حسن خليل اسماعيل الروازق

This work investigates the convergence of the EM approximation for neutral-type stochastic FDEs with infinite delay in the space Cu. To ensure numerical accuracy and stability for systems with memory effects, we extend the analysis to demonstrate the squared mean convergence of the EM scheme to the exact solution under Lipschitz conditions. Asker’s work on distribution stability is expanded upon here. A numerical illustration of the impact of time step size on accuracy empirically supported the theoretical conclusions

This paper focuses on numerical analysis of Newtonian compressible fluid flow in a square cavity driven by a lid, where the motion of the fluid is generated by moving the upper wall with prescribed velocity. The numerical solution of the governing equations is performed within a Taylor–Galerkin/Pressure–Correction finite element framework (TGPC). The pattern of the flow is governed by Navier–Stokes equations, consisting of the continuity equation describing mass conservation and unsteady momentum equations written in Cartesian coordinates. The objective of this work is to investigate the role of low dynamic viscosity (μ) on some significant flow properties. The influence of the viscosity change on the velocity field, shear stress, pressure and density distributions is investigated in details. Numerical solutions show that, when (μ) is increased, the trajectory of velocity becomes smaller in magnitude, whereas the other hydrodynamic profiles become larger. In general, the expected tendencies are in qualitative agreement with basic physical principles of viscous compressible flows.

This work introduces an absorbing Discrete-Time Markov Chain (DTMC) model for a one-out-of-three cold-warm standby Cloud Server System (CSS) with a repairable switching mechanism. The model quantifies the degradation of the system leading to total failure—represented by five absorption stages—in scenarios where restoration of standby units is unfeasible (e.g., during disasters). A 17-state DTMC is formulated and analytically solved: the fundamental matrix N and absorption-probability matrix B are calculated to derive precise Absorption Probabilities (APs) and Mean Time to Absorption (MTTA) for each transient state. Monte-Carlo simulation, comprising 10,000 iterations, is used to generate the empirical distribution of Time-To-Absorption (TTA). Three key conclusions are drawn: (1) Absorbing State 13—defined as total server failure coupled with a functional switch—represents the predominant failure mode, accounting for up to 88% of the probability mass from various temporary states. (2) Transient States 1 (primary active) and 4 (warm failed) demonstrate the longest MTTA of around 24.5 time steps, while States 11 and 14 experience quick collapse with an MTTA of approximately 3.3 steps. (3) Dependability indicates that while State 14 exhibits strong initial dependability, it deteriorates significantly within five steps, whereas States 11 and 12 sustain over 80% survival probability up to step 15. These findings provide designers with clear directions: reinforce the switch path to State 13, prioritize predictive maintenance for States 11 and 14, and initiate mission-critical workloads in States 1 or 4 to prolong the grace period before irreversible failure

In this paper, we propose a new formula for backward fuzzy stochastic differential equations (BFSDEs) and prove that the approximation solutions of BFSDEs converge to the exact solution under Lipschitz conditions

In this paper, we study formula for the backward doubly fuzzy stochastic differential equations (BFDSDEs), In the beginning, we study some basic concepts, definitions, and Hypotheses to obtain the numerical scheme for BFDSDEs, as our scheme depends on the partition of interval [0, T]. In our work, we prove that under Lipschitz conditions, the approximation solution for the backward fuzzy doubly stochastic differential equations converges to the exact solution by using mean square error, and prove the existence and uniqueness of approximations solutions to BFDSDEs