In this paper, we present the mathematical and numerical analysis of a numerical method for a normalized time-fractional diffusion equation. The method is unconditionally stable and preserves the maximum principle. The normalized derivative was defined using a weighting function. The integral of this function is always equal to one for all fractional orders and times. This property makes it possible to compare the diffusion behavior under different fractional orders in a fair way. We considered a recently proposed finite difference scheme with an implicit time discretization for the equation. The resulting algebraic system has a tridiagonal matrix. The Thomas algorithm was applied to solve the system efficiently. We proved that the numerical scheme is unconditionally stable, satisfies the maximum principle, and converges correctly. The convergence order is $O(\mathrm{\Delta }{t}^{2-\alpha })$ in time and $O(h^2)$ in space. Several numerical examples were also presented. The results agree with the theoretical analysis and further validate the influence of different fractional orders on the diffusion behavior.

Image segmentation is the process of partitioning an image into homogenous regions, and represents one of the most fundamental and important procedures in image processing. Image denoising is a process to remove unwanted noise from a digital image, enhancing its visual quality. Various algorithms, like non-local means and deep learning-based approaches, have been developed to remove noise while preserving important image details. Currently, the prevalent application of pattern recognition technology is achieved through the implementation of image segmentation algorithms. In this study, we present a new, highly efficient, and fast computational scheme specifically developed for a phase-field mathematical model of image segmentation. The numerical methodology is based on an operator splitting method (OSM). The split operators are solved by using closed-form analytic solutions and a finite difference method (FDM) with an alternating direction explicit (ADE) method. To show the notable efficiency and rapid computational performance of the proposed computational algorithm, we conduct a series of numerical experiments. Through these computational tests, we confirm a significant contribution to the advancement of methodologies employed in the critical domain of image processing.

In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.

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.