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Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered. [ABSTRACT FROM AUTHOR]

Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the L^\infty (0, T; L^2(\Omega ; L^2)) norm all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of O(\tau ^{1/2}+ h^2) in the L^\infty (0, T; L^2(\Omega ; L^2)) norm for approximating the velocity, and strong convergence of O(\tau ^{1/2}+ h) in the L^{\infty }(0, T;L^2(\Omega ;L^2)) norm for approximating the time integral of pressure, where \tau and h denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates. [ABSTRACT FROM AUTHOR]

We present distinct types of analytical waveform solutions of the dispersive concatenation model which involves spatio-temporal dispersion, Hamiltonian perturbation terms and multiplicative white noise. The including high-order dispersion terms have important practical applications in nonlinear fiber optics, ocean waves and the Heisenberg spin chain. The obtained waveforms comprise solutions involving the Weierstrass function, periodic solutions, doubly periodic solutions, and soliton solutions, illustrating the potentially rich set of localized and periodic wave solutions of the studied model. It is found that the system support the existence of different kinds of soliton solutions including bright, dark, singular, dark-singular combo soliton, and bright-dark combo solitons. The physical meanings of these solutions and constraint conditions for their existence in the nonlinear medium are also determined. By choosing suitable values of the parameters, the obtained structures are also illustrated graphically. The results demonstrate that the phase component of the newly found wave solutions exhibit a nontrivial form containing the white noise. [ABSTRACT FROM AUTHOR]

Abstract.Accelerated degradation test plays a prominent role in reliability assessment and lifetime prediction for highly reliable products. The literature on accelerated degradation modeling primarily focuses on single stress factor situations. Therefore, this article proposes a nonlinear Wiener process-based dual stress factors accelerated degradation model with interaction, which simultaneously accounts for temporal variability, individual variability, and measurement variability. The maximum likelihood estimates (MLEs) of the model parameters are obtained using the profile likelihood approach and the Nelder-Mead algorithm, along with the MLEs for the reliability metrics of interest under normal operating conditions. We then provide bootstrap confidence intervals of the model parameters using the parametric percentile bootstrap method. The performance of the proposed method is assessed through the Monte Carlo simulation. Finally, a real-world example is presented to illustrate the application of our method. [ABSTRACT FROM AUTHOR]

The paper considers the nonparametric Poisson regression problem with a regular equidistant design on the unit interval. The nonparametric estimation of the Poisson regression function is studied. The uniform consistency conditions are established and the limit theorems are proved for continuous functionals on C [ a , 1 - a ] , 0 < a < 1 2 . [ABSTRACT FROM AUTHOR]

We have proposed the maintenance effort assessment model based on two Wiener processes for the operation of open source software (OSS) used in the edge computing in the past. In particular, we consider that this proposed model can assess the reliability by using three dimensional graph. Then, we have proposed two-dimensional modeling based on the effort management in the past. In this paper, we propose new expanded maintenance model considering OSS edge computing by expanding the existing two Wiener processes model in order to consider the network environment under the edge OSS operation. Especially, it is important to control the amount of maintenance effort expense in the long-term phase. Then, we propose the optimization method based on the past two-dimensional Wiener processes model. Thereby, it will be helpful to assess the operation effort expenditures with network environment of edge OSS service. Moreover, actual effort data sets are analyzed to show numerical examples of the proposed optimization method considering the network environment under the edge OSS operation. [ABSTRACT FROM AUTHOR]

We utilize the unified approach and He's semi-inverse method to derive novel stochastic optical solutions for the (2 + 1)-dimensional nonlinear Schrödinger equation (2D-NLSE) in the context of Itôcalculus. The solutions obtained encompass distinct structures such as super solitons and collapsing dissipative waves. These solutions hold significant potential for elucidating physical phenomena across various domains, including stochastic plasma media, ocean waves, and optical fiber. We investigate the dependence of the 2D-NLSE wave solutions on the physical model parameters, namely the group velocity dispersion, nonlinearity and linear coefficients. These parameters play a crucial role in regulating the amplitude and phase of optical communication waves during propagation. Graphical representations of selected solutions are provided to visually demonstrate their dynamic characteristics. The presented solutions expand the possibilities for optical manipulation and offer prospects for addressing practical challenges. [ABSTRACT FROM AUTHOR]

This article investigates the nonlinear Maccari model with multiplicative noise using the unified technique. Numerous new important solitary wave solutions are presented with free physical parameters. These solutions play a vital role in various domains, including nonlinear optics, plasma physics, and hydrodynamics. The investigation shows that the solution process is quick and clear, where a comparatively higher number of novel solutions are obtained. The performance of the used approach is compared with that of other methods. We create 2D and 3D graphs for certain solutions of the study, utilizing suitably selected values for the physical parameters. We also address the impact of model parameters on the solution characteristics. We observe that our results may help to resolve some physical problems in the actual world by determining the motion of a single wave in a tiny region. Finally, the outcomes show how simple and effective this method is at producing rich, accurate solutions to nonlinear models in mathematical physics as well as complex nonlinear wave structures.

The assessment of rolling bearing reliability is vital for ensuring mechanical operational safety and minimizing maintenance costs. Due to the difficulty in obtaining data on the performance degradation and failure time of rolling bearings, traditional methods for reliability assessment are challenged. This paper introduces a novel hybrid method for the reliability assessment of rolling bearings, combining the convolutional neural network (CNN)‐convolutional block attention module (CBAM)‐ bidirectional long short‐term memory (BiLSTM) network with the Wiener process. The approach comprises three distinct stages: Initially, it involves acquiring two‐dimensional time‐frequency representations of bearings at various operational phases using Continuous Wavelet Transform. Subsequently, the CNN‐CBAM‐BiLSTM network is employed to establish health index (HI) for the bearings and to facilitate the extraction of deep features, serving as input for the Wiener process. The final stage applies the Wiener process to evaluate the bearings' reliability, characterizing the HI and quantifying uncertainties. The experiment is performed on bearing degradation data and the results indicate the effectiveness and superiority of the proposed hybrid method. [ABSTRACT FROM AUTHOR]

In this paper, we prove that stochastic porous media equations over σ -finite measure spaces (E , B , μ) , driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator L and the diffusivity function given by a maximal monotone multi-valued function Ψ of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions Ψ , for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an L p (μ) -Itô formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular self-organized criticality models) and cases where E is a manifold or a fractal, and to non-local operators L, as e.g. L = - f (- Δ) , where f is a Bernstein function. [ABSTRACT FROM AUTHOR]

LED lamp beads (hereinafter referred to as LEDs) are complex electronic components, and their degradation process shows multi-stage characteristics. Ignoring the effects of multi-stage degradation and stress coupling will lead to a higher theoretical lifespan. In this paper, a Wiener process model based on generalized coupling is proposed for the staged degradation of LEDs. This paper first conducts accelerated degradation tests on LEDs under different temperature, humidity, and current stress combinations to obtain three index parameters of LEDs. Light output performance (LOP) is selected as the degradation characteristic quantity, and the Shapiro–Wilk test is used to determine whether the parameters conform to the normal distribution. Then, the unknown parameters of the multi-stage Wiener process are estimated and a generalized coupling model is established using the unknown parameters and accelerated degradation test data. Finally, the LED life under standard stress is extrapolated based on the multiple stress acceleration factors. The analysis of LED reliability experimental data shows that the proposed method can realize reliability assessment and has higher lifetime prediction accuracy compared with the multi-stage model without considering stress coupling. [ABSTRACT FROM AUTHOR]

AbstractDegradation of many products in practical applications is often subject to unit-to-unit heterogeneity. Such heterogeneity can be attributed to the heterogeneous quality of the raw materials and the fluctuation of the manufacturing process, as well as the heterogeneous usage conditions and environments. The heterogeneity leads to the scattering of the degradation rates and diffusion intensities in the population. To model this phenomenon, this study proposes a general random-effects Wiener process model that accounts for the unit-to-unit heterogeneity in the degradation drift and the volatility simultaneously. In particular, the drift of the Wiener process is characterized by a normal distribution and the diffusion parameter is characterized by an independent inverse Gaussian distribution. The proposed model is flexible for characterization of heterogeneous degradation, and permits an analytically tractable model inference. An EM algorithm incorporating the variational Bayesian method is developed to estimate the model parameters, and a parametric bootstrap approach is proposed to construct confidence intervals. The performance of the proposed model and the estimation algorithm is validated by Monte Carlo simulations. The degradation data of an infrared LED device and the wear data of the magnetic head of a hard disk drive are studied based on the proposed model. With comprehensive comparative studies, the good performance of the proposed model in fitting the real degradation data is validated. [ABSTRACT FROM AUTHOR]

In conventional remaining useful life (RUL) prediction approaches grounded on maintenance, the maintenance threshold is typically established as a stationary value. However, the actual maintenance threshold may exceed its preset value due to the uncertainty of degradation and other factors. Therefore, it is necessary to consider the dynamic maintenance threshold to improve the precision of remaining useful life prediction. By considering the Wiener process, the maintenance threshold error is introduced to reflect the dynamic nature of the maintenance threshold. The influence of maintenance on degradation amount, degradation rate, and degradation path are comprehensively considered to establish a multi‐stage maintenance‐affected degradation process model. The RUL formula of the equipment is derived using the first hitting time (FHT). The maximum likelihood estimation (MLE) approach and Bayesian theory are employed to estimate the model's parameters. The proposed approach is validated using simulation data and gyroscope degradation data. The outcomes reveal that the proposed approach can significantly enhance the precision of life prediction for the equipment. [ABSTRACT FROM AUTHOR]

This paper proposes a new way of modelling imperfect maintenance in degradation models, by assuming that maintenance affects only a part of the degradation process. More precisely, the global degradation process is the sum of two dependent Wiener processes with drift. Maintenance has an effect of the ARD1$ARD_1$‐type on only one of these processes: it reduces the degradation level of a quantity which is proportional to the amount of degradation of this process accumulated since previous maintenance. Two particular cases of the model are considered: perturbed ARD1$ARD_1$ and partial replacement models. The usual ARD1$ARD_1$ model is also a specific case of this new model. The system is regularly inspected in order to measure the global degradation level. Two observation schemes are considered. In the complete scheme, the degradation levels are measured both between maintenance actions and at maintenance times (just before and just after). In the general scheme, the degradation levels are measured only between maintenance actions. The maximum likelihood estimation of the model parameters is studied for both observation schemes in both particular models. The quality of the estimators is assessed through a simulation study. [ABSTRACT FROM AUTHOR]

Electromagnets are often used in indirect control for industrial applications. The ability of an electromagnet to control objects should decrease with performance degradation. And electromagnets product poses a danger to people and objects in the working environment. So, it is very difficult to analyze the reliability of electromagnetic performance degradation because of the complicated working condition. Failure Mode and Effect Analysis (FMEA) is the most commonly used tool for product reliability analysis. The new version of FMEA uses integer as evaluation value, which cannot represent the hesitation psychology of the evaluator. The Action Priority (AP) table of the FMEA describes the relationship between the evaluation of influencing factors and the risk level of the failure mode, which provides rules for determining the risk level of the failure mode. However, the AP table may result in multiple failure modes having the same ranking, which does not align with the intention of FMEA to prevent failures. Therefore, this paper proposes a reliability analysis method for electromagnetic performance degradation based on FMEA and FIS. Firstly, the Double Hierarchy Hesitant Fuzzy Linguistic Term Set (DHHFLTS) is used as the evaluation language to describe the hesitation psychology of evaluators. Secondly, the AP table of FMEA is used as FIS fuzzy inference rule. In this way, the idea of FMEA to determine the risk level of failure mode is retained and the problem of FIS fuzzy rule making is overcome. Then, FIS defuzzification AP table inference results to determine the risk ranking of failure modes. This avoids situations where the order of failure modes is equal. Finally, a performance degradation model of the electromagnet is constructed based on the Wiener process, and the calculation results of the new method are verified. [ABSTRACT FROM AUTHOR]

This article focuses on the correlation between the degradation levels of the two components that form a system. The degradation evolution of each component is modeled using Wiener processes. Both components are dependent and this dependence is described using the trivariate reduction method. To reduce the degradation and extend the system lifetime, preventive maintenance actions are periodically performed. These preventive maintenance actions are imperfect and they are modeled by using an arithmetic reduction of degradation of infinite order model with a determined maintenance efficiency parameter. The evolution of the maintained system is analysed by assessing the expectation and variance of both degradation processes at successive maintenance times. The novelty of this work is the analysis of the Pearson correlation coefficient between the degradation levels of the two components. Different properties of the monotonicity of the Pearson correlation coefficient between the two degradation paths are obtained by considering equal maintenance efficiency and equal general time scales functions for the two Wiener degradation processes associated to each degrading component. [ABSTRACT FROM AUTHOR]

Hot rolling work rolls are essential components in the hot rolling process. However, they are subjected to high temperatures, alternating stress, and wear under prolonged and complex working conditions. Due to these factors, the surface of the work rolls gradually degrades, which significantly impacts the quality of the final product. This paper presents an improved degradation model based on the Wiener process for predicting the remaining useful life (RUL) of hot rolling work rolls, addressing the critical need for accurate and reliable RUL estimation to optimize maintenance strategies and ensure operational efficiency in industrial settings. The proposed model integrates pulsed eddy current testing with VMD-Hilbert feature extraction and incorporates a Gaussian kernel into the standard Wiener process to effectively capture complex degradation paths. A Bayesian framework is employed for parameter estimation, enhancing the model's adaptability in real-time prediction scenarios. The experimental results validate the superiority of the proposed method, demonstrating reductions in RMSE by approximately 85.47% and 41.20% compared to the exponential Wiener process and the RVM model based on a Gaussian kernel, respectively, along with improvements in the coefficient of determination (CD) by 121% and 19.76%. Additionally, the model achieves reductions in MAE by 85.66% and 42.61%, confirming its enhanced predictive accuracy and robustness. Compared to other algorithms from the related literature, the proposed model consistently delivers higher prediction accuracy, with most RUL predictions falling within the 20% confidence interval. These findings highlight the model's potential as a reliable tool for real-time RUL prediction in industrial applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider two different types of numerical schemes for the nonstationary stochastic Stokes–Darcy equations with multiplicative noise. Firstly, we consider the Chorin-type time-splitting scheme for the Stokes equation in the free fluid region. The Darcy equation and an elliptic equation for the intermediate velocity of free fluid coupled with the interface conditions are solved, and then the velocity and pressure in free fluid region are updated by an elliptic system. Secondly, we further consider the artificial compressibility method (ACM) which separates the fully coupled Stokes–Darcy model into two smaller subphysics problems. The ACM reduces the storage and the computational time at each time step, and allows parallel computing for the decoupled problems. The pressure in free fluid region only needs to be updated at each time step without solving an elliptic system. We utilize the RT 1 -P 1 pair finite element space and the interior penalty discontinuous Galerkin (IPDG) scheme based on the BDM 1 -P 0 finite element space in the spatial discretizations. Under usual assumptions for the multiplicative noise, we prove that both of the Chorin-type scheme and the ACM are unconditionally stable. We present the error estimates for the time semi-discretization of the Chorin-type scheme. Numerical examples are provided to verify the stability estimates for both of schemes. Moreover, we test the convergence rate for the velocity in time for both of schemes, and the convergence rate for the pressure approximation in time average is also tested. [ABSTRACT FROM AUTHOR]

In this paper we establish uniqueness of the weak solutions of nonlinear stochastic functional differential equations of neutral type in Hilbert spaces. We also study the continuous dependence of such solutions on the initial data

Abstract The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion.

Abstract In this paper, we apply stochastic differential equations with the Wiener process to investigate the soliton solutions of the Chaffee–Infante (CI) equation. The CI equation, a fundamental model in mathematical physics, explains concepts such as wave propagation and diffusion processes. Exact soliton solutions are obtained through the application of the modified extended tanh (MET) method. The obtained wave figures in 3D, 2D, and contour are highly localized and determine an individual frequency shift under the behavior of sharp peak, periodic wave, and singular soliton. The MET method shows to be a valuable analytical tool for obtaining soliton solutions, essential for understanding the dynamics of nonlinear wave phenomena. Numerical simulations enable us to explore soliton solutions in two and three dimensions, shedding light on their properties over time. Our results have wide applications in various domains, including stochastic processes and nonlinear dynamics, impacting advancements in physics, engineering, finance, biology, and beyond.

The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion. [ABSTRACT FROM AUTHOR]

We study a triple of stochastic processes: a Wiener process W t , t ≥ 0 , its running maxima process M t = sup { W s : s ∈ [ 0 , t ] } , and its running minima process m t = inf { W s : s ∈ [ 0 , t ] } . We derive the analytical formula for the corresponding copula and show that it is supported on the hemicube, a convex hexahedron with seven vertices. As an application, we draw out an analytical formula for pricing of a double barrier option. [ABSTRACT FROM AUTHOR]

Several variations of stochastic processes have been studied in the literature to obtain reliability estimations of products and systems from degradation data. As the degradation trajectories may have different degradation rates, it is necessary to consider alternatives to characterize their individual behavior. Some stochastic processes have a constant drift parameter, which defines the mean rate of the degradation process. However, for some cases, the mean rate must not be considered as constant, which means that the rate varies in the different stages of the degradation process. This poses an opportunity to study alternative strategies that allow to model this variation in the drift. For this, we consider the Hjorth rate, which is a failure rate that can define different shapes depending on the values of its parameters. In this paper, the integration of this hazard rate with the Wiener process is studied to individually identify the degradation rate of multiple degradation trajectories. Random effects are considered in the model to estimate a parameter of the Hjorth rate for every degradation trajectory, which allows us to identify the type of rate. The reliability functions of the proposed model is obtained through numerical integration as the function results in a complex form. The proposed model is illustrated in two case studies based on a crack propagation and infrared LED datasets. It is found that the proposed approach has better performance for the reliability estimation of products based on information criteria. [ABSTRACT FROM AUTHOR]

In reliability engineering, different types of accelerated degradation tests have been used to obtain reliability information for evaluating highly reliable or expensive products. The step‐stress accelerated degradation test (SSADT) is one of the useful experimental schemes that can be used to save the resources of an experiment. Motivated by the SSADT data for operational amplifiers collected in Xi'an Microelectronic Technology Institute, in which the underlying degradation mechanism of the operational amplifiers is unknown, we propose a semiparametric approach for SSADT data analysis that does not require strict distributional assumptions. Specifically, the empirical saddlepoint approximation method is proposed to estimate the items' lifetime (first‐passage time) distribution at both stress levels included and not included in the SSADT experiment. Monte Carlo simulation studies are used to evaluate the performance and illustrate the advantages of the proposed approach. Finally, the proposed semiparametric approach is applied to analyze the motivating data set. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to demonstrate the uniform large deviation principle (UDLP) for the solutions of two-dimensional stochastic Navier–Stokes equations (SNSE) in the vorticity form when perturbed by two distinct types of noises. We first consider an infinite-dimensional additive noise that is white in time and colored in space and then consider a finite-dimensional Wiener process with linear growth coefficient. In order to obtain the ULDP for 2D SNSE in the vorticity form, where the noise is white in time and colored in space, we utilize the existence and uniqueness result from B. Ferrario et. al., Stochastic Process. Appl.,129 (2019), 1568–1604, and the uniform contraction principle. For the finite-dimensional multiplicative Wiener noise, we first prove the existence of a unique local mild solution to the vorticity equation using a truncation and fixed point arguments. We then establish the global existence of the truncated system by deriving a uniform energy estimate for the local mild solution. By applying stopping time arguments and a version of Skorokhod’s representation theorem, we conclude the global existence and uniqueness of a solution to our model. We employ the weak convergence approach to establish the ULDP for the law of the solutions in two distinct topologies. We prove ULDP in the C ([ 0 , T ] ; L p (T 2)) topology, for p > 2 , taking into account the uniformity of the initial conditions contained in bounded subsets of L p (T 2) . Finally, in C ([ 0 , T ] × T 2) topology, the uniformity of initial conditions lying in bounded subsets of C (T 2) is considered. [ABSTRACT FROM AUTHOR]

Proton exchange membrane fuel cell (PEMFC) is considered as one of the most promising green energy devices. Although fruitful results can be available for the remaining useful life (RUL) prediction of PEMFC, stochastic uncertainties have never been considered. To tackle this problem, a hybrid method is proposed in this paper. Wiener process with temporal uncertainty and individual uncertainty is adopted to model the degradation of the state of health (SOH), which is then estimated from monitoring voltage with measurement noise using the unscented Kalman filter (UKF), where unknown filtering and model parameters are jointly identified by expectation-maximization (EM) algorithm and Rauch-Tung-Striebel (RTS) smoother. Finally, gated recurrent unit (GRU) network is employed to realize the RUL prediction with the prediction uncertainty quantified by the Bayesian variational inference technology. The proposed method is verified on the experimental data. Results indicate that smaller mean absolute percentage error (MAPE), root mean square error (RMSE), and mean absolute error (MAE) values can be obtained compared with other methods. When 60% data are used for prediction, the proposed method can achieve a RUL prediction accuracy with 1.63% and 2.17% relative errors under static and dynamic conditions, respectively, which illustrates the feasibility and superiority of the proposed method. [ABSTRACT FROM AUTHOR]