digital twin simulation of structural systems and presents a strong and scalable computational tool for the study of plate–foundation dynamics. Full article

In this paper, we consider a mathematical model for the flow of a dilute aqueous polymer solution through a heterogeneous porous medium. On the boundary of the flow region, the impermeability condition and a nonlinear Navier-type slip condition are prescribed. Our goal is to investigate the existence of weak solutions to the governing equations of this model, which is a challenging problem for both steady and unsteady flows. To prove the weak solvability, we use a modified Galerkin scheme with special basis elements in appropriate Sobolev spaces. We obtain the existence results without assuming smallness of the model data for the boundary value problem related to steady flows and prove the global-in-time solvability of the initial-boundary value problem describing unsteady flows. Moreover, energy equalities are established for weak solutions possessing additional regularity. Our results can serve as a starting point for further research on the problems under consideration, including numerical analysis and optimal control of polymer fluid flows. Full article