In this study, we present an efficient and novel unconditionally stable Monte Carlo simulation (MCS) for solving the multi-dimensional Allen–Cahn (AC) equation, which can model the motion by mean curvature flow of a hypersurface. We use an operator splitting method, where the diffusion and nonlinear terms are solved separately. The diffusion term is calculated using MCS for the stochastic differential equation, while the nonlinear term is locally computed for each particle in a virtual grid. Several numerical experiments are presented to demonstrate the performance of the proposed algorithm. The computational results confirm that the proposed algorithm can solve the AC equation more efficiently as the dimension of space increases.

In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.

A fingerprint is the unique, complex pattern of ridges and valleys on the surface of an individual's fingertip. Fingerprinting is one of the most popular and widely used biometric authentication methods for personal identification because of its reliability, acceptability, high level of security, and low cost. When using fingerprints as a biometric, restoring poor-quality or damaged fingerprints is an essential process for accurate verification. In this study, we present a semi-automatic fingerprint image restoration method using a partial differential equation to repair damaged fingerprint images. The proposed algorithm is based on the Cahn-Hilliard (CH) equation with a source term, which was developed for simulating pattern formation during the phase separation of diblock copolymers in chemical engineering applications. In previous work, in order to find an optimal model and numerical parameter values in the governing equation, we had to make several trial and error preliminary attempts. To overcome these problems, the proposed novel algorithm minimizes user input and automatically computes the necessary model and numerical parameter values of the governing equation. Computational simulations on various damaged fingerprint samples are presented to demonstrate the superior performance of the proposed method.

We propose a normalized Caputo–Fabrizio (CF) fractional diffusion equation. The CF fractional derivative replaces the power-law kernel in the Caputo derivative with an exponential kernel, which avoids singularities. Compared to the Caputo derivative, the CF derivative is better suited for systems where memory effects decay smoothly rather than following a power law. However, the kernel is not normalized in the sense that its weighting function does not integrate to unity. To resolve this limitation, we develop a modified formulation that ensures proper normalization. To investigate the fractional order's effect on evolution dynamics, we perform computational tests that highlight memory effects.

In this paper, we present accurate and well-designed benchmark problems for evaluating the effectiveness and precision of physics-informed neural networks (PINNs). The presented problems were generated using the Allen–Cahn (AC) equation, which models the mechanism of phase separation in binary alloy systems and simulates the temporal evolution of interfaces. The AC equation possesses the property of motion by mean curvature, which means that, in the sharp interface limit, the evolution of the interface described by the equation is governed by its mean curvature. Specifically, the velocity of the interface is proportional to its mean curvature, which implies the tendency of the interface to minimize its surface area. This property makes the AC equation a powerful mathematical model for capturing the dynamics of interface motion and phase separation processes in various physical and biological systems. The benchmark source codes for the 1D, 2D, and 3D AC equations are provided for interested researchers.

We introduced a fully explicit finite difference method (FDM) designed for numerically solving the conservative Allen–Cahn equation (CAC) on a cubic surface. In this context, the cubic surface refers to the combined areas of the six square faces that enclose the volume of a cube. The proposed numerical solution approach is structured into two sequential steps. First, the Allen–Cahn (AC) equation was solved by applying the fully explicit FDM, which is computationally efficient. Following this, the conservation term is resolved using the updated solution from the AC equation to ensure consistency with the underlying conservation principles. To evaluate the effectiveness of the proposed scheme, computational tests are performed to verify that the resulting numerical solution of the CAC equation successfully conserves the discrete mass. Additionally, the solution is examined for its ability to exhibit the property of constrained motion by mass conserving mean curvature, a critical characteristic of the CAC equation. These two properties are fundamental to the integrity and accuracy of the CAC equation.

In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments.

The Allen-Cahn (AC) model is a mathematical equation that represents the phase separation process. The AC equation has numerous applications in various disciplines, such as image processing, physics, and biology. It models phase transitions, such as solidification and grain growth in materials, pattern formation in chemical reactions, and domain coarsening in biological systems like lipid membranes. Numerical methods are crucial for solving the AC equation due to its complexity and nonlinear nature. Analytical solutions are often extremely difficult to obtain. Therefore, the development of efficient numerical techniques is indispensable for approximating solutions and studying phase transitions, material behavior, and pattern formation accurately. We investigate the stability of an explicit finite difference method (FDM) used to numerically solve the two-dimensional (2D) AC model with a high-order polynomial potential, which was recently proposed to preserve a more intricate structure of interfaces. To demonstrate the precision and optimal estimate of our stability constraints, we conduct various computational tests using the derived time step formulas that ensure the maximum principle.

In this paper, we investigated the influence of climatic factors, particularly low temperature and humidity, on the grounding resistance of transmission towers in severely cold regions. Drawing on a full year of field monitoring data, we first identified a clear negative linear correlation between temperature and grounding resistance. Notably, the inclusion of precipitation as an additional variable led to a significant increase in the model's explanatory power and indicated that the resistance behavior was influenced by multiple environmental factors. To further explore the nonlinear and dynamic aspects of this relationship, we used multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DCCA). The computational results showed that the resistance time series exhibits strong multifractal characteristics and suggests high variability across temporal scales. Moreover, the combined influence of temperature and precipitation demonstrated a markedly stronger cross-correlation with resistance than temperature alone. These findings emphasize the complex and multiscale nature of grounding behavior in harsh climates and underscore the importance of moving beyond simplified linear models. Our research offers both methodological insights and practical implications for the design, maintenance, and risk assessment of power transmission infrastructure operating under extreme weather conditions, particularly in high-latitude or alpine environments.

This paper presents a numerically stable, time-accurate algorithm for simulating the gradient flow associated with the Modica–Mortola functional with a uniformly spaced multi-well potential. The scheme uses operator splitting; the nonlinear component is updated analytically, while the linear part is advanced by a Fourier spectral discretization. The method is unconditionally stable, preserves pointwise boundedness independently of the time step size, and attains spectral accuracy in space and first-order accuracy in time. We provide a theoretical analysis establishing unconditional stability and boundedness, and present comprehensive numerical experiments that demonstrate the accuracy and robustness of the proposed approach.