We propose a practically stable explicit finite difference method (FDM) for the ternary Cahn–Hilliard (CH) system, which involves challenging fourth-order nonlinear terms. Therefore, if we use a fully explicit numerical method such as the explicit Euler scheme, then the excessively restrictive time-step limitation makes it impractical to use. The proposed algorithm is based on the alternating direction explicit (ADE) scheme, which ensures both simplicity of implementation and stability of the computational solutions for the ternary CH system. To show the high performance of the proposed algorithm, we present multiple numerical experiments. The numerical tests demonstrate that the proposed method overcomes the stringent time-step limitations inherent in the explicit Euler method.

We have presented a maximum principle preserving the unconditionally stable scheme for the Allen–Cahn (AC) equation with a high-order polynomial potential. The proposed method ensures the preservation of the maximum principle, a critical characteristic for accurately modeling phase transitions and maintaining physical consistency in simulations. The proposed method uses an operator splitting technique, a numerical approach that decomposes a complex problem into simpler subproblems, solved sequentially, to improve computational efficiency and stability. The operator splitting method applied to the AC equation yields one nonlinear equation and several linear equations. To solve the nonlinear equation, we applied the frozen coefficient method, which approximates variable coefficients in differential equations by treating them as constants within small regions, simplifies the problem, and enables more efficient numerical solutions. For several linear equations, which are diffusion equations, we applied a fully implicit finite difference scheme to obtain unconditional stability. By using these methods, we achieved unconditional stability for the AC equation. To validate the superior performance of the developed algorithm, we performed computational tests. Computational experiments demonstrated its unconditional stability, particularly in handling high-order polynomial potentials. Furthermore, we highlighted a distinctive feature of the AC equation in modeling phase separation under noisy data conditions.

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.