Mogtaba Mohammed

In this paper, we address the transport of a solute through a porous media that involves both convection and diffusion, along with a linear chemical reaction (desorption/adsorption) occurring on the pore surfaces, all of which are influenced by external nonlinear random forces. The mathematical representation of this model is a system of a non-linear stochastic convection-diffusion equation in the saturated fluid phase and a linear stochastic convection-diffusion equation on the surface of porosity coupled with a linear reaction term. We use the method of two-scale convergence with drift and probabilistic compactness results to obtain a homogenized model consisting of a nonlinear stochastic diffusion equation where the concentration of the fluid on the pore surface contributes to the diffusion coefficient of the homogenized model.

In this paper, we introduce a novel competitive stochastic predator-prey model by means of Holling type II with small random immigration. A Series of analytical results are conducted on the model such as the boundedness of the model’s solution, stochastic mean square stability, and the conditions of the persistence and extinction of the predator-prey population. All analytical results were validated with numerical simulations by choosing different sets of small random immigration. Conditions on the random immigration under which the system is stochastically stable are specified. In addition, small immigration has been shown to have a great impact on persistence and extinction of the system, which can play an important role in species conservation, especially by controlling the conditions of stochastic small immigration.

The aim of this research is to explore new homogenization results for a stochastic linear coupled thermoelastic model. We focus on a stochastic equation of motion coupled with a stochastic heat equation, representing the linear thermoelastic behavior of composite anisotropic materials with a periodic heterogeneous structure. These materials are subjected to random external forces and heat sources. By employing the periodic unfolding method and leveraging probabilistic compactness results from Prokhorov and Skorokhod, we obtain homogenization, convergence of the associated stochastic energies, and corrector results.

In this paper, we considered a stochastic model of chemical reactive flows acting through porous media under the influence of nonlinear external random fluctuations, where the interchanges of chemical flow across the skeleton's surface are represented by a nonlinear function. We studied the existence and uniqueness of strong probabilistic solutions for the model under consideration. We also show the positivity for the concentration of the solute in the fluid face as well as the concentration of reactants on the surface of the skeleton under extra reasonable assumptions on the data. Initially, we approximated the solution of the nonlinear stochastic diffusion equation using Galerkin's approximation, and obtained important bound estimates along with probabilistic compactness results. Thereafter, we passed the limit and obtained a weak probabilistic solution. This was followed by the path-wise uniqueness of the solution, which leads to the existence and uniqueness of strong probabilistic solutions as a result of Yamada-Watanabe's theorem. Finally, we discuss some important numerical applications such as Langmuir and Freundlich kinetics using the extended stochastic non-conforming finite element method to illustrate the efficiency of this approach and compare it to the deterministic approach in both cases. Let us mention that well-posedness, positivity, and numerical simulations have not been considered so far for such a nonlinear stochastic model.

In this paper, we introduce a novel stochastic prey-predator model under random small immigration. Mainly, we prove boundedness for the solution of the model using probabilistic and analytic types of inequalities. Furthermore, possible conditions on the immigration for achieving stochastic square stability are obtained. The immigration of both prey and predator is assumed to be either constant and stochastically perturbed or proportional to the population and stochastically perturbed. In all cases, we arrived at the fact that stability can only be achieved if the immigration is small enough. We also show that as random immigration increases, the dynamic becomes destabilized and could lead to chaos. Lastly, we perform a computational analysis in order to verify the obtained theoretical results.

In this paper, we construct a new model based on the coupling of thermoelasticity, plasma, and microelongation effect under volumetric absorption of laser pulse. Three different thermoelasticity theories are applied to construct the new model in a 2D thermoelastic semiconducting medium whose properties are temperature-dependent. The medium surface is exposed to laser radiation having spatial and temporal Gaussian distributions; in addition, the surface is considered traction-free. The general solutions were obtained analytically via Laplace and Fourier transformations; for Laplace inverse, we use the well-known Riemann sum approximation. As an application and consistency validation, silicon material is used.

In this work, we look at homogenization results for nonlinear hyperbolic problem with a non-local boundary condition. We use the periodic unfolding method to obtain a homogenized nonlinear hyperbolic equation in a fixed domain. Due to the investigation's peculiarity, the unfolding technique must be developed with special attention, creating an unusual two-scale model. We note that the non-local boundary condition caused a damping on the homogenized model.