Identification of Hydrodynamic Dispersion Tensor by Optimization Algorithm Based LBM/CMA-ES Combination

The hydrodynamic dispersion tensor (HDT) of a porous medium is a key parameter in engineering and environmental sciences. Its knowledge allows for example, to accurately predict the propagation of a pollution front induced by a surface (or subsurface) flow. This paper proposes a new mathematical model based on inverse problem-solving techniques to identify the HDT (noted D=) of the studied porous medium. We then showed that in practice, this new model can be written in the form of an integrated optimization algorithm (IOA). The IOA is based on the numerical solution of the direct problem (which solves the convection–diffusion type transport equation) and the optimization of the error function between the simulated concentration field and that observed at the application site. The partial differential equations of the direct model were solved by high resolution of (Δx=Δy=1 m) Lattice Boltzmann Method (LBM) whose computational code is named HYDRODISP-LBM (HYDRO-DISpersion by LBM). As for the optimization step, we opted for the CMA-ES (Covariance Matrix Adaptation-Evolution Strategy) algorithm. Our choice for these two methods was motivated by their excellent performance proven in the abundant literature. The paper describes in detail the operation of the coupling of the two computer codes forming the IOA that we have named HYDRODISP-LBM/CMA-ES. Finally, the IOA was applied at the Beauvais experimental site to identify the HDT D=. The geological analyzes of this site showed that the tensor identified by the IOA is in perfect agreement with the characteristics of the geological formation of the site which are connected with the mixing processes of the latter.

Effects of Ambience on Thermal-Diffusion Type Ga-doping Process for ZnO Nanoparticles

Various annealing atmospheres were employed during our unique thermal-diffusion type Ga-doping process to investigate the surface, structural, optical, and electrical properties of Ga-doped zinc oxide (ZnO) nanoparticle (NP) layers. ZnO NPs were synthesized using an arc-discharge-mediated gas evaporation method, followed by Ga-doping under open-air, N2, O2, wet, and dry air atmospheric conditions at 800 °C to obtain the low resistive spray-coated NP layers. The I–V results revealed that the Ga-doped ZnO NP layer successfully reduced the sheet resistance in the open air (8.0 × 102 Ω/sq) and wet air atmosphere (8.8 × 102 Ω/sq) compared with un-doped ZnO (4.6 × 106 Ω/sq). Humidity plays a key role in the successful improvement of sheet resistance during Ga-doping. X-ray diffraction patterns demonstrated hexagonal wurtzite structures with increased crystallite sizes of 103 nm and 88 nm after doping in open air and wet air atmospheres, respectively. The red-shift of UV intensity indicates successful Ga-doping, and the atmospheric effects were confirmed through the analysis of the defect spectrum. Improved electrical conductivity was also confirmed using the thin-film-transistor-based structure. The current controllability by applying the gate electric-field was also confirmed, indicating the possibility of transistor channel application using the obtained ZnO NP layers.

Two-phase problem with a free boundary for systems of parabolic equations with a nonlinear term of convection

Эта статья посвящена задаче со свободной границей для полулинейных параболических уравнений, в которой описывается феномен сегрегации местообитаний в популяционной экологии. Основная цель — показать глобальное существование, единственность решений проблемы. Предлагается двухфазная математическая модель со свободными границами для параболических уравнений типа реакция-диффузия. Установлены априорные оценки щаудеровского типа, на основе которых доказана однозначная разрешимость задачи. Неустойчивость каждого решения полностью определяется с помощью теоремы сравнения. This article is concerned with a free boundary problem for semilinear parabolic equations, wbich describes the habitat segregation phenomenon in population ecology. The main goal is to show global existence, the uniqueness of solutions to the problem. A two-phase mathematical model with free boundaries for parabolic equations of the reaction-diffusion type is proposed. A priori estimates of Schauder type are established, on the basis of which the unique solvability of the problem is proved. The instability of each solution is fully determined using the comparison theorem.

On the spread of ultrafine particulate matter: A mathematical model for motor vehicle emissions and their effects as an asthma trigger

Asthma is a respiratory disease that affects the lungs, with a prevalence of 339.4 million people worldwide [G. Marks, N. Pearce, D. Strachan, I. Asher and P. Ellwood, The Global Asthma Report 2018, globalasthmareport.org (2018)]. Many factors contribute to the high prevalence of asthma, but with the rise of the industrial age, air pollutants have become one of the main Ultrafine particles (UFPs), which are a type of air pollutant that can affect asthmatics the most. These UFPs originate primarily from the combustion of motor vehicles [P. Solomon, Ultrafine particles in ambient air. EM: Air and Waste Management Association’s Magazine for Environmental Managers (2012)] and although in certain places some regulations to control their emission have been implemented they might not be enough. In this work, a mathematical model of reaction–diffusion type is constructed to study how UFPs grow and disperse in the environment and in turn how they affect an asthmatic population. Part of our focus is on the existence of traveling wave solutions and their minimum asymptotic speed of pollutant propagation [Formula: see text]. Through the analysis of the model it was possible to identify the necessary threshold conditions to control the pollutant emissions and consequently reduce the asthma episodes in the population. Analytical and numerical results from this work prove how harmful the UFEs are for the asthmatic population and how they can exacerbate their asthma episodes.

Quadrinomial trees with stochastic volatility to value real options

PurposeThe purpose of this article is to propose a detailed methodology to estimate, model and incorporate the non-constant volatility onto a numerical tree scheme, to evaluate a real option, using a quadrinomial multiplicative recombination.Design/methodology/approachThis article uses the multiplicative quadrinomial tree numerical method with non-constant volatility, based on stochastic differential equations of the GARCH-diffusion type to value real options when the volatility is stochastic.FindingsFindings showed that in the proposed method with volatility tends to zero, the multiplicative binomial traditional method is a particular case, and results are comparable between these methodologies, as well as to the exact solution offered by the Black–Scholes model.Originality/valueThe originality of this paper lies in try to model the implicit (conditional) market volatility to assess, based on that, a real option using a quadrinomial tree, including into this valuation the stochastic volatility of the underlying asset. The main contribution is the formal derivation of a risk-neutral valuation as well as the market risk premium associated with volatility, verifying this condition via numerical test on simulated and real data, showing that our proposal is consistent with Black and Scholes formula and multiplicative binomial trees method.

A Computational Technique for Solving Singularly Perturbed Delay Partial Differential Equations

In this work, a matrix method based on Laguerre series to solve singularly perturbed second order delay parabolic convection-diffusion and reaction-diffusion type problems involving boundary and initial conditions is introduced. The approximate solution of the problem is obtained by truncated Laguerre series. Moreover convergence analysis is introduced and stability is explained. Besides, a test case is given and the error analysis is considered by the different norms in order to show the applicability of the method.

Instability of Traveling Pulses in Nonlinear Diffusion-Type Problems and Method to Obtain Bottom-Part Spectrum of Schrödinger Equation with Complicated Potential

The instability of traveling pulses in nonlinear diffusion problems is inspected on the example of Gunn domains in semiconductors. Mathematically, the problem is reduced to the calculation of the “energy” of the ground state in the Schrödinger equation with a complicated potential. A general method to obtain the bottom-part spectrum of such equations based on the approximation of the potential by square wells is proposed and applied. Possible generalization of the approach to other types of nonlinear diffusion equations is discussed.

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

In this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.