Search Results (19)

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular and nonlocal Mittag–Leffler kernel (MLk) and provides a more accurate depiction of memory and heredity effects, to examine the dynamic behavior of the models. Using nonlinear analysis, the uniqueness of the suggested models is investigated, and distinct wave profiles are created for various fractional orders. The accuracy and effectiveness of the suggested approach are validated by a number of example cases, which also support the approximate solutions of the nonlinear FCRWPEs. This work provides significant insights into the modeling of anomalous diffusion and complex dynamic processes in fields such as phase transitions, biological transport, and population dynamics. The inclusion of the ABFD enhances the model’s ability to capture nonlocal effects and long-range temporal correlations, making it a powerful tool for simulating real-world systems where classical derivatives may be inadequate. Full article

This study, grounded in traveling wave theory, develops a cross-scale reaction-diffusion model to describe nutrient-dependent bacterial growth on agar surfaces and applies it to in silico investigations of microbial population dynamics. The approach is based on the coupling of a modified Allen–Cahn equation with the formulation of quorum sensing signal dynamics, incorporating a nutrient-dependent regulatory threshold and stochastic diffusion. A closed-loop model of bacterial growth regulated by quorum sensing is developed through theoretical analysis, numerical simulations, and computational experiments.The model is implemented using Yanenko’s computational scheme, which incorporates corrective refinement via Heun’s method to account for nonlinear components. Numerical simulations are carried out in MATLAB, allowing for accurate computation of spatio-temporal patterns and facilitating the identification of key mechanisms governing the collective behavior of bacterial communities. Full article

This paper is focused on developing a Galerkin finite element framework for the fractional Allen–Cahn equation with regularized logarithmic potential over the ${\mathbb{R}}^d$ ($d=1,2,3$) domain, where the regularization of the singular potential extends beyond the classical double-well formulation. A fully discrete finite element scheme is developed using a k-th-order finite element space for spatial approximation and a backward Euler scheme for the temporal discretization of a regularized system. The existence and uniqueness of numerical solutions are rigorously established by applying Brouwer’s fixed-point theorem. Moreover, the proposed numerical framework is shown to preserve the discrete energy dissipation law analytically, while a priori error estimates are derived. Finally, numerical experiments are conducted to verify the theoretical results and the inherent physical property, such as phase separation phenomenon and coarsening processes. The results show that the fractional Allen–Cahn model provides enhanced capability in capturing phase transition characteristics compared to its classical equation. Full article

We propose an unconditionally stable computational algorithm that preserves the maximum principle for the three-dimensional (3D) high-order Allen–Cahn (AC) equation. The presented algorithm applies an operator-splitting technique that decomposes the original equation into nonlinear and linear diffusion equations. To guarantee the unconditional stability of the numerical solution, we solve the nonlinear equation using the frozen coefficient technique, which simplifies computations by approximating variable coefficients by constants within small regions. For the linear equation, we use an implicit finite difference scheme under the operator-splitting method. To validate the efficiency of the proposed algorithm, we conducted several computational tests. The numerical results confirm that the scheme achieves unconditional stability even for large time step sizes and high-order polynomial potential. In addition, we analyze motion by mean curvature in three-dimensional space and show that the numerical solutions closely match the analytical solutions. Finally, the robustness of the method is evaluated under noisy data conditions, and its ability to accurately classify complex data structures is demonstrated. These results confirm the efficiency and reliability of the proposed computational algorithm for simulating phase-field models with a high-order polynomial potential. Full article

In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such as sixth, eighth, or any even order greater than four, can model more complex dynamics in phase transition problems. Our study aims to explore how the order of these polynomial potentials affects the speed and behavior of front propagation in the AC framework. By systematically varying the polynomial order, we observe significant changes in front dynamics. Higher-order polynomials tend to influence the sharpness and speed of moving fronts, leading to modifications in the overall pattern formation process. These results have implications for understanding the role of polynomial potentials in phase transition phenomena and offer insights into the broader application of AC equations for modeling complex systems. This work demonstrates the importance of considering higher-order polynomial potentials when analyzing front propagation and phase transitions, as the choice of polynomial order can dramatically alter system behavior. Full article

Different researchers have analyzed effective computational methods that maintain the precision of Allen–Cahn (AC) equations and their constant security. This article presents a method known as the reduced-order model technique by utilizing kernel principle component analysis (KPCA), a nonlinear variation of traditional principal component analysis (PCA). KPCA is utilized on the data matrix created using discrete solution vectors of the AC equation. In order to achieve discrete solutions, small variations are applied for dividing up extraterrestrial elements, while Kahan’s method is used for temporal calculations. Handling the process of backmapping from small-scale space involves utilizing a non-iterative formula rooted in the concept of the multidimensional scaling (MDS) method. Using KPCA, we show that simplified sorting methods preserve the dissipation of the energy structure. The effectiveness of simplified solutions from linear PCA and KPCA, the retention of invariants, and computational speeds are shown through one-, two-, and three-dimensional AC equations. Full article

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes’ versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics. Full article

Within recent years, considerable progress has been made regarding high-performance solvers for partial differential equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely remains the status quo for scientists and engineers focusing on applying simulation tools to specific problems in practice. We attribute this growing technical gap to the increasing complexity and knowledge required to pick and assemble state-of-the-art methods. Thus, with this work, we initiate an effort to build a common taxonomy for the most popular grid-based approximation schemes to draw comparisons regarding accuracy and computational efficiency. We then build upon this foundation and introduce a method to systematically guide an application expert through classifying a given PDE problem setting and identifying a suitable numerical scheme. Great care is taken to ensure that making a choice this way is unambiguous, i.e., the goal is to obtain a clear and reproducible recommendation. Our method not only helps to identify and assemble suitable schemes but enables the unique combination of multiple methods on a per-field basis. We demonstrate this process and its effectiveness using different model problems, each comparing the resulting numerical scheme from our method with the next best choice. For both the Allen–Cahn and advection equations, we show that substantial computational gains can be attained for the recommended numerical methods regarding accuracy and efficiency. Lastly, we outline how one can systematically analyze and classify a coupled multiphysics problem of considerable complexity with six different unknown quantities, yielding an efficient, mixed discretization that in configuration compares well to high-performance implementations from the literature. Full article

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs. Full article

Due to the significant value that hotel reviews hold for both consumers and businesses, the development of an accurate sentiment classification method is crucial. By effectively distinguishing the authenticity of reviews, consumers can make informed decisions, and businesses can gain insights into customer feedback to improve their services and enhance overall competitiveness. In this paper, we propose a partial differential equation model based on phase-field for sentiment analysis in the field of hotel comment texts. The comment texts are converted into word vectors using the Word2Vec tool, and then we utilize the multifractal detrended fluctuation analysis (MF-DFA) model to extract the generalized Hurst exponent of the word vector time series to achieve dimensionality reduction of the word vector data. The dimensionality reduced data are represented in a two-dimensional computational domain, and the modified Allen–Cahn (AC) function is used to evolve the phase values of the data to obtain a stable nonlinear boundary, thereby achieving automatic classification of hotel comment texts. The experimental results show that the proposed method can effectively classify positive and negative samples and achieve excellent results in classification indicators. We compared our proposed classifier with traditional machine learning models and the results indicate that our method possesses a better performance. Full article

In this article, the new iterative transform method is applied to evaluate the time-fractional Cahn–Allen model solution. In this technique, Elzaki transformation is a mixture of the new iteration technique. Two problems are studied to demonstrate and confirm the accuracy of the proposed technique. The current technique’s mathematical analysis showed that the method is simple to understand and reliable. These solutions indicate that the proposed technique is advantageous and simple to apply in science and engineering problems. Full article

In this paper, we consider an important problem for modeling complex coupled phenomena in porous media at multiple scales. In particular, we consider flow and transport in the void space between the pores when the pore space is altered by new solid obstructions formed by microbial growth or reactive transport, and we are mostly interested in pore-coating and pore-filling type obstructions, observed in applications to biofilm in porous media and hydrate crystal formation, respectively. We consider the impact of these obstructions on the macroscopic properties of the porous medium, such as porosity, permeability and tortuosity, for which we build an experimental probability distribution with reduced models, which involves three steps: (1) generation of independent realizations of obstructions, followed by, (2) flow and transport simulations at pore-scale, and (3) upscaling. For the first step, we consider three approaches: (1A) direct numerical simulations (DNS) of the PDE model of the actual physical process called BN which forms the obstructions, and two non-DNS methods, which we call (1B) CLPS and (1C) LP. LP is a lattice Ising-type model, and CLPS is a constrained version of an Allen–Cahn model for phase separation with a localization term. Both LP and CLPS are model approximations of BN, and they seek local minima of some nonconvex energy functional, which provide plausible realizations of the obstructed geometry and are tuned heuristically to deliver either pore-coating or pore-filling obstructions. Our methods work with rock-void geometries obtained by imaging, but bypass the need for imaging in real-time, are fairly inexpensive, and can be tailored to other applications. The reduced models LP and CLPS are less computationally expensive than DNS, and can be tuned to the desired fidelity of the probability distributions of upscaled quantities. Full article

The lattice Boltzmann method, now widely used for a variety of applications, has also been extended to model multiphase flows through different formulations. While already applied to many different configurations in low Weber and Reynolds number regimes, applications to higher Weber/Reynolds numbers or larger density/viscosity ratios are still the topic of active research. In this study, through a combination of a decoupled phase-field formulation—the conservative Allen–Cahn equation—and a cumulant-based collision operator for a low-Mach pressure-based flow solver, we present an algorithm that can be used for higher Reynolds/Weber numbers. The algorithm was validated through a variety of test cases, starting with the Rayleigh–Taylor instability in both 2D and 3D, followed by the impact of a droplet on a liquid sheet. In all simulations, the solver correctly captured the flow dynamics andmatched reference results very well. As the final test case, the solver was used to model droplet splashing on a thin liquid sheet in 3D with a density ratio of 1000 and kinematic viscosity ratio of 15, matching the water/air system at We = 8000 and Re = 1000. Results showed that the solver correctly captured the fingering instabilities at the crown rim and their subsequent breakup, in agreement with experimental and numerical observations reported in the literature. Full article

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A. Full article

In this paper, we propose a computationally fast and accurate explicit hybrid method for image segmentation. By using a gradient flow, the governing equation is derived from a phase-field model to minimize the Chan–Vese functional for image segmentation. The resulting governing equation is the Allen–Cahn equation with a nonlinear fidelity term. We numerically solve the equation by employing an operator splitting method. We use two closed-form solutions and one explicit Euler’s method, which has a mild time step constraint. However, the proposed scheme has the merits of simplicity and versatility for arbitrary computational domains. We present computational experiments demonstrating the efficiency of the proposed method on real and synthetic images. Full article