. In this paper, we introduce the concept of m-superquadraticity, a natural generalization of classical superquadraticity that incorporates a parameter m to provide adjustable control over the growth and curvature of functions. Utilizing this framework, we establish new forms of Jensen's and (Hermite-Hadamard) HH type inequalities and extend them to their fractional counterparts via (Riemann-Liouville) R.L fractional integrals. The parameter m allows the inequalities to flexibly adapt to a wider class of functions, offering tighter bounds and greater applicability in analytical and applied contexts. The theoretical findings are substantiated with graphical illustrations and tabular analyses from representative examples. Furthermore, the msuperquadratic framework is applied in information theory to formulate novel classes of divergence measures. Overall, this approach enhances the classical theory of superquadraticity by providing additional flexibility, precision, and avenues for applications in stochastic fractional modeling, optimization under uncertainty, and entropy-based information measures.
In this paper, a modified version of the KL method algorithm (a geometric iterative method for solving systems of linear equations) is presented. In this modified algorithm, we obtain the intersection point of the line passing through the two previously obtained points with one of the system planes and consider it as the next point obtained by the algorithm. The algorithm obtained with this modification requires only a fraction of the number of iterations of the previous method to reach the system solution. With this trick, the new algorithm shows much better performance.
This paper proposes a robust numerical framework for estimating the implied volatility of equity-linked securities (ELS), which are characterized by complex path-dependent features such as early redemption and knock-in barriers. By extending the classical Black-Scholes model and applying finite difference methods, we solve the pricing problem for ELS products with non-linear and discontinuous payoff structures. The proposed algorithm calibrates a constant implied volatility over the life of the product by minimizing the error between market-observed fair prices and model-generated prices using a least-squares optimization approach. Numerical experiments were conducted on real ELS contracts linked to the KOSPI 200 index, Samsung Electronics stock, and Tesla, Inc. stock. The results show that the method is applicable across different underlying assets and captures distinct volatility characteristics for each asset. For each underlying asset, implied volatilities estimated from ELS issued on different dates remain within a consistent range, demonstrating the method's practical effectiveness in estimating ELS implied volatility.
The surface Allen-Cahn (AC) equation with variable mobility on smooth closed curved geometries is numerically investigated. The governing model is formulated as an L2-gradient flow of the classical double-well energy, where a spatially and state-dependent mobility function modulates the relaxation dynamics along the surface. A fully explicit scheme based on a finite difference discretization with a cotangent-type Laplace-Beltrami operator on triangulated surfaces is used. Numerical experiments on a sphere demonstrate the predicted convergence behavior and illustrate the nontrivial impact of heterogeneous mobility on interface motion and pattern formation. These results provide a systematic framework for explicit, geometry-informed simulation of phase-field dynamics with variable mobility on curved surfaces.
. This study presents a numerical investigation of the Allen-Cahn equation incorporating a domain-dependent free energy functional. We adopt a quartic polynomial potential to approximate the logarithmic free energy, effectively capturing the thermodynamic characteristics of phase separation while maintaining analytical tractability. Through three-dimensional numerical simulations, we analyze the morphological evolution of the phase field under spatially varying temperature parameters, 91. A comparative analysis of three distinct cases reveals that a marginal difference of 0.02 in the maximum value of 91 is sufficient to induce distinctly different final topological states, despite the simulations sharing identical initial conditions and concentric isosurface structures. Furthermore, the study successfully reproduces the Ostwald ripening phenomenon, validating the system's tendency to minimize surface energy. We also confirm the strict preservation of the order parameter's boundedness within the physically admissible interval of [-1, 1] throughout the evolution, ensuring both physical validity and numerical stability. These findings highlight the significant sensitivity of microstructural formation to domain-dependent parameters and demonstrate the robustness of the proposed numerical work.
Image segmentation divides an image into homogeneous regions and constitutes a fundamental procedure in image processing. This study presents a numerical approach for image segmentation that derives from a modified Allen-Cahn equation. The temporal discretization is performed using the explicit Euler scheme. The explicit Euler method reduces computational complexity and memory requirements, and it simplifies the treatment of nonlinear terms. We solve the equation numerically using zero Neumann boundary conditions. Numerical experiments show that the proposed method achieves accurate segmentation results, and with enough iterations, it also performs well in capturing complex boundaries and sharp regions.
We study the pricing of European power options in a market where the underlying asset is driven by a fast mean-reverting stochastic volatility factor, which captures empirically observed features such as volatility clustering and mean reversion that are absent in the classical Black-Scholes framework. Using singular perturbation techniques, we derive an asymptotic pricing formula for the price of the power option. The resulting approximation yields a semianalytic pricing formula that preserves much of the tractability of the constant-volatility power option formula while reflecting more realistic volatility dynamics. Furthermore, to evaluate the performance of the approximation, we conduct Monte Carlo simulations of the stochastic volatility model and use the simulated prices as benchmark values. The numerical results show that the correction significantly improves accuracy, with the approximate prices closely matching the Monte Carlo benchmarks. This demonstrates that the proposed asymptotic method offers an efficient and reliable tool for pricing power options under fast mean-reverting stochastic volatility.
This study introduces a modified conservative Allen-Cahn (CAC) equation that preserves the zero-level interface throughout the temporal evolution. Conventional CAC formulations that use a Lagrange multiplier tend to shift the zero level during the mass-correction process, which leads to unintended distortion of the interface. The proposed model incorporates nonstandard variable mobility functions M-1(phi) = |phi|(alpha) and M-2(phi) =|phi|(beta)(1-phi(2)), together with a newly designed correction term that redistributes mass away from the zero level. This structure enables strict preservation of the initial zero-level contour while maintaining mass conservation. A numerical algorithm based on an operator-splitting strategy and an explicit finite-difference method is developed to solve the resulting equation efficiently. Numerical experiments on two-dimensional test problems demonstrate that the proposed formulation preserves the interface shape more accurately than previous CAC models and guarantees stable mass conservation. These results confirm the effectiveness of the modified approach as a zero-level interface preserving model.
Restoring color images degraded by blurring and mixed noise-specifically randomvalued (RV) impulse and Gaussian noise-is a challenging task due to the distinct statistical characteristics of the two noise types. To address this problem, we propose a group-based sparse representation (GSR) model for restoring color images degraded by such mixed noise and blurring. The model incorporates a combined l(0)-l(2) data-fidelity term to effectively separate and suppress RV impulse and Gaussian noise. A multi-channel, patch-based GSR model is employed to preserve fine details and textures while effectively eliminating noise in homogeneous regions, yielding significantly fewer color artifacts and outliers than total variation-based methods. The resulting nonconvex and nonsmooth optimization problem is solved using an alternating minimization scheme combined with the alternating direction method of multipliers, leading to an efficient iterative algorithm. Numerical experiments demonstrate that the proposed model outperforms existing methods in both visual quality and quantitative metrics.
The Allen-Cahn equation incorporating a high-order potential plays an essential role in modeling phase separation and interface motion. We propose a hybrid method for solving the AC equation with a high-order potential, which combines the operator splitting method with the finite element method. Specifically, the Allen-Cahn equation is divided into a nonlinear reaction equation and a diffusion (heat) equation. The frozen coefficient method is employed to treat the nonlinear term by locally approximating variable coefficients as constants, which simplifies the computation. The diffusion term is then discretized and solved using an implicit finite element formulation. Finally, various numerical results are presented to verify the accuracy of the proposed method.
We propose an operator splitting method for efficiently solving two-dimensional diffusion equations with spatially varying diffusion coefficients. The method decomposes the two-dimensional problem into a sequence of one-dimensional subproblems, enhancing computational efficiency. Each subproblem is solved using the Crank-Nicolson method in time and a conservative finite difference scheme in space, achieving second-order accuracy in both time and space. Theoretical analysis establishes the stability of the method for heterogeneous diffusion problems, and numerical experiments confirm its second-order convergence. Owing to the operator splitting framework, each directional solve reduces to a tridiagonal system, ensuring computational efficiency. Moreover, the method is applicable to the Fisher-KPP equation with heterogeneous diffusivity, indicating its potential for practical use in reaction-diffusion models.
This paper presents a finite element approximation for solving a scalar second-order elliptic boundary value problem using the first-order system least-squares (FOSLS) method. While traditional FOSLS approaches, such as the div-curl method, are limited to problems with smooth domains and coefficients, the div least-squares method offers a robust alternative for problems with less regularity, including discontinuous coefficients and corner singularities. We utilize the lowest Raviart-Thomas element space, RT0h x P1h, to approximate the first-order system. Numerical experiments are conducted on various singularity problems to demonstrate the method's effectiveness. A key finding is a significant reduction in the number of degrees of freedom when compared to the general mixed finite element method, RT0h x P0h, making this approach more computationally efficient.
Tridiagonal linear systems arise ubiquitously in finite-difference discretizations of diffusion-type partial differential equations (PDEs). The Thomas algorithm is a workhorse direct solver for such systems and is widely embedded in time-integration frameworks. Boundary conditions, however, can alter the algebraic structure significantly. In particular, periodic boundaries induce cyclic couplings that violate strict tridiagonality. Leveraging rank-one corrections via the Sherman-Morrison identity, we extend a two-dimensional boundary-treatment strategy to three dimensions within an operator-splitting framework. We detail implementations for periodic, Dirichlet, and Neumann conditions, and assess accuracy and stability through heat-equation benchmarks and Allen-Cahn dynamics. The results furnish a practical blueprint for robust boundary enforcement in Thomas-based solvers for 3D time-dependent PDEs.
We develop a new marketing model that incorporates referral effects, purchase behavior, and repeated actions inspired by infection processes in epidemiology. The equilibria of the system are derived explicitly, and their stability is analyzed. Numerical computations are then carried out to quantitatively examine the time evolution of the system. The results obtained under various initial conditions are further analyzed and interpreted from a marketing perspective.
In this article, we consider the initial-value problem of the two-dimensional (2D) Boussinesq equations in a bounded domain. Unlike the well-studied 2D Navier-Stokes equations, the Boussinesq system poses greater challenges due to its temperature coupling and buoyancy effects, our work establishes its long-time dynamics by constructing absorbing sets and proving asymptotic compactness in higher-order spaces. First, using the Galerkin approximation method, we establish the global well-posedness and the higher regularity. Furthermore, the existence of bounded absorbing sets and the global attractor are given by means of the energy method and theory of the semigroup.
The main objective of the article is to examine the role of the order of the fractional derivatives in the rate of heat transfer and related thermal stresses. The time fractional Cattaneo equation results from a time-nonlocal generalization of the classical Fourier law with the Short-tail memory exponential kernel. The time fractional Cattaneo equation with nonmoving and moving time harmonic sources under zero initial conditions is studied on a line, and the equation with a moving time harmonic source subject to initial and zero boundary conditions is investigated on a half-line. Also, the time fractional Cattaneo equation is studied under the Dirichlet, Neumann, and Fractional order boundary conditions varying harmonically in time on a half-line. The fundamental solutions are acquired by employing integral transforms. The corresponding thermal stresses are obtained by using the displacement potential. Graphical representations of numerical outcomes are shown for various nondimensional parameter values.
In this paper, we explore finite-time stochastic flocking in a Cucker-Smale type model influenced by multiplicative white noise. To this end, we derive several preparatory results and apply stochastic analysis to reduce the model to suitable stochastic differential inequalities. We then numerically verify that the proposed model exhibits the desired finite-time stochastic flocking under appropriate sufficient conditions on the initial data and system parameters.
. The Thomas algorithm is widely used for efficiently solving tridiagonal systems that arise from finite difference discretizations of partial differential equations. While the core algorithm is well established, the practical treatment of boundary conditions, especially periodic boundary, often requires careful implementation to maintain numerical stability and accuracy. This paper presents a comprehensive discussion of practical strategies for incorporating various boundary conditions into the Thomas algorithm. We compare different implementation approaches, analyze their computational implications, and provide code-level insights to facilitate integration into numerical solvers. Benchmark tests on prototypical diffusion-type equations demonstrate the effectiveness and robustness of the proposed methods.
A discrete model of interaction timing optimization is studied. The model is a finite two-person zero-sum game called a silent duel with linear accuracy symmetry. Each of the duelists has a single bullet to shoot during the duel time span standardized to unit interval [0; 1]. The number of shooting moments is finite, where the very beginning and the very end of the duel are always included. Preliminarily the shooting moments are distributed uniformly corresponding to the perfectly timed potential shots. In real practice, the uniformly distributed shooting moments have a jitter, apart from the very beginning and the very end of the duel. This case is solved and gathered along with the duel without jitter. The duel by linear singlebullet accuracy symmetry and shooting uniform jitter can have a pure strategy solution only if the number of shooting moments is seven or fewer, but duelist's single-action optimal behavior exists only in the 7 x 7 duel without jitter. The vastest variety and changeability of the singleaction optimal behavior exist in the 4 x 4 duel, where more than 90% of jitter is rendered to a duelist's optimal pure strategy, which is not single only if the negative jitter is half its maximum magnitude. Besides, the 4 x 4 duel has an optimal pure solution by any negative jitter.
. This paper, to the best of the author's knowledge, is the first to propose an efficient finite-difference method (FDM) for valuing a convertible bond (CB) whose issuer call option has priority over the holder's conversion right. The study aims to establish a clear valuation framework for CBs in which the call dominates the conversion. When the call is exercised, the call's credit risk varies with the state of the underlying CB. To capture this feature, new intermediate conditions are introduced. The numerical scheme is based on the Tsiveriotis-Fernandes (TF) model, which formulates the CB value and its bond component as coupled equations. An implicit scheme is adopted to enhance the temporal flexibility. Furthermore, a linear boundary condition together with the projected successive over-relaxation (PSOR) technique is employed to handle the complex payoff structure arising from multiple early-exercise features. This approach ensures a stable and convergent numerical solution.
