Search Results (22)

Post-disaster underground (UG) mine environments are characterized by complex and rapidly changing conditions, adding extra attenuation to propagating electromagnetic (EM) waves. One such complex condition is the extreme generation of dust and sudden rise in humidity contributing to extra attenuation effects to propagating waves, especially under varying airborne humidity and dust levels. The existing wave propagation prediction models, especially those that factor in the effect of dust particles, are deterministic in nature, limiting their ability to account for uncertainties, especially during emergency conditions. In this work, the vector parabolic equation (VPE) model is modified to include dust attenuation effects. Using the complex permittivity of dust as a random variable, the Karhunen–Loève (KL) expansion is used to generate random samples of permittivity along the drifts for which each realization is solved using deterministic VPE method. The model is validated using a modified Friis method and experimentally obtained data from literature. The findings show that accounting for dust and humidity effects stochastically captures the extra losses that would have otherwise been lost using deterministic methods. The proposed framework offers key insights for designing resilient underground wireless systems, strengthening miner tracking, and improving safety during emergencies. Full article

We study a stochastic differential model for the dynamics of institutional corruption, extending a deterministic three-variable system—corruption perception, proportion of sanctioned acts, and policy laxity—by incorporating Gaussian perturbations into key parameters. We prove global existence and uniqueness of solutions in the physically relevant domain, and we analyze the linearization around the asymptotically stable equilibrium of the deterministic system. Explicit mean square bounds for the linearized process are derived in terms of the spectral properties of a symmetric matrix, providing insight into the temporal validity of the linear approximation. To investigate global behavior, we relate the first exit time from the domain of interest to backward Kolmogorov equations and numerically solve the associated elliptic and parabolic PDEs with FreeFEM, obtaining estimates of expectations and survival probabilities. An application to the case of Mexico highlights nontrivial effects: while the spectral structure governs local stability, institutional volatility can non-monotonically accelerate global exit, showing that highly reactive interventions without effective sanctions increase uncertainty. Policy implications and possible extensions are discussed. Full article

The main purpose of the present paper is to investigate the problem of estimating the time-varying coefficient in a stochastic parabolic equation driven by a sub-fractional Brownian motion. More precisely, we introduce a kernel-type estimator for the time-varying coefficient $\theta \left(t\right)$ in the following evolution equation:$$\left\{\begin{array}{c}{\displaystyle du(t,x)=(A_0+\theta \left(t\right)A_1)u(t,x)dt+d{\xi }^H(t,x),x\in [0,1],t\in (0,T],}\hfill \\ {\displaystyle u(0,x)=u_0\left(x\right),}\hfill \end{array}\right.$$ where ${\xi }^H(t,x)$ is a cylindrical sub-fractional Brownian motion in ${\mathbb{L}}^2\left([0,T]\times [0,1]\right)$, and $A_0+\theta \left(t\right)A_1$ is a strongly elliptic differential operator. We obtain the asymptotic mean square error and the limiting distribution of the proposed estimator. These results are proved under some standard conditions on the kernel and some mild conditions on the model. Finally, we give an application for the confidence interval construction. Full article

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume. Full article

In this paper, a structure-preserving local discontinuous Galerkin (LDG) method is proposed for parabolic stochastic partial differential equations with periodic boundary conditions and multiplicative noise. It is proven that under certain conditions, this numerical method is stable in the $L^2$ sense and can preserve energy conservation. The optimal spatial error estimate in the mean square sense can reach $n+1$ if the degree of the polynomial is n. The correctness of the theoretical results is verified through numerical examples. Full article

This paper defines stubbornness as an optimal feedback Nash equilibrium within a dynamic setting. Stubbornness is treated as a player-specific parameter, with the team’s coach initially selecting players based on their stubbornness and making substitutions during the game according to this trait. The payoff function of a soccer player is evaluated based on factors such as injury risk, assist rate, pass accuracy, and dribbling ability. Each player aims to maximize their payoff by selecting an optimal level of stubbornness that ensures their selection by the coach. The goal dynamics are modeled using a backward parabolic partial stochastic differential equation (BPPSDE), leveraging its theoretical connection to the Feynman–Kac formula, which links stochastic differential equations (SDEs) to partial differential equations (PDEs). A stochastic Lagrangian framework is developed, and a path integral control method is employed to derive the optimal measure of stubbornness. The paper further applies a variant of the Ornstein–Uhlenbeck BPPSDE to obtain an explicit solution for the player’s optimal stubbornness. Full article

The Feynman–Kac formula establishes a link between parabolic partial differential equations and stochastic processes in the context of the Schrödinger equation in quantum mechanics. Specifically, the formula provides a solution to the partial differential equation, expressed as an expectation value for Brownian motion. This paper demonstrates that the Feynman–Kac formula does not produce a unique solution but instead carries infinitely many solutions to the corresponding partial differential equation. Full article

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are $O({\lambda }_N^{-\gamma })$ in space and $O({\tau }^{\gamma })$ in time, where $\gamma \in (0,1]$ from the assumption $\parallel A^{\frac{\gamma -1}{2}}{Q}^{\frac{1}{2}}{\parallel }_{{\mathcal{L}}_2}$ is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings. Full article

In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications. Full article

In the present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on a bounded or unbounded domain and driven by a nonlinear noise. Since the theory of pathwise random dynamical systems cannot be applied directly to the equation with nonlinear noise, we first establish the existence of weak pullback mean random attractors for the equation by applying the theory of mean-square random dynamical systems; then, we prove the existence of (pathwise) pullback random attractors for the Wong–Zakai approximate system of the equation. In addition, we establish the upper semicontinuity of pullback random attractors for the Wong–Zakai approximate system of the equation under consideration driven by a linear multiplicative noise. Full article

The main aim of this contribution is to construct a numerical scheme for solving stochastic time-dependent partial differential equations (PDEs). This has the advantage of solving problems with positive solutions. The scheme provides conditions for obtaining positive solutions, which the existing Euler–Maruyama method cannot do. In addition, it is more accurate than the existing stochastic non-standard finite difference (NSFD) method. Theoretically, the suggested scheme is more accurate than the current NSFD method, and its stability and consistency analysis are also shown. The scheme is applied to the linear scalar stochastic time-dependent parabolic equation and the nonlinear auto-catalytic Brusselator model. The deficiency of the NSFD in terms of accuracy is also shown by providing different graphs. Many observable occurrences in the physical world can be traced back to certain chemical concentrations. Examining and understanding the inter-diffusion between chemical concentrations is important, especially when they coincide. The Brusselator model is the gold standard for describing the relationship between chemical concentrations and other variables in chemical systems. A computational code for the proposed model scheme may be made available to readers upon request for convenience. Full article

Model predictive control (MPC) heavily relies on the accuracy of the system model. Nevertheless, process models naturally contain random parameters. To derive a reliable solution, it is necessary to design a stochastic MPC. This work studies the chance constrained MPC of systems described by parabolic partial differential equations (PDEs) with random parameters. Inequality constraints on time- and space-dependent state variables are defined in terms of chance constraints. Using a discretization scheme, the resulting high-dimensional chance constrained optimization problem is solved by our recently developed inner–outer approximation which renders the problem computationally amenable. The proposed MPC scheme automatically generates probability tubes significantly simplifying the derivation of feasible solutions. We demonstrate the viability and versatility of the approach through a case study of tumor hyperthermia cancer treatment control, where the randomness arises from the thermal conductivity coefficient characterizing heat flux in human tissue. Full article

Stochastic simulations enable researchers to incorporate uncertainties beyond numerical discretization errors in computational fluid dynamics (CFD). Here, the authors provide examples of stochastic simulations of incompressible flows and numerical solutions for validating these newly emerging stochastic modeling methods. A numerical scheme is constructed for finding solutions to stochastic parabolic equations. The scheme is second-order accurate in time for the constant coefficient of the Wiener process term. The stability analysis of the scheme is also provided. The scheme is applied to the dimensionless heat and mass transfer model of mixed convective non-Newtonian nanofluid flow over oscillatory sheets. Both the deterministic and stochastic energy equations use temperature-dependent thermal conductivity. The stochastic model is more general than the deterministic model. The results are calculated for both flat and oscillatory plates. Casson parameter, mixed convective parameter, thermophoresis, Brownian motion parameter, Prandtl number, Schmidt number, and reaction rate parameter all impact the velocities, temperatures, and concentrations shown in the graphs. Under the influence of the oscillating plate, the results reveal that the concentration profile decreases with increasing Brownian motion parameters and increases with increasing thermophoresis parameters. The behavior of the velocity profile for the deterministic and stochastic models is provided, and contour plots for the stochastic model are also displayed. This article aims to provide a state-of-the-art overview of recent achievements in the field of stochastic computational fluid dynamics (SCFD) while also pointing out potential future avenues and unresolved challenges for the computational mathematics community to investigate. Full article

The work is dedicated to mathematical modelling of random diffusion flows of admixture particles in a two-phase stratified strip with stochastic disposition of phases and random thickness of inclusion-layers. The study of such models are especially important during the creation of composite layered materials, in the research of the transmission properties of filters, and in the prediction of the spread of pollutants in the environment. Within the model we consider one case of uniform distribution of coordinates of upper boundaries of the layers of which the body is made up and two more cases, i.e., of uniform and triangular distributions of the inclusion thickness. The initial-boundary value problems of diffusion are formulated for flux functions; the boundary conditions at one of the body’s surfaces are set for flux and, at the other boundary, the conditions are given for admixture concentration; the initial condition being concerned with zero and non-zero constant initial concentrations. An equivalent integro-differential equation is constructed. Its solution is found in terms of Neumann series. For the first time it was obtained calculation formulae for diffusion flux averaged over the ensemble of phase configurations and over the inclusion thickness. It allowed to investigate the dependence of averaged diffusion fluxes on the medium’s characteristics on the basis of the developed software. The simulation of averaged fluxes of admixture in multilayered $Fe-Cu$ and $\alpha Fe-Ni$ materials is made. Comparative analysis of solutions, depending on the stage of averaging procedure over thickness, is carried out. It is shown that for some values of parameters the stage of averaging procedure over thickness has almost no effect on the diffusion flow value. Full article

The current article studies optical solitons solutions for the dimensionless form of the stochastic resonant nonlinear Schrödinger equation (NLSE) with both spatio-temporal dispersion (STD) and inter-modal dispersion (IMD) having multiplicative noise in the itô sense. We will discuss seven laws of nonlinearities, namely, the Kerr law, power law, parabolic law, dual-power law, quadratic–cubic law, polynomial law, and triple-power law. The new auxiliary equation method is investigated. We secure the bright, dark, and singular soliton solutions for the model. Full article