Search Results (1,095)

This study introduces a Caputo fractional-order version of the SEIRS-V model to investigate the spreading dynamics of worms within wireless sensor networks. Traditional integer-order worm propagation models describe the instantaneous evolution of network states; however, they do not adequately account for memory and hereditary characteristics that may influence the transmission dynamics. Consequently, their ability to represent realistic network behavior can be limited in systems where past states affect current propagation patterns. The framework divides sensor nodes into susceptible, exposed, infectious, recovered, and vaccinated classes, while explicitly incorporating worm transmission rates, temporary loss of immunity, and the impact of preventive security measures under limited resource conditions. A detailed theoretical examination is performed, covering the existence, boundedness, and uniqueness of solutions of the fractional-order system. The coupled nonlinear fractional system is solved semi-analytically by means of the Fractional Reduced Differential Transform (FRDT) technique. To confirm accuracy and robustness, the identical system is also discretized and solved using the finite difference scheme (FDS). Unlike previous studies on worm propagation models in wireless sensor networks, which are mainly limited to equilibrium point analysis and qualitative investigations without deriving explicit solutions, the present work develops an approximate semi-analytical solution for the fractional-order SEIRS-V system using the FRDTM. Comparisons between the two solution sets demonstrate excellent agreement and high precision. Numerical outcomes are presented through a series of 2D graphical profiles that illustrate the time-dependent behavior of each compartment and reveal the sensitivity of worm propagation and suppression to variations in the fractional order and key model parameters. The integrated theoretical and computational findings underscore the strong protective role of vaccination in mitigating worm outbreaks and offer valuable guidelines for strengthening cybersecurity measures in wireless sensor networks. Full article

Heat transport in heterogeneous materials can deviate markedly from classical Fourier behavior when microstructural disorder, trapping effects, nonlinear mobility, and long-range temporal correlations interact across multiple spatial and temporal scales. These mechanisms may produce delayed relaxation, persistent thermal footprints, front deformation, and non-classical spreading patterns that are not adequately represented by conventional integer-order diffusion models. In this study, a modeling and simulation framework is developed for anomalous heat transport in heterogeneous media using a two-dimensional time-fractional porous medium equation. The model combines a Caputo fractional time derivative, which represents thermal memory, with nonlinear degenerate porous-medium diffusion, spatially heterogeneous conductivity, localized volumetric heating, and Robin-type convective boundary exchange. A conservative fully discrete numerical scheme is constructed using flux-based finite differences for the heterogeneous nonlinear diffusion operator and an $L1$ approximation for the Caputo derivative. The nonlinear algebraic system at each time level is solved using an under-relaxed Picard frozen-coefficient iteration with non-negativity enforcement and sparse direct solution of the resulting linear systems. The numerical implementation is verified through a manufactured-solution convergence study, and additional analyses are performed to examine computational cost, Picard iteration behavior, coefficient-regularization sensitivity, strong-source effects, heterogeneous conductivity structures, and long-time thermal-footprint persistence. The results show that heterogeneous conductivity mainly redirects heat through preferential pathways and enlarges the spatial footprint while producing negligible changes in global heat content. Stronger fractional memory, represented by smaller fractional order, increases the persistence and spatial reach of moderate heating, whereas larger porous-medium exponents confine heat near the source and preserve higher local peaks. Source amplitude increases the thermal burden and footprint monotonically over the tested range, including strong forcing, without producing an abrupt localization-spreading transition. Boundary exchange remains secondary in the short-time interior-heating regime considered. These findings demonstrate that the proposed two-dimensional time-fractional porous medium framework provides a verified and physically interpretable model for non-Fourier heat transport in heterogeneous materials, where local intensity, global heat retention, and spatial thermal exposure must be assessed jointly. Full article

This paper proposes a novel hybrid numerical scheme that augments the classical L1 finite-difference approximation of the Caputo fractional derivative of order $\alpha \in (0,1]$ with a selective shifted Grünwald–Letnikov correction (controlled by a shift parameter $\beta \in [0,1)$) applied only to the most recent time increment. When $\beta =0$, the scheme reduces exactly to the classical L1 scheme and retains its optimal convergence rate $O\left(h^{2-\alpha }\right)$, where h denotes the uniform time-step size. For $\beta >0$ (optimally chosen as $\beta =1-\alpha /2$), extra numerical damping is introduced at the cost of a mildly reduced convergence order $O\left(h^{1-\alpha }\right)$, while long-term stability is significantly improved. The scheme is applied to the fractional Kelvin-Voigt and Standard Linear Solid models to analyze creep and relaxation responses. Numerical simulations demonstrate that the proposed hybrid scheme achieves improved accuracy, long-term stability, and computational efficiency compared to classical integer-order models and several existing fractional schemes reported in the recent literature. Results show that fractional orders capture anomalous creep behavior more accurately, aligning with experimental data from recent studies. The proposed method offers improved computational performance for real-time structural health monitoring applications. Full article

We develop a systematic framework for incorporating perturbative correction terms into classical finite difference schemes for Allen–Cahn type stochastic partial differential equations. Three distinct correction approaches are introduced, conceptually motivated by perturbative quantum field theory, quantum coherent state evolution, and WKB (Wentzel–Kramers–Brillouin) barrier penetration theory. These quantum-inspired perturbative method (QIPM) corrections function as classical perturbations executing entirely on conventional hardware; quantum-mechanical formalism serves only as a design principle for constructing specific functional forms of correction terms. The primary novelty of this work lies in (i) a generic convergence-preservation theorem establishing sufficient conditions on correction magnitude for any perturbative correction to maintain the base scheme’s accuracy order, and (ii) a systematic translation methodology from quantum-mechanical analogies to explicit, implementable finite difference corrections with rigorous parameter bounds. Through convergence analysis, we demonstrate that appropriately parametrized corrections preserve the accuracy of the underlying numerical scheme, provided the solution possesses sufficient regularity and the parabolic scaling constraint $\Delta t=\mathcal{O}\left(h^2\right)$ holds. Numerical experiments on a spatially discretized domain over a finite time horizon using spatially correlated noise reveal that the anharmonic oscillator correction achieves exceptional accuracy with modest computational overhead, while the amplitude encoding correction provides intermediate accuracy with negligible timing cost. The tunneling-inspired correction exhibits higher error for smooth initial conditions, indicating strong problem-dependence. While these methods enhance accuracy in specific scenarios, genuine speedups on classical hardware are not achieved. The primary value lies in establishing systematic methodologies for perturbative correction design and developing theoretical foundations for future hybrid computational approaches. Full article

This study presents a comparative sensitivity analysis of the Hartmann number (Ha) and Brinkman number (Br) on magnetohydrodynamic (MHD) flow in rectangular channels and circular pipes. Normalized sensitivity coefficients quantify the response of key metrics, including velocity, wall shear stress, temperature, and convective heat transfer, with validation against recent experimental and numerical studies. The system equations were solved through a coupled analytical–numerical method coded in Python 3.14; velocity field was solved analytically whereas temperature field was discretized using a finite differences scheme and solved numerically using the Thomas algorithm. The entire code was written by the authors. The results show that Ha predominantly governs hydrodynamics, inducing velocity suppression, flow flattening, and enhanced wall shear stress. Rectangular channels experience stronger Hartmann layer effects, while circular pipes exhibit smoother velocity profiles. Conversely, Br primarily controls thermal behavior, with higher values intensifying internal heat generation and elevating centerline temperature, potentially attenuating the average Nusselt number at high Br levels. Nonlinear Ha–Br interactions define distinct operational regimes, from heat transfer enhancement to thermal degradation. Optimal performance windows are identified: Ha ≈ 8–12 and Br ≈ 0.05–0.3 for channels, and Ha ≈ 10–15 and Br ≈ 0.1–0.4 for pipes, balancing thermal and hydraulic efficiency. Deviations from benchmark studies remain within ±5%, confirming predictive reliability. This work provides practical design guidance for advanced MHD thermal systems and establishes a foundation for future studies on temperature-dependent properties, three-dimensional effects, and complex flow regimes. Full article

Fractional differential operators provide an effective approach for modeling complex technological processes, particularly physical phenomena in continuum mechanics characterized by memory and non-local effects. Different types of fractional derivatives require different numerical approximation schemes; in this study, the Caputo and Grünwald–Letnikov derivatives are considered. The aim of this work was to develop and validate a fractional differential model of longitudinal oscillations in a suspended monorail system that accounts for nonlinear and memory-dependent effects. In contrast to classical integer-order approaches, the proposed framework incorporates multiscale surface irregularity effects, including rail roughness, friction, and other disturbances influencing system dynamics, through a fractional-order formulation. A fractional differential mathematical model describing the motion of longitudinal oscillations of a large-sized cargo transported along a suspended monorail is proposed. A numerical algorithm based on finite-difference approximation of fractional operators was developed for its implementation. The scientific contribution lies in integrating multiscale surface irregularity effects into a fractional-order modeling framework to improve the accuracy of dynamic response prediction. Numerical experiments demonstrated the effectiveness of the approach, and the results were validated through comparison with existing models of monorail dynamics. Additionally, statistical validation based on correlation analysis confirmed good agreement with the experimental data. The proposed model can be applied to the design and optimization of suspended transport systems, improving vibration control, reliability, and operational safety under real dynamic loading conditions. Full article

High-order numerical methods are essential for achieving predictive fidelity in modern computational fluid dynamics, yet many existing schemes face significant trade-offs between accuracy, robustness, and computational efficiency. This study introduces the Compact-Corrected MUSCL (CCMUSCL) scheme, a novel framework that enhances the traditional MUSCL approach by incorporating localized information from high-order, compact finite-difference formulas. Unlike classical compact schemes that require solving global linear systems, this method applies corrections locally to the MUSCL flux. This strategy allows the scheme to maintain spectral-like resolution while preserving the robustness and locality of the original MUSCL framework. The performance of CCMUSCL is evaluated using a series of rigorous 1D and 2D benchmark cases. Numerical results demonstrate that CCMUSCL achieves accuracy comparable to or exceeding that of traditional high-order WENO schemes, particularly in resolving intricate, small-scale flow structures and sharp discontinuities. Furthermore, efficiency analysis reveals that CCMUSCL is significantly more cost-effective than WENO, requiring substantially fewer arithmetic operations in 1D and offering an even more pronounced reduction in operations for 3D flux evaluations. By offering a tunable balance between robustness and accuracy through the use of the van Albada limiter as a localized indicator, the CCMUSCL scheme provides a highly efficient and flexible alternative for large-scale high-speed flow simulations. Full article

This study provides a numerical solution for a two-dimensional axisymmetric problem involving a deformable cylindrical body impacting a rigid barrier at a constant velocity. Under the specified velocity conditions, the cylindrical body undergoes both elastic and plastic deformation. The two-dimensional axisymmetric nonstationary wave problem is solved using the finite difference method, employing the modified Wilkins scheme. The analysis focuses on changes in the shape of the cylindrical body post-impact and examines the parameters of elastic–plastic wave deformation in both axial and radial directions at fixed points. The numerical solutions are compared with established experimental results, demonstrating good agreement. Additionally, the wave parameters within the cylindrical body are analyzed as functions of the impact velocity. The findings indicate that the finite difference method utilizing the Wilkins scheme yields reliable results for a two-dimensional axisymmetric elastic–plastic nonstationary wave problem. The numerical results obtained can be used to assess wave parameters and the strength of similar bodies impacting a rigid barrier at high velocities. Full article

Magnetic targeting of drug carriers is commonly studied at macroscopic scales, while its impact on drug transport across individual cell membranes remains poorly quantified. Here, we present a theoretical and numerical model of magnetically assisted drug transport across the membrane of a single tumor cell exposed to magnetic field gradients. Extracellular transport is described by an advection–diffusion equation that couples passive diffusion with magnetophoretic drift, whereas intracellular transport is governed by diffusion and first-order uptake kinetics. The cell membrane is modeled as a semi-permeable interface with finite permeability, providing explicit coupling between extracellular and intracellular domains. Assuming spherical symmetry, the coupled transport equations are solved using finite-difference schemes, with magnetic forcing represented through an effective drift velocity $v_{\mathrm{m}\mathrm{a}\mathrm{g}}$ and interpreted using the magnetic Peclet number. To enable a controlled comparison between healthy and tumor cells, identical geometric, diffusive, and magnetic parameters are used, while biological differences are introduced solely through membrane permeability and intracellular uptake rates. By separating cumulative membrane delivery from cumulative intracellular uptake, the model resolves ambiguities arising from heterogeneous uptake kinetics. The results show that magnetophoretic drift enhances near-membrane drug accumulation and effective transmembrane flux without modifying intrinsic membrane properties. Magnetic targeting therefore acts as a transport amplifier, magnifying pre-existing biological differences and producing a larger model-predicted delivery advantage in tumor cells. Overall, the framework identifies the magnetic Peclet number as the key parameter governing the transition from diffusion-dominated to drift-enhanced cellular drug transport. Full article

This paper investigates numerical modeling of transport and diffusion processes of harmful impurities governed by the advection–diffusion–reaction equation, along with the solution of corresponding direct and inverse problems. Particular emphasis is placed on identifying pollution source parameters and reconstructing spatiotemporal concentration distributions from limited and noisy observational data. Classical numerical methods, including stable finite-difference schemes, are employed for solving direct problems. Inverse problems are tackled using modern approaches such as regularization techniques, global optimization, and machine learning methods. In particular, evolutionary optimization algorithms and physics-informed neural networks (PINNs) are considered, enabling the integration of physical laws, observational data, and prior information within a unified computational framework. Computational experiments demonstrate that hybrid approaches combining classical numerical methods with machine learning significantly enhance the accuracy and stability of inverse problem solutions, especially under incomplete or noisy data conditions. Neural network-based methods exhibit strong approximation capabilities and effectively recover unknown model parameters. The results highlight the potential of integrating numerical and intelligent methods for environmental monitoring and pollutant dispersion forecasting, and can be applied in the development of operational analysis and environmental management systems. Full article

The inverse problem under consideration concerns a one-dimensional parabolic equation whose thermal diffusivity takes the quadratic-in-space form $a_s\left(\tau \right){\kappa }^2+b_s\left(\tau \right)\kappa +c_s\left(\tau \right)$. The unknowns are three time-dependent coefficients $a_s\left(\tau \right),b_s\left(\tau \right),c_s\left(\tau \right)$ together with the temperature field $T(\kappa ,\tau )$. The direct problem supplies initial data, Neumann boundary conditions, and three over-determination conditions: two boundary temperatures and the spatial integral of T. We prove two theorems. The first theorem establishes the local-in-time existence of a solution under explicit regularity and sign conditions on the given data $\xi ,{\nu }_k,{\delta }_{\ell },\theta$ and compatibility at $\tau =0$. The second theorem guarantees the uniqueness of this solution. Despite uniqueness, the inverse reconstruction remains ill-posed: small perturbations in the over-specified data can cause large deviations in the recovered coefficients. For the forward model, we implement two numerical schemes: (i) a Crank–Nicolson finite difference methodology (CN-FDM) on a uniform grid and (ii) a semi-discretized Crank–Nicolson approach combined with Bell spectral collocation in space (CN–Bell). The inverse step minimizes a Tikhonov-regularized least-squares functional using MATLAB’s (R2026a) lsqnonlin. Two numerical examples (smooth and non-smooth), tested with both exact synthetic data and artificially added noise, demonstrate stable and accurate coefficient reconstructions. The framework applies directly to heat conduction and porous media flow where diffusivity varies quadratically in space. Full article

Compact finite difference schemes approximate spatial derivatives through implicit relations between neighboring grid points. Despite using compact stencils and relatively simple algebraic structures, these schemes achieve high-order accuracy and spectral-like resolution, reducing dispersion errors while maintaining low numerical dissipation. These properties make them particularly attractive for problems requiring accurate spatial derivatives and computational efficiency, such as wave propagation, aeroacoustics, and turbulent flow simulations. This review presents the main ideas behind compact finite difference schemes, including their derivation from Taylor expansions and Padé approximations, their accuracy properties, and their resolution characteristics through modified wavenumber analysis. The manuscript is intended as a review and practical synthesis, rather than as the proposal of a new numerical scheme, and aims to connect the theoretical construction of compact schemes with their numerical behavior, practical implementation, and representative applications. To support reproducibility, we provide a fully documented open-source Python 3.11 notebook with a reference implementation of the schemes discussed in the paper. The examples include first- and second-order derivative calculations and representative one- and two-dimensional boundary-value problems, including Helmholtz-type equations. Finally, we survey applications across computational fluid dynamics, acoustics, geophysical flows, structural mechanics, biology, electromagnetism, and quantitative finance. Full article

We present and analyze a weighted family of iterative methods for solving systems of nonlinear equations. The proposed schemes are constructed as a generalization of the Singh–Sharma fifth-order method by incorporating suitable weight functions into the correction step, thereby generating a flexible class of methods that includes the original scheme as a special case. Sufficient conditions on the weight functions are established to guarantee fifth-order local convergence, and the resulting error equation shows how the weights influence the leading error term. Several admissible choices are presented to illustrate the versatility of the family. The practical performance of the proposed variants is investigated on a collection of large-scale nonlinear systems. Furthermore, the family is applied to the nonlinear algebraic system obtained from the finite-difference discretization of a stationary one-dimensional viscous Burgers problem. Numerical experiments indicate that the proposed methods provide a competitive and accurate alternative for solving nonlinear systems of this type. Full article

Retinal vessel segmentation remains challenging for thin vessels and low-contrast bifurcations. We evaluate a Swin-UNet-style model family that conditions decoder features with a single-channel local fractal dimension prior refined by a short learnable anisotropic diffusion model and injected through Spatially-Adaptive Normalization (SPADE). On Fundus Image Vessel Segmentations (FIVES), the strongest no-test-time-augmentation result was obtained by OPT-I v2 at 200 epochs, reaching Dice 0.8899, clDice 0.8517, and Area Under the ROC Curve (AUC) 0.9904, compared with 0.8643, 0.8125, and 0.9856 for the matched 200-epoch baseline. In a matched Neural Cellular Automata (NCA)/no-NCA ablation using the same seed, data, 200-epoch budget, and evaluation pipeline, enabling NCA improved the test Dice from 0.8813 to 0.8907 and the test clDice from 0.8325 to 0.8518, with NCA winning on all 80 paired test images for both metrics. The results support PDE (partial differential equation)-SPADE fractal prior conditioning and NCA topology refinement as ablation-grounded improvements over the tested baseline family, while broader matched external validation requires future work. Full article

In this study, a mathematical model based on nonlinear differential equations was developed to describe nitrate (NO₃⁻) removal in a woodchip denitrification bioreactor treating tile drainage water. The model captures temperature-dependent denitrification kinetics and transport processes under variable operating conditions. The model was validated using pilot-scale experimental data collected at different inflow water temperatures. The results indicated a strong temperature dependence of nitrate removal efficiency, with higher performance at elevated temperatures due to increased microbial activity and reaction rates. After validation, numerical simulations using a finite difference scheme were performed to evaluate bioreactor performance under varying hydraulic and geometric conditions. The analysis focused on the effect of bioreactor length, assuming constant width and depth (1.0 m each). Results showed that increasing reactor length enhances NO₃⁻ removal by extending hydraulic retention time, although the effect becomes nonlinear due to substrate limitation along the flow path. Simulations further demonstrated that a target NO₃⁻ removal efficiency of approximately 40% can be achieved through different combinations of temperature, bioreactor length, and hydraulic loading, indicating a compensatory relationship between kinetic and design parameters. Overall, this study provides a predictive framework for optimizing bioreactor design and operation, offering practical guidance for improving nitrate removal in agricultural drainage systems. Full article