Search Results (55)

The large time step (LTS) explicit scheme relaxes the Courant–Friedrichs–Lewy (CFL) restriction by propagating waves across multiple cells per time step and has shown clear efficiency gains for the homogeneous shallow water equations. Earlier Godunov-type LTS constructionscombined an exact Riemann solver with a multi-wave approximation of the rarefaction fan and a random choice method to suppress spurious oscillations. The present work proposes a simpler yet equally accurate variant. The star state is obtained from Toro’s closed-form two-rarefaction approximation (TRA) and then refined by a short Newton iteration; this combination is exact for two-rarefaction configurations, removes the systematic bias of TRA in two-shock configurations, and converges in two to three iterations from the TRA initial guess in all other cases. The rarefaction fan is split into sub-waves with linearly interpolated speeds, and each sub-wave is dispatched to either a left-moving or a right-moving propagator according to the sign of its mean speed. Fractional contributions are resolved based on a deterministic threshold with θ = 0.5. The scheme is validated based on seven one-dimensional flat-bottom Riemann problems covering dry-bed, wet-bed, two-rarefaction, and two-shock configurations, each at CFL numbers 0.9, 3, 8, and 15. The L1 error in water depth remains below 3% for all cases and is as low as 0.2% in the two-shock configuration at CFL = 15. The Xu 2014 dam break is reproduced with an error of 0.4% at CFL = 15 using only three time steps. Full article

This paper adapts a semi-analytical framework the Laplace-Residual Power Series Method (LRPSM) to solve the time-fractional Allen–Cahn equation under the Caputo derivative. While the classical Allen–Cahn model successfully describes phase separation, its fractional counterpart is essential for capturing sub-diffusive memory effects in complex heterogeneous materials. However, the interplay between the non-local fractional temporal operator and the cubic nonlinearity of the bistable double-well potential creates significant computational bottlenecks for conventional time-domain series solvers. The proposed approach projects the governing fractional partial differential equation into the Laplace domain, systematically replacing the computation of iterative fractional derivatives with the algebraic evaluation of asymptotic limits at infinity. Furthermore, the nonlinear cubic interactions are managed through Laplace-space convolution theorems. The structural convergence of this approach is evaluated against multi-scenario one-dimensional phase transitions. Graphical analyses, featuring 2D profile trajectories and 3D spatiotemporal surface mappings, visually illustrate the retarded interfacial propagation driven by fractional memory. Ultimately, this study presents the LRPSM as an applicable, continuous mathematical tool for approximating anomalous diffusion in the specific phase-field dynamics evaluated herein. Full article

Understanding how sentiment propagates in signed networks is crucial for uncovering mechanisms behind opinion polarization, trust formation, and information cocoons in digital communities. This paper investigates the generation of signed edges, representing positive or negative sentiments, in online social networks. We propose an analytical framework that models the dynamic growth of sentiment as a diffusion process. By introducing a walker on an infinite one-dimensional lattice, we derive a time-fractional diffusion equation that captures subdiffusive, normal diffusive, and superdiffusive behaviors. The model is empirically validated using two large-scale temporal signed networks: RedditHyperlinks and Bitcoin OTC. Our findings reveal that sentiment diffusion exhibits distinct regimes depending on the stage of network evolution, providing a foundation for further theoretical analysis and applications in signed social networks. Full article

This work presents a new and efficient numerical framework for solving three-dimensional time-fractional diffusion and mobile–immobile equations in the Caputo sense. The method is formulated using four-variable Jacobi polynomials, constructed systematically via the Kronecker product of one-dimensional Jacobi bases to accurately represent the multidimensional nature of the governing equations. Within a pseudo-operational collocation formulation, these polynomials enable a highly accurate and computationally efficient approximation of the fractional operators in both temporal and spatial directions. From the theoretical standpoint, the existence and uniqueness of the approximate solution are rigorously established through Schauder’s fixed-point theorem. Furthermore, the Ulam–Hyers stability of the numerical solution is verified, demonstrating the robustness of the method with respect to perturbations in the input data. To reinforce the reliability of the approach, an explicit error bound for the residual function is derived in a Jacobi-weighted Sobolev space, offering a firm analytical basis for assessing convergence. Numerical experiments confirm that the proposed approach achieves superior accuracy and efficiency, highlighting its potential as a powerful tool for high-dimensional fractional partial differential equations. Full article

This work develops and analyzes a novel fractional-order chemostat system (FOCS) with a Caputo fractional derivative (CFD) featuring a sliding memory window and periodic boundary conditions (PBCs), designed to model microbial pollutant degradation in sustainable water treatment. By incorporating the Caputo fractional derivative with sliding memory (CFDS), the model captures time-dependent behaviors and memory effects in biological systems more realistically than classical integer-order formulations. We reduce the two-dimensional fractional differential equations (FDEs) governing substrate and biomass concentrations to a one-dimensional FDE by utilizing the PBCs. The existence and uniqueness of non-trivial, periodic solutions are established using the Carathéodory framework and fixed-point theorems, ensuring the system’s well-posedness. We prove the positivity and boundedness of solutions, demonstrating that substrate concentrations remain within physically meaningful bounds and biomass concentrations stay strictly positive, with solution trajectories confined to a biologically feasible invariant set. Additionally, we analyze non-trivial equilibria under constant dilution rates and derive their stability properties. The rigorous mathematical results confirm the viability of FOCS models for representing memory-driven, periodic bioprocesses, offering a foundation for advanced water treatment strategies that align with Sustainable Development Goal 6 (Clean Water and Sanitation). This work establishes a comprehensive mathematical framework that bridges fractional calculus with sustainable water treatment applications, providing both theoretical foundations and practical implications for optimizing bioreactor performance in environmental biotechnology. Full article

Accurate modeling of multiphase flow in porous media remains challenging due to the nonlinear transport and sharp displacement fronts described by the Buckley–Leverett (B-L) equation. Although Fourier Neural Operators (FNOs) have recently emerged as powerful surrogates for parametric partial differential equations, they exhibit limited robustness when extrapolating beyond the training regime, particularly for shock-dominated fractional flows. This study aims to enhance the extrapolative performance of FNOs for one-dimensional B-L displacement. Analytical solutions were generated using Welge’s graphical method, and datasets were constructed across a range of mobility ratios. A baseline FNO was trained to predict water saturation profiles and evaluated under both interpolation and extrapolation conditions. While the standard FNO accurately reconstructs saturation profiles within the training window, it misestimates shock positions and saturation jumps when extended to longer times or higher mobility ratios. To address these limitations, we develop Physics-Informed FNOs (PI-FNOs), which embed PDE residuals and boundary constraints, and Transfer-Learned FNOs (TL-FNOs), which adapt pretrained operators to new regimes using limited data. Comparative analyses show that both approaches markedly improve extrapolation accuracy, with PI-FNOs achieving the most consistent and physically reliable performance. These findings demonstrate the potential of combining physics constraints and knowledge transfer for robust operator learning in multiphase flow systems. Full article

The present research offers reliable analytical solutions for time-fractional linear and nonlinear dispersive Korteweg–de Vries (dKdV)-type equations by employing the Natural Generalized Laplace Transform Decomposition Method (NGLTDM). The nonlinear differential dispersive Korteweg–de Vries (dKdV) equation involves a nonlinear derivative term that depends on $\varphi$ and its partial derivative with respect to x. We employ Adomian polynomials to deal with this nonlinear part, and we utilize the Caputo derivative to illustrate the fractional part of the equation. The work provides exact theorems regarding the stability, convergence, and accuracy of the generated solutions. Illustrative examples demonstrate the effectiveness and precision of the method by delivering solutions for quickly converging series with easily calculable coefficients. We use Maple 2021 software to show graphical comparisons between the approximate and exact solutions to show how rapidly the method converges. Full article

A class of initial boundary value problems is here considered for a one-dimensional diffusion equation with a general time-fractional derivative with the Sonin kernel. One of the boundary conditions is in a general non-classical form, which includes no-nlocal cases of integral or multi-point boundary conditions. The problem is studied here by applying spectral projection operators to convert it to a system of relaxation equations in generalized eigenspaces. The uniqueness of the solution is established based on the uniqueness property of the spectral expansion. An algorithm is given for constructing the solution in the form of spectral expansion in terms of the generalized eigenfunctions. Estimates for the time-dependent components in this expansion are established and applied to prove the existence of a solution in the classical sense. The obtained results are applied to a particular case in which the specified boundary conditions lead to two sequences of eigenvalues, one of which consists of triple eigenvalues. Full article

The time-dependent creep behavior of coal is essential for assessing long-term structural stability and operational safety in deep coal mining. Therefore, this work develops a full-stage creep constitutive model. By integrating fractional calculus theory with statistical damage mechanics, a nonlinear fractional-order (FO) damage creep model is constructed through serial connection of elastic, viscous, viscoelastic, and viscoelastic–plastic components. Based on this model, both one-dimensional and three-dimensional (3D) fractional creep damage constitutive equations are acquired. Model parameters are identified using experimental data from deep coal samples in the mining area. The result curves of the improved model coincide with experimental data points, accurately describing the deceleration creep stage (DCS), steady-state creep stage (SCS), and accelerated creep stage (ACS). Furthermore, a sensitivity analysis elucidates the impact of model parameters on coal creep behavior, thereby confirming the model’s robustness and applicability. Consequently, the proposed model offers a solid theoretical basis for evaluating the sustained stability of deep coal mining and has great application potential in deep underground engineering. Full article

This work presents and compares different methodologies for the joint optimisation of the fractional derivative order and the parameters of the right-hand-side neural network in Neural Fractional Differential Equation models. The proposed strategies aim to tackle the training difficulties typically encountered when learning the fractional order $\alpha$ together with the network weights. One approach is based on regulating the gradient magnitude of the loss function with respect to $\alpha$, encouraging more stable and effective updates. Another strategy introduces an online pre-training scheme, where the network parameters are initially optimised over progressively longer time intervals, while $\alpha$ is updated more conservatively using the full time trajectory. The study focuses only on a foundational setting with one-dimensional problems, and numerical experiments demonstrate that the proposed techniques improve both training stability and accuracy. Nonetheless, the issue of non-uniqueness in the optimal derivative order remains, particularly in less well-posed scenarios, suggesting the need for further research in data-driven modelling of fractional-order systems. Full article

In this article, we introduce and investigate two classes of fuzzy fractional two-dimensional continuous-time (FFTDCT) linear systems to deal with uncertainty and fuzziness in system parameters. First, we analyze FFTDCT linear systems based on the Roesser model, incorporating fuzzy parameters into the state-space equations. The potential solution of the fuzzy fractional system is obtained using a two-dimensional granular Laplace transform approach. Second, we examine FFTDCT linear systems described by the second Fornasini–Marchesini (FM) model, where the state-space equations involve two-dimensional and one-dimensional partial fractional-order granular Caputo derivatives. We determine the fuzzy solution for this model by applying the two-dimensional granular Laplace transform. To enhance the validity of the proposed approaches, real-world applications, including signal processing systems and wireless sensor network data fusion, are solved to support the theoretical framework and demonstrate the impact of uncertainty on the system’s behavior. Full article

We consider the melting of a one-dimensional domain ($x\ge 0$), initially at the melting temperature $u=0$, by fixing the boundary temperature to a value $u(0,t)=U_0>0$—the so called Stefan melting problem. The governing transient heat-conduction equation involves a time derivative and the spatial derivative of the temperature gradient. In the general case the order of the time derivative and the gradient can take values in the range $(0,1]$. In these problems it is known that the advance of the melt front $s\left(t\right)$ can be uniquely determined by a specified prefactor multiplying a power of time related to the order of the fractional derivatives in the governing equation. For given fractional orders the value of the prefactor is the unique solution to a transcendental equation formed in terms of special functions. Here, our main purpose is to provide efficient numerical schemes with low computational complexity to compute these prefactors. The values of the prefactors are obtained through a dimensionalization that allows the recovery of the solution for the quasi-stationary case when the Stefan number approaches zero. The mathematical analysis of this convergence is given and provides consistency to the numerical results obtained. Full article

This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. $L^2$-norm stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstrate optimal convergence order in both spatial and temporal directions. The numerical results confirm that the proposed method achieves an accuracy of the order $O\left(\tau +h^{k+1}\right)$, where $\tau$ and h represent the time step and spatial mesh size, respectively. This work extends previous studies on one-dimensional problems to higher spatial dimensions, providing a robust framework for handling evolution equations with a weakly singular kernel. Full article

Time-fractional evolution equations for probability distributions provide a means of representing an important class of stochastic processes. Their solutions have features that are important in modeling anomalous diffusion and a variety of other real-world applications, like the search patterns of predators or queuing problems. However, these equations are usually not included in current physics education, as the underlying mathematics might be considered too advanced. In the following, we present a novel approach to understanding time-fractional diffusion equations and their solutions for practical applications. This approach shifts the focus to the physical rather than the in-depth mathematical properties typically studied for this topic. We introduce the fractional differential operator simply as an identity operation on generalizations of exponential functions. This shifts the emphasis to the actual functions of fractional derivatives instead of what they are. This concept is applied to a discrete one-dimensional time-fractional diffusion equation on a finite interval modeling anomalous subdiffusion. Examples and tasks are provided for readers to allow interactive learning. Full article

With the increasing construction of engineering structures on soft soils, accurately assessing their consolidation behavior has become crucial. To address this, Terzaghi’s one-dimensional consolidation model was revisited. The elastic behavior of soil skeleton was modified by incorporating viscous effects using the fractional derivative Merchant model (FDMM), while the linear Darcy’s law governing flux–pressure relations was extended by introducing time memory formalism through the fractional derivative Darcy model (FDDM). The governing equation is derived by incorporating the resulting constitutive behavior of both the soil skeleton and water flow into the Terzaghi’s formulation of the poroelasticity problem. The proposed rheological consolidation model is solved by a forward time-centered space scheme (FTCS). After verifying the numerical procedure with published data, the influence of parameters on both the average degree of settlement and the pressure was comprehensively studied. Full article