Abstract In this work, we consider the model of a mass transport problem through polymeric membranes which is important in many applications especially drug delivery. The aim of this paper is to present a more accurate mathematical model to describe this phenomenon. We propose adopting fractional diffusion modeling for the concentration by using spatial fractional-order Riesz derivative. The numerical simulations for studying this phenomenon were carried out using an approach of finite element method and finite difference method. The error analysis and stability condition for this technique are derived. For the order of the fractional derivative, we studied three cases: the constant-order case, the piecewise continuous case, and the time-varying-order case. The results obtained show that using time-varying Riesz fractional derivative yields a more accurate description of the considered problem than both the classical diffusion model reported in the literature and the other fractional derivatives considered.

Abstract The GBS code (Ricci et al. 2012 PPCF 54 124047) for boundary plasma turbulence is extended to handle flexible first wall geometry. This capability is enabled by the use of a finite difference scheme on a curvilinear structured grid. Grid generation and optimization leverage a spline-based elliptic grid generation framework originally developed in the context of isogeometric analysis applications (Hinz and Buffa. 2024 Eng. Comput. 40, 3735–3764). First turbulence simulations of the Tokamak à Configuration Variable (TCV) with a realistic first wall geometry including baffles are presented and highlight a reduction of upstream fluctuations compared to unbaffled cases.

This article proposes a numerical scheme based on finite difference approximation for solving the time-fractional advection-diffusion equation on a metric star graph comprising initial, boundary, and transmission conditions on the nodes of the graph. The choice of considering a star graph is motivated by the fact that any arbitrary graph can be decomposed into a star graph. The fractional derivative is considered in the Caputo sense and the well-known L 1 method is applied for its discrete approximation. The unconditional stability of the difference scheme for Δ x i ≤ 2 is analyzed by employing the discrete energy method. The unique solution behaviour of the scheme is also studied. An overall convergence of order O ( Δ t 2 − α + ∑ i = 1 k Δ x i 2 ) is obtained from the proposed scheme. Finally, a few experiments over test examples are demonstrated to verify the accuracy and convergence of the proposed scheme.

An enhanced method called FreeViBe+ for moving target segmentation is proposed in this paper, addressing limitations in the ViBe algorithm such as ghosting, shadows, and holes. To eliminate ghosts, multi-frame background modeling is introduced. Shadows are detected and removed based on their characteristics in the HSV color space, while holes are filled by merging GrabCut segmentation results with the ViBe extraction output. Furthermore, the Structure-measure is tuned to optimize image fusion, enabling improved foreground–background separation. Comprehensive experiments on the UCF101 and Weizmann datasets demonstrate the effectiveness of FreeViBe+ in comparison with Finite Difference, Gaussian Mixture Model, and ViBe methods. Ablation studies confirm the individual contributions of multi-frame modeling, shadow removal, and GrabCut refinement, while sensitivity analysis verifies the robustness of key parameters. Quantitative evaluations show that FreeViBe+ achieves superior performance in precision, recall, and F-measure compared with existing approaches.

A new iterative multi-step method for solving nonlinear equation.

Thermotactic microorganisms play a vital role in bioremediation, biodegradation, and the food and agriculture industries. Their thermostable and thermoactive enzymes make them particularly suitable for moderate-temperature bioprocesses, thereby enhancing efficiency and overall performance. Building on these characteristics and practical relevance, this work explores how linear and nonlinear internal heat sources influence the onset of bioconvection in a nanofluid layer populated with thermotactic microorganisms. A mathematical model is developed using Buongiorno’s model for nanofluid together with Pedley’s model for microorganisms. Using normal-mode analysis, the dimensional flow equations are converted into a system of non-dimensional ordinary differential equations. The eigenvalue problem obtained is numerically solved using the finite difference scheme with MATLAB’s bvp4c solver, under the constraints of zero nanoparticle flux and rigid–rigid boundaries. The influence of key flow parameters on the onset of bioconvection is analyzed and presented in graphically and tabular form. A key novel insight of this study is that, under top heating in a nanofluid containing thermotactic microorganisms, linear internal heat generation destabilizes the system by driving microorganisms upward toward the hotter boundary, whereas nonlinear internal heat generation exerts a stabilizing influence by suppressing their accumulation near the heated wall. This contrasting behavior, not explicitly reported in earlier works, provides a new comparative perspective on the stability of bioconvective nanofluids. It is further observed that, in both cases, an increase in the Péclet number consistently promotes destabilization.

A local weak form of the generalized finite difference method (GFDM) with control volume in heat conduction problems

The general finite difference method with the Krylov deferred correction technique for dynamic 2D and 3D piezoelectric analysis

Modeling bioconvective mixed convection of non-newtonian nanofluids using finite difference approach: A Jeffrey fluid model

On the numerical approximation of a phase-field volume reconstruction model: Linear and energy-stable leap-frog finite difference scheme

This study numerically derived the higher order convergence for a class of singularly perturbed Fredholm integro differential equations with reaction diffusion and convection diffusion type problems. A non-standard finite difference approach is used to approximate the derivatives. The trapezoidal rule determines the integral term. The suggested numerical technique achieves a uniform convergence rate independently of the perturbation parameter. Implementing the Richardson extrapolation technique achieves a fourth order convergence rate for reaction diffusion type problems and a second order convergence rate for convection diffusion type problems. Specific numerical examples are provided to corroborate in practice the effectiveness of the theoretical findings.

A structure-preserving reduced-order finite difference approach for a class of semilinear stochastic partial differential equations driven by white noise

Predicting thermal transport of blood-based penta-hybrid nanofluid in Fin geometries using deep neural networks and finite difference approach

ABSTRACT This study investigates the flow of a non‐Newtonian viscoelastic fluid over a stretched sheet within a porous medium, accounting for magnetic force, thermophoresis, thermal radiation, heat and mass transfer, Brownian motion, and chemical reactions. The unsteady governing equations are derived using the boundary layer approximation and are subsequently transformed into a dimensionless form. The explicit finite difference scheme is consequently executed to numerically solve the equations with the help of the programming code FORTRAN 6.6.a. Additionally, specific parameters for stability and convergence are established to ensure that the solution converges. The physical behavior of significant parameters has been widely described on a variety of flow fields using both the tabular and graphical representations. With the use of sophisticated streamline and isotherm visualization, the variation in the depth of the momentum and thermal border layer is identified. As the fluid's enhanced elasticity enables it to store and release energy more effectively, a higher elastic value for the fluid improves the velocity distribution and stimulates quicker flow. It has been found that raising elastic parameter builds up the fluid's resistance to deformation and promotes skin friction at the surface. The findings give new insights on the changing behavior of the thermophoresis ( Nt ) parameter, as viscoelastic fluids exhibit complex flow features such as elasticity or shear‐thinning qualities. These characteristics inhibit heat transmission, resulting in a comparably lower Nusselt number than Newtonian fluids. At last, a favorable consensus is reached as part of guarantee the validity of the current investigation.

High-order bound-preserving finite difference methods for incompressible two-phase flow in porous media

A parallel iterative algorithm for solving the two-dimensional Helmholtz equation with discontinuous coefficients using a high-order compact finite difference scheme

Singularly perturbed reaction-diffusion differential equations with large negative and advanced shifts are significant challenges in mathematical modeling across various scientific disciplines, particularly in control theory and neuroscience. This study addresses the numerical treatment of these complex equations where the presence of a small perturbation parameter combined with large shift terms creates boundary and interior layers that standard numerical approaches fail to capture accurately. We develop a robust nonstandard finite difference scheme (NSFDS) specifically designed to handle the difficulties arising from the singular perturbation parameter and mixed large shift arguments. Our approach incorporates a carefully constructed denominator function that maintains the essential qualitative properties of a continuous model. Through rigorous convergence analysis, we established that the proposed scheme is uniformly convergent with respect to the perturbation parameter, achieving second-order accuracy. Extensive numerical experiments are performed to validate the theoretical predictions, demonstrating the scheme’s superior convergence rate, efficiency, and ability to resolve layer phenomena without spurious oscillations, even for extremely small perturbation parameters. The results demonstrate that our method effectively resolves layer phenomena and maintains stability and accuracy even for extremely small values of the perturbation parameter, outperforming existing standard approaches as well as other existing previous results. This study provides a reliable computational framework for this challenging class of equations, significantly advancing state-of-the-art numerical methods for singularly perturbed problems with shift terms.

Abstract In this study, an Accelerated Discrete Ordinate Method (ADOM) is proposed to enhance the computational efficiency of multi‐layer radiative transfer (RT) simulations while maintaining a high accuracy. ADOM applies the Discrete Ordinate Method (DOM) only in scattering layers, while the radiances for the adjacent clear‐sky layers are merged and computed by non‐scattering RT theory. The merging process significantly reduces the number of layers involved, enhancing computational efficiency while the vertical structure of both the Planck function and the Rayleigh scattering single‐scattering albedo are fully accounted. This hybrid RT approach enables ADOM to be applicable across the visible to microwave spectrum. For satellite radiance assimilation, tangent linear and adjoint modules of ADOM are also developed to compute the Jacobians of all relevant parameters. Although ADOM merges adjacent clear‐sky atmospheric layers during RT calculations, the Jacobians of properties in each merged clear‐sky layer can still be accurately computed by constructing an adjoint module of the merging process. The accuracy of both the forward and adjoint modules of ADOM is evaluated against 128‐stream DOM and the finite difference results based on DOM. Notably, the computational efficiency gain of ADOM is influenced by the ratio of clear‐sky layers to cloud layers. As the number of cloud layers decreases, the efficiency of ADOM increases. In fully cloudy conditions, the runtime of ADOM converges to that of DOM.

Purpose The shape factor of nanoparticles is a parameter of interest in the variation of the thermophysical properties of nanofluids, and it affects their fluid flow and temperature distribution. Hence, this study aims to focus on analysing the influence of the shape factor on the convective heat and mass transfer over a nonlinear stretching sheet under the influence of magnetohydronamics. Design/methodology/approach By using similarity transformations, the governing system of partial differential equations was simplified to a nonlinear ordinary differential equation system, which was solved numerically using an explicit finite difference method (Keller box method). The behaviour of the fluid velocity and thermal profile at the boundary, as a result of slip conditions, is studied through a comprehensive parameter exploration. Findings The behaviour of the fluid velocity and thermal profile at the boundary, as a result of slip conditions, is studied by a very extensive parameter exploration. The major findings reveal that increasing values of magnetic parameters promote Lorentz’s force, which slows down the velocity profile. Increasing the stretching rate parameter causes deformation reducing both the velocity and temperature profile. It is found that the temperature rise for elevated values of a thermophoretic parameter is greater for brick-shaped nanoparticles. It is also observed that the heat transfer (Nusselt number) for augmented values of Eckert number is lower in brick-shaped nanoparticles compared to platelet-shaped nanoparticles. Research limitations/implications The main limitation of this work has been the two-dimensional nature of the modelling, the analysis of a certain range of parameters that may not suit all the reader interests and the assumption of specific functions for the stretched sheet velocity and temperature, as well as the magnetic field. Also, regarding the nanofluid characteristics, it has been considered as single-phase, with Fe3O4 particles only and Newtonian. Originality/value This manuscript analyzes mathematically important aspects of the behaviour of a nanofluid with nanoparticles in magnetohydrodynamics of a non-linearly stretched sheet. This work is a detailed parametric exploration (magnetic parameter, stretching parameter, slip parameters, Brownian motion parameter, thermophoretic parameter, Ecker number, Lewis number and solid volume fraction) of the behaviour of two different nanoparticle shapes (brick and platelet), which sheds light on relevant aspects such as, skin friction, heat transfer and mass transfer. These are valuable results for the scientific community for either perform their numerical analysis upon these results and methodology, or perform experimental prototyping according to the behaviour described in this manuscript.

Abstract This paper explores using fractional-order methods to model chaotic dynamics and spatiotemporal Turing-type patterns in complex systems. By employing fractional calculus, which captures noninteger order dynamics, the study provides insights into the mechanisms driving pattern formation and chaos in systems such as chemical reactions, ecological models, and biological processes. A mathematical framework is developed to investigate the emergence, stability, and influence of these patterns. The integer-order spatial derivative is replaced with two-sided Riemann–Liouville fractional operators, and a fourth-order finite difference scheme is introduced for approximating fractional diffusion-like problems in one and two dimensions. Stability and convergence analyses of the proposed methods are conducted. To further explore pattern formation, the study extends its analysis to a multicomponent system, solving for spatiotemporal Turing-like patterns in both one and two dimensions. The results demonstrate the generation of several novel and existing patterns. Understanding chaotic behavior and pattern formation is essential for various scientific and engineering applications. The insights gained from this study contribute to a deeper comprehension of complex systems and may aid in controlling or utilizing these patterns across disciplines such as physics, chemistry, biology, and ecology.