Search Results (34)

This paper examines temporal symmetry breaking and structural duality in a class of time-changed McKean–Vlasov neutral stochastic differential equations. The system features super-linear drift coefficients satisfying a one-sided local Lipschitz condition and incorporates a fundamental duality: one drift component evolves under a random time change $E_t$, while the other progresses in regular time t. Within the symmetric framework of mean-field interacting particle systems, where particles exhibit permutation invariance, we establish strong convergence of the tamed Euler–Maruyama method over finite time intervals. By replacing the one-sided local condition with a globally symmetric Lipschitz assumption, we derive an explicit convergence rate for the numerical scheme. Two numerical examples validate the theoretical results. Full article

Conventional integer-order models fail to adequately capture non-local memory effects and constrained nonlinear interactions in emotional dynamics. To address these limitations, we propose a coupled framework that integrates Caputo fractional derivatives with hyperbolic tangent–based interaction functions. The fractional-order term quantifies power-law memory decay in affective states, while the nonlinear component regulates connection strength through emotional difference thresholds. Mathematical analysis establishes the existence and uniqueness of solutions with continuous dependence on initial conditions and proves the local asymptotic stability of network equilibria ($W_{ij}^{*}={\displaystyle \frac{1}{\delta }}{\sec \mathrm{h}}^2(\parallel {\mathrm{E}}_i-{\mathrm{E}}_j\parallel )$, e.g., $W^{*}\approx 1.40$ under typical parameters $\eta =0.5$, $\delta =0.3$). We further derive closed-form expressions for the steady-state variance under stochastic perturbations ($\mathrm{Var}\left(W_{ij}\right)={\displaystyle \frac{{\sigma }_{\zeta }^2}{2\eta \delta }}$) and demonstrate a less than 6% deviation between simulated and theoretical values when ${\sigma }_{\zeta }=0.1$. Numerical experiments using the Euler–Maruyama method validate the convergence of connection weights toward the predicted equilibrium, reveal Gaussian features in the stationary distributions, and confirm power-law scaling between noise intensity and variance. The numerical accuracy of the fractional system is further verified through L1 discretization, with observed error convergence consistent with theoretical expectations for $\mu =0.5$. This framework advances the mechanistic understanding of co-evolutionary dynamics in emotion-modulated social networks, supporting applications in clinical intervention design, collective sentiment modeling, and psychophysiological coupling research. Full article

This research introduces an innovative probabilistic method for examining torsional stress behavior in spherical shell structures through Monte Carlo simulation techniques. The spherical geometry of these components creates distinctive computational difficulties for conventional analytical and deterministic numerical approaches when solving torsion-related problems. The authors develop a comprehensive mesh-free Monte Carlo framework built upon the Feynman–Kac formula, which maintains the geometric symmetry of the domain while offering a probabilistic solution representation via stochastic processes on spherical surfaces. The technique models Brownian motion paths on spherical surfaces using the Euler–Maruyama numerical scheme, converting the Saint-Venant torsion equation into a problem of stochastic integration. The computational implementation utilizes the Fibonacci sphere technique for achieving uniform point placement, employs adaptive time-stepping strategies to address pole singularities, and incorporates efficient algorithms for boundary identification. This symmetry-maintaining approach circumvents the mesh generation complications inherent in finite element and finite difference techniques, which typically compromise the problem’s natural symmetry, while delivering comparable precision. Performance evaluations reveal nearly linear parallel computational scaling across up to eight processing cores with efficiency rates above 70%, making the method well-suited for multi-core computational platforms. The approach demonstrates particular effectiveness in analyzing torsional stress patterns in thin-walled spherical components under both symmetric and asymmetric boundary scenarios, where traditional grid-based methods encounter discretization and convergence difficulties. The findings offer valuable practical recommendations for material specification and structural design enhancement, especially relevant for pressure vessel and dome structure applications experiencing torsional loads. However, the probabilistic characteristics of the method create statistical uncertainty that requires cautious result interpretation, and computational expenses may surpass those of deterministic approaches for less complex geometries. Engineering analysis of the outcomes provides actionable recommendations for optimizing material utilization and maintaining structural reliability under torsional loading conditions. Full article

This study presents a comprehensive analysis of zoonotic disease transmission dynamics between baboon and human populations using both deterministic and stochastic modeling approaches. The model is constructed with a symmetric compartmental structure for each species—susceptible, infected, and recovered—which reflects a biological and mathematical symmetry between the two interacting populations. Public health control strategies such as sterilization, restricted food access, and reduced human–baboon interaction are incorporated symmetrically, allowing for a balanced evaluation of their effectiveness across species. The basic reproduction number (${\mathcal{R}}_0$) is derived analytically and examined through sensitivity indices to identify critical epidemiological parameters. Numerical simulations, implemented via the Euler–Maruyama method, explore the influence of stochastic perturbations on disease trajectories. Statistical tools including Maximum Likelihood Estimation (MLE), Mean Squared Error (MSE), and Power Spectral Density (PSD) analysis validate model predictions and assess variability across noise levels. The results provide probabilistic confidence intervals and highlight the robustness of the proposed control strategies. This symmetry-aware, dual-framework modeling approach offers novel insights into zoonotic disease management, particularly in ecologically dynamic regions with frequent human–wildlife interactions. Full article

The score-based diffusion model has made significant progress in the field of computer vision, surpassing the performance of generative models, such as variational autoencoders, and has been extended to applications such as speech enhancement and recognition. This paper proposes a U-Net architecture using a score-based diffusion model and an efficient multi-scale attention mechanism (EMA) for the speech enhancement task. The model leverages the symmetric structure of U-Net to extract speech features and captures contextual information and local details across different scales using the EMA mechanism, improving speech quality in noisy environments. We evaluate the method on the VoiceBank-DEMAND (VB-DMD) dataset and the DARPA TIMIT Acoustic-Phonetic Continuous Speech Corpus–TUT Sound Events 2017 (TIMIT-TUT) dataset. The experimental results show that the proposed model performed well in terms of speech quality perception (PESQ), extended short-time objective intelligibility (ESTOI), and scale-invariant signal-to-distortion ratio (SI-SDR). Especially when processing out-of-dataset noisy speech, the proposed method achieved excellent speech enhancement results compared to other methods, demonstrating the model’s strong generalization capability. We also conducted an ablation study on the SDE solver and the EMA mechanism, and the results show that the reverse diffusion method outperformed the Euler–Maruyama method, and the EMA strategy could improve the model performance. The results demonstrate the effectiveness of these two techniques in our system. Nevertheless, since the model is specifically designed for Gaussian noise, its performance under non-Gaussian or complex noise conditions may be limited. Full article

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the $L^2$ sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the $L^2$ sense. Full article

The truncated Euler–Maruyama (EM) method for stochastic differential equations with Poisson jumps (SDEwPJs) has been proposed by Deng et al. in 2019. Although the finite-time $L^r$-convergence theory has been established, the strong convergence theory remains absent. In this paper, the strong convergence refers to the use of an $L^2$ measure and places the supremum over time inside the expectation operation. Our version can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. Noting that the conditions imposed are too strict, this paper presents an existence and uniqueness theorem for SDEwPJs under general conditions and proves the convergence of the truncated EM method for these equations. Finally, two examples are considered to illustrate the application of the truncated EM method in option price calculation. Full article

The conditional Gaussian nonlinear system (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise. Full article

The strong convergence of numerical solutions is studied in this paper for stochastic Volterra integral differential equations (SVIDEs) with a Hölder diffusion coefficient using the truncated Euler–Maruyama method. Firstly, the numerical solutions of SVIDEs are obtained based on the Euler–Maruyama method. Then, the pth moment boundedness and strong convergence of truncated the Euler–Maruyama numerical solutions are proven under the local Lipschitz condition and the Khasminskii-type condition. Finally, the convergence rate of the truncated Euler–Maruyama method of the numerical solutions is also discussed under some suitable assumptions. Full article

In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis. Full article

Two novel crossover models for breast cancer that incorporate $\Psi$-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function $\Psi \left(t\right)$ in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter $\zeta$ is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo–Katugampola derivatives as special cases from the proposed $\Psi$-Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The $\Psi$-nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik–Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler–Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions. Full article

A vector–host model of dengue with multiple stages and independent fluctuations is investigated in this paper. Firstly, the existence and uniqueness of the positive solution are shown by contradiction. When the death rates of aquatic mosquitoes, adult mosquitoes, and human beings respectively control the intensities of white noises, and if ${\mathcal{R}}_0^s>1$, then the persistence in the mean for both infective mosquitoes and infective human beings is derived. When ${\mathcal{R}}_0^s>1$ is valid, the existence of stationary distribution is derived through constructing several appropriate Lyapunov functions. If the intensities of white noises are controlled and $\phi <0$ is valid, then the extinction for both infective mosquitoes and infective human beings is obtained by applying the comparison theorem and ergodic theorem. Further, the main findings are verified through numerical simulations by using the positive preserving truncated Euler–Maruyama method (PPTEM). Moreover, several numerical simulations on the infection scale of dengue in Fuzhou City were conducted using surveillance data. The main results indicate that the decrease in the transfer proportion from aquatic mosquitoes to adult mosquitoes reduces the infection scale of infective human beings with dengue virus, and the death rates of aquatic mosquitoes and adult mosquitoes affect the value of the critical threshold ${\mathcal{R}}_0^s$. Further, the controls of the death rates of mosquitoes are the effective routes by the decision-makers of the Chinese mainland against the spread of dengue. Full article

This work focuses on a class of regime-switching diffusion processes with both continuous components and discrete components. Under suitable conditions, we adopt the Euler–Maruyama method to deal with the convergence of numerical solutions of the corresponding stochastic differential equations. More precisely, we first show the convergence rates in the $L^p$-norm of the stochastic differential equations with regime switching under Lipschitz conditions. Then, we also discuss $L^1$ and $L^2$ convergence under non-Lipschitz conditions. Full article

This work focuses on the convergence of the numerical invariant measure for a stochastic age-dependent population–toxicant model with Markov switching. Considering that Euler–Maruyama (EM) has the advantage of fast computation and low cost, explicit EM was used to discretize the time variable. With the help of the p-th moment boundedness of the analytical and numerical solutions of the model, the existence and uniqueness of the corresponding invariant measures were obtained. Under suitable assumptions, the conclusion that the numerical invariant measure converges to the invariant measure of the analytic solution was proven by defining the Wasserstein distance. A numerical simulation was performed to illustrate the theoretical results. Full article

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study’s originality is the use of fresh data simulation techniques and different solution methodologies. Full article