A Mathematical Approach to the Property of One-dimensional Steady Solution of Reverse Smolder Waves

The mathematical property of one-dimensional steady solution for reverse smolder waves in the context of a model that permits both fuel-rich and fuel-lean has been studied using the method of analysis. Based on the equations and the boundary conditions some asymptotic properties of the solution at infinity are proved. It is shown that the value of oxygen or the mass of fuel （corresponding to the fuel-rich case and the fuel-lean case, respectively） tends to zero, and the temperature approaches to a fixed value. This is confirmed by other authors using large activation energy asymptotic methods.

TRANSONIC SHOCK SOLUTIONS TO THE EULER SYSTEM IN DIVERGENT-CONVERGENT NOZZLES

In this paper,we study the transonic shock solutions to the steady Euler system in a quasi-one-dimensional divergent-convergent nozzle.For a given physical supersonic inflow at the entrance,we obtain exactly two non-isentropic transonic shock solutions for the exit pressure lying in a suitable range.In addition,we establish the monotonicity between the location of the transonic shock and the pressure downstream.

POSITIVE STEADY STATES AND DYNAMICS FOR A DIFFUSIVE PREDATOR-PREY SYSTEM WITH A DEGENERACY

In this article, we consider positive steady state solutions and dynamics for a spatially heterogeneous predator-prey system with modified Leslie-Gower and Holling-Type II schemes. The heterogeneity here is created by the degeneracy of the intra-specific pressures for the prey. By the bifurcation method, the degree theory, and a priori estimates, we discuss the existence and multiplicity of positive steady states. Moreover, by the comparison argument, we also discuss the dynamical behavior for the diffusive predator-prey system.

Global Stability to Steady Supersonic Rayleigh Flows in One-Dimensional Duct

Heat exchange plays an important role in hydrodynamical systems,which is an interesting topic in theory and application.In this paper,the authors consider the global stability of steady supersonic Rayleigh flows for the one-dimensional compressible Euler equations with heat exchange,under the small perturbations of initial and boundary conditions in a finite rectilinear duct.

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.

Remarks on Classical Solutions to Steady Quantum Navier-Stokes Equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier- Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier-Stokes equations, when the nonzero boundary value u0 satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u0. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier-Stokes equations.

Improvement of Convergence to Steady State Solutions of Euler Equations with Weighted Compact Nonlinear Schemes

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes （WCNS） [Deng X. and Zhang H. （2000）, J. Comput. Phys. 165, 22–44 and Zhang S., Jiang S. and Shu C.-W. （2008）, J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. （2007）, J. Sci. Comput. 31, 273–305 and Zhang S., Jiang S and Shu. C.-W. （2011）, J. Sci. Comput. 47, 216–238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.

A brief review on the convergence to steady state solutions of Euler equations with high-order WENO schemes

Weighted essentially non-oscillatory(WENO)schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions.However,like many other high-order shock capturing schemes,WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave.This is a long-standing difficulty for high-order shock capturing schemes.In recent years,this non-convergence problem has been studied extensively for WENO schemes.Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations,which are at the small local truncation error level but prevent the residue to settle down to machine zero.Several strategies have been proposed to reduce these slight post shock oscillations,including the design of new smoothness indicators for the fifth-order WENO scheme,the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy,and the design of a new type of WENO schemes.With these strategies,the convergence to steady states is improved significantly.Moreover,the strategies are applicable to other types of weighted schemes.In this paper,we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.

A Single Server Queue with Coxian-2 Service and One-Phase Vacation (M/C-2/M/1 Queue)

In this paper, we study a single server queueing system with Coxian-2 service. In Particular, we study M/C-2/M/1 queue with Coxian-2 service and exponential vacation. We assume that units (customers) arrive at the system one by one in a Poisson process and the server provides one-by-one service based on first in first out (FIFO) rule. We obtained the steady state queue size distributions in terms of the probability generating functions, the average number of customers and their average waiting time in the system as well as in the queue.

ASYMPTOTIC BEHAVIOR OF TIME-DEPENDENT SOLUTIONS TO SEMICONDUCTOR EQUATIONS

This paper considers the relationship between the time dependent solutions and the steady state solutions of semiconductor equations under the thermal equilibrium conditions. The asymptotic behavior of the time dependent solution is obtained.

Analysis of a Nutrient-phytoplankton Model in the Presence of Viral Infection

In this paper,a system of reaction-diffusion equations arising in a nutrient-phytoplankton populations is investigated.The equations model a situation in which phytoplankton population is divided into two groups,namely susceptible phytoplankton and infected phytoplankton.A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given.If the diffusion coefficient of the infected phytoplankton is treated as bifurcation parameter,non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.

A Semi-Implicit Fractional Step Method Immersed Boundary Method for the Numerical Simulation of Natural Convection Non-Boussinesq Flows

The paper presents a novel pressure-corrected formulation of the immersed boundary method(IBM)for the simulation of fully compressible non-Boussinesq natural convection flows.The formulation incorporated into the pressure-based fractional step approach facilitates simulation of the flows in the presence of an immersed body characterized by a complex geometry.Here,we first present extensive grid independence and verification studies addressing incompressible pressure-driven flow in an extended channel and non-Boussinesq natural convection flow in a differentially heated cavity.Next,the steady-state non-Boussinesq natural convection flow developing in the presence of hot cylinders of various diameters placed within a cold square cavity is thoroughly investigated.The obtained results are presented and analyzed in terms of the spatial distribution of path lines and temperature fields and of heat flux values typical of the hot cylinder and the cold cavity surfaces.Flow characteristics of multiple steady-state solutions discovered for several configurations are presented and discussed in detail.