The numerical simulation of the noncharring pyrolysis process and fire development within a compartment

A pyrolysis model for noncharring solid fuels is presented in this paper. Model predictions are compared with experimental data for the mass loss rates of polymethylmethacrylate (PMMA) and very good agreement is achieved. Using a three-dimensional CFD environment, the pyrolysis model is then coupled with a gas-phase combustion model and a thermal radiation model to simulate fire development within a small compartment. The numerical predictions produced by this coupled model are found to be in very good agreement with experimental data. Furthermore, numerical predictions of the relationship between the air entrained into the fire compartment and the ventilation factor produce a characteristic post-flashover linear correlation with constant of proportionality 0.38 kg/sm^5/2. The simulation results also suggest that the model is capable of predicting the onset of “flashover” and “post-flashover” type behaviour within the fire compartment.     

Analytical and numerical investigations of a batch crystallization model

This article is concerned with the analytical and numerical investigations of a one-dimensional population balance model for batch crystallization processes. We start with a one-dimensional batch crystallization model and prove the local existence and uniqueness of the solution of this model. For this purpose Laplace transformation is used as a basic tool. A semi-discrete high resolution finite volume scheme is proposed for the numerical solution of the current model. The issues of positivity (monotonicity), consistency, stability and convergence of the proposed scheme for the current model are analyzed and proved. Finally, we give a numerical test problem. The numerical results of the proposed high resolution scheme are compared with the solution of the reduced four-moments model and the first-order upwind scheme.     

Dynamics of a viral infectiology under treatment

This paper deals with a nonlinear model of the viral dynamics which describes the interactions between the human immune system and the virus. The novelty of this work is the introduction of combined treatments in the dynamics to modify the model. We investigate the qualitative behavior of the model and find a threshold parameter that guarantees the asymptotic stability of the equilibrium points, this parameter is known as the basic reproduction number. We estimated the parameters of the model by least-squares minimization between the numerical solution of the system and clinical data of cell cultures. It is also demonstrated that critical drug efficacy in terms of the model parameter is greatly useful to curtail the spreading of the disease.     

Two-compartment modeling and dynamics of top-sprayed fluidized bed granulator

A top Spray Fluidized Bed Granulator (SFBG) is being modeled and analyzed with the help of population balance equation (PBE) and analytical as well as numerical results. The mathematical model for SFBG is derived using the concept of compartment modeling. The granulator is divided into two compartments, a wet compartment in which aggregation is the dominant process, and a dry compartment that is dominated by breakage. A new discretization is given to solve the model which is based on the idea of conserving the important properties of the system. The numerical results of the moments derived by the new discretization are validated against the developed exact results for different combinations of the aggregation and breakage kernels. The model is also tested for physical tractable kernel and the numerical results are authenticated with the results of constant volume Monte Carlo. The two-compartment model is shown to behave dynamically in a distinctly different manner to the simpler one-compartment model. The most critical parameter is the exchange flow between the compartments. When the characteristic time for this flow is low relative to the rates of aggregation and breakage, the effect of breakage is amplified disproportionally relative to breakage, and whereas the single-compartment granulator always reaches steady state between the rates of aggregation and breakage, the two-compartment model may, under some conditions, lead to continuously decreasing size under the dominance of breakage.     

On a thermomechanical milling model

This paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions.     

Three-dimensional non-Darcy flow analysis using a hybrid numerical model

A hybrid numerical model is developed for the simulation of three-dimensional, unsteady non-Darcy flow through an unconfined aquifer. The major problem in analysing flow through unconfined aquifers is that they involve two boundaries, namely a surface of seepage and a free surface, the location of which is not known beforehand. The model that is presented here determines these boundaries via a two stage modelling technique. In the first stage a one-dimensional finite difference model is used to estimate the surface of seepage height whereas in the second stage a vertically integrated finite element model determines the free surface solution within the flow domain. A comparison between numerical and experimental results is included which indicates the sensitivity of the numerical solution to the selected aquifer parameters, particularly to those associated with the determination of the height of the surface of seepage.     

Stability and Asymptotic Behaviour for the Numerical Solution of a Reaction--Diffusion Model for a Deterministic Diffusive Epidemic

The aim of this paper is to outline the numerical solution ofa reaction—diffusion system describing the evolution ofan epidemic in an isolated habitat. The model we consider isdescribed by two weakly coupled semi-linear parabolic equationsand we introduce a finite difference scheme for its numericalsolution. We study the behaviour of the exact solution by meansof the numerical scheme. We show the positivity, the decreaseand the decay to extinction of the numerical solution. Finallywe report the results of the numerical tests; in these simulationswe observe that the asymptotic behaviour of the reaction-diffusionsystem is the same as that of the associated ODE system (Kermack—McKendrickmodel).     

Numerical schemes for a size-structured cell population model with equal fission

We study numerically the evolution of a size-structured cell population model, with finite maximum individual size and minimum size for mitosis. We formulate two schemes for the numerical solution of such a model. The schemes are analysed and optimal rates of convergence are derived. Some numerical experiments are also reported to demonstrate the predicted accuracy of the schemes. We also consider the behaviour of the methods with respect to the different discontinuities that appear in the solution to the problem and the stable size distribution. In addition, the numerical schemes are used to study asynchronous exponential growth.     

A stabilized method for a secondary consolidation Biot's model

This work deals with the numerical solution of a secondary consolidation Biot's model. A family of finite difference methods on staggered grids in both time and spatial variables is considered. These numerical methods use a weighted two‐level discretization in time and the classical central difference discretization in space. A priori estimates and convergence results for displacements and pressure in discrete energy norms are obtained. Numerical examples illustrate the convergence properties of the proposed numerical schemes, showing also a non‐oscillatory behavior of the pressure approximation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Numerical identification of the leading coefficient of a parabolic equation

For a multidimensional parabolic equation, we study the problem of finding the leading coefficient, which is assumed to depend only on time, on the basis of additional information about the solution at an interior point of the computational domain. For the approximate solution of the nonlinear inverse problem, we construct linearized approximations in time with the use of ordinary finite-element approximations with respect to space. The numerical algorithm is based on a special decomposition of the approximate solution for which the transition to the next time level is carried out by solving two standard elliptic problems. The capabilities of the suggested numerical algorithm are illustrated by the results of numerical solution of a model inverse two-dimensional problem.     

A numerical study on a nonlinear conservation law model pertaining to manufacturing system

This paper deals with a numerical scheme applied to a conservation law model of manufacturing system incorporating yield loss. Yield loss involving a singular term has been considered. Even though an explicit form of the material density in a production system can be obtained under certain assumptions, in general, it is difficult to get an explicit form of the material density. On the other hand, the singular term in a conservation law model imposes severe challenges for the numerical approximations on regular grids. Moreover, an approximate solution often converges to a wrong weak solution. A finite volume type numerical scheme has been studied. The convergence of the numerical solution towards entropy solution (in the Kruzkov sense) is proved. Numerical experiments are presented to get the overview of density distribution and outflux of the system.     

Numerical analysis of a corrected Smagorinsky model

The classical Smagorinsky model's solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This model has recently been corrected to incorporate this flow and still be well-posed. Herein we first develop some basic properties of the corrected model. Next, we perform a complete numerical analysis of two algorithms for its approximation. They are tested and proven to be effective.     

Numerical resolution for a compressible two-phase flow model based on the theory of thermodynamically compatible systems

This paper is concerned with the numerical solution of the equations governing two-phase gas-solid mixture in the framework of thermodynamically compatible systems theory. The equations constitute a non-homogeneous system of nonlinear hyperbolic conservation laws. A total variation diminishing (TVD) slope limiter centre (SLIC) numerical scheme, based on the splitting approach, is presented and applied for the solution of the initial-boundary value problem for the equations. The model equations and the numerical methods are systematically assessed through a series of numerical test cases. Strong numerical evidence shows that the model and the methods are accurate, robust and conservative. The model correctly describes the formations of shocks and rarefactions in two-phase gas-solid flow.     

A Differential Games Solution to a Logarithmic Advertising Model

A logarithmic excess-advertising model of a duopoly is presented, and Nash optimal open-loop advertising strategies are determined. It turns out that if the two firms use different discount rates, then the optimal strategies will be exponentially decreasing. However, in this case the state equation has no nice solution and must be solved by numerical methods. When both firms use the same discount rate, then the state equation has a simple solution. This solution is also valid for the case where no discounting is performed. Furthermore, when no discounting is performed, the optimal strategies will be simple time-linear decreasing strategies. Finally, it is studied how the optimal strategies and trajectories depend on the parameters of the model.     

The reconstruction of a solely time-dependent load in a simply supported non-homogeneous Euler–Bernoulli beam

In this paper, the theoretical and numerical determination of a solely time-dependent load distribution is investigated for a simply supported non-homogeneous Euler–Bernoulli beam. The missing source is recovered from an additional “local” integral measurement. The existence and uniqueness of a solution to the corresponding variational problem is proved by employing Rothe’s method. This method also reveals a time-discrete numerical scheme based on the backward Euler method to approximate the solution. Corresponding error estimates are proved and assessed by two numerical experiments.     

Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

On a coupled system of reaction–diffusion-transport equations arising from catalytic converter

This paper is concerned with two mathematical models which describe the transient behavior of a catalytic converter in automobile engineering. The first model consists of a coupled system of a heat-conduction equation and two integral equations while the second model involves only one integral equation. It is shown that for any nonnegative initial and boundary functions the three-equation model has a unique bounded global solution while the solution of the two-equation model blows up in finite time. The proof for the global existence and finite-time blow-up property of the solution is by the method of upper and lower solutions and its associated monotone iteration. This method can be used to develop computational algorithms for numerical solutions of the coupled systems.     

Numerical analysis of a mathematical model for propagation of an electrical pulse in a neuron

This article is devoted to the study of a mathematical model arising in the mathematical modeling of pulse propagation in nerve fibers. A widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations. When considered as part of a particular initial boundary value problem the equation models the electrical activity in a neuron. A small perturbation parameter ε is introduced to the highest order derivative term. The parameter if decreased, speeds up the fast variables of the model equations whereas it does not affect the slow variables. In order to formally reduce the problem to a discussion of the moment of fronts and backs we take the limit ε → 0. This limit is singular and is therefore the solution tends to a slowly moving solution of the limiting equation. This leads to the boundary layers located in the neighborhoods of the boundary of the domain where the solution has very steep gradient. Most of the classical methods are incapable of providing helpful information about this limiting solution. To this effort a parameter robust numerical method is constructed on a piecewise uniform fitted mesh. The method consists of standard upwind finite difference operator. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives. A parameter uniform error estimate for the numerical scheme so constructed is established in the maximum norm. It is then proven that the numerical method is unconditionally stable and provides a solution that converges to the solution of the differential equation. A set of numerical experiment is carried out in support of the predicted theory, which validates computationally the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

一类具有脉冲免疫治疗的HIV-1感染模型的动力学分析

该文基于一类HIV-1感染免疫治疗模型，研究了一类具有脉冲免疫治疗的HIV-1感染模型.借助脉冲微分方程理论，研究了脉冲免疫治疗模型解的非负性和一致有界性.利用Floquet乘子理论和微分方程的比较定理，推导出脉冲免疫模型无感染周期解局部和全局渐近稳定以及HIV-1一致持续的阈值条件.通过数值模拟，比较了3种不同治疗方案的治疗效果，验证了脉冲免疫治疗的有效性.数值模拟结果表明，当药物输入量足够大或服药间隔适当短时，从理论上可以有效控制甚至根除病毒.     

Global analysis of a delayed epidemic dynamical system with pulse vaccination and nonlinear incidence rate

This study explores the influence of epidemics by numerical simulations and analytical techniques. Pulse vaccination is an effective strategy for the treatment of epidemics. Usually, an infectious disease is discovered after the latent period, H1N1 for instance. The vaccinees (susceptible individuals who have started the vaccination process) are different from both susceptible and recovered individuals. So we put forward a SVEIRS epidemic model with two time delays and nonlinear incidence rate, and analyze the dynamical behavior of the model under pulse vaccination. The global attractivity of ‘infection-free’ periodic solution and the existence, uniqueness, permanence of the endemic periodic solution are investigated. We obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. The main feature of this study is to introduce two discrete time delays and impulse into SVEIRS epidemic model and to give pulse vaccination strategies.