An effective model of the dynamics of a barotropic gas with fast oscillating initial data

Under study are the classical three-dimensional Navier-Stokes equations of a compressible inhomogeneous viscous fluid in a smooth bounded domain endowed with no-slip conditions on the boundary of the domain and fast oscillating initial density distributions. The state equation of the medium is the state equation for a barotropic gas. We assume that the adiabatic constant is greater than 3. We give a rigorous derivation of the homogenization procedure as the frequencies of fast oscillations tend to infinity and obtain a limit effective model of the dynamics of a compressible viscous gas with fast oscillating initial data.     

Suboptimal control of a 2D molten carbonate fuel cell PDAE model

Numerical results for suboptimal boundary control of an integro partial differential algebraic equation system of dimension 28 are presented. The application is a molten carbonate fuel cell power plant. The technically and economically important fast tracking of the new stationary cell voltage during a load change is optimized. The nonstandard optimal control problem s.t. degenerated PDE is discretized by the method of lines yielding a very large DAE constrained standard optimal control problem. An index analysis is performed to identify consistent initial conditions.     

Global existence and blowup of solutions to a free boundary problem for mutualistic model

This article is concerned with a system of semilinear parabolic equations with a free boundary, which arises in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained. The asymptotic behavior of the free boundary problem is studied. Our results show that the free problem admits a global slow solution if the inter-specific competitions are strong, while if the inter-specific competitions are weak there exist the blowup solution and global fast solution.     

Efficient stochastic model for the point kinetics equations

A new stochastic model for the point kinetics equations with I-delayed neutron precursor groups is presented. In this stochastic model, the point kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and external neutrons source. The matrix form of the efficient stochastic model is solved by a semi-analytical method. The semi-analytical method is based on the exponential function of the coefficient matrix. The eigenvalues of the coefficient matrix and Gaussian elimination are used to calculate this exponential function. The mean and standard deviation of neutron and precursor populations of the efficient stochastic model with step, ramp, and sinusoidal reactivities are computed. The results of the efficient stochastic model are compared with the results of Allen's stochastic model for the point kinetics equations. This comparison confirms that the efficient stochastic model is an accurate model compared with the deterministic point kinetics equations. This stochastic model is efficient to study the natural behavior of neutron and precursor populations in the nuclear reactor dynamics.     

Global weak solutions for a viscous liquid–gas model with transition to single-phase gas flow and vacuum

This work deals with a viscous two-phase liquid–gas model relevant to the flow in wells and pipelines. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. The model is rewritten in terms of Lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially, and continuous at the boundary. Consequently, the initial masses involve a transition to single-phase gas flow and vacuum at the boundary. An appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses. These estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary. By combining these estimates with basic energy type of estimates, higher order regularity estimates are obtained. The existence of global weak solutions is then proved by showing compactness for a class of semi-discrete approximations.     

Sensitivity of a shallow-water model to parameters

An adjoint based technique is applied to a shallow water model in order to estimate the influence of the model’s parameters on the solution. Among parameters, the bottom topography, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients, Coriolis parameter and the amplitude of the wind stress tension are considered. Their influence is analyzed from three points of view:

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flexibility of the model with respect to a parameter that is related to the lowest value of the cost function that can be obtained in the data assimilation experiment that controls this parameter;

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possibility to improve the model by the parameter’s control, i.e., whether the solution with the optimal parameter remains close to observations after the end of control;

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sensitivity of the model solution to the parameter in a classical sense. That implies the analysis of the sensitivity estimates and their comparison with each other and with the local Lyapunov exponents that characterize the sensitivity of the model to initial conditions.

Two configurations have been analyzed: an academic case of the model in a square box and a more realistic case simulating Black sea currents. It is shown in both experiments that the boundary conditions near a rigid boundary highly influence the solution. This fact points out the necessity to identify optimal boundary approximation during a model development.     

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     

Boundary stabilization of a microbeam model

In this paper, we study the boundary stabilization of the deflection of a clamped-free microbeam, which is modeled by a sixth-order hyperbolic equation. We design a boundary feedback control, simpler than the one designed in Vatankhah et al,² that forces the energy associated to the deflection to decay exponentially to zero as the time goes to infinity. The rate in which the energy exponentially decays is explicitly given.     

Stochastic model for the nonlinear point reactor kinetics equations in the presence Newtonian temperature feedback effects

A stochastic model for the nonlinear point reactor kinetics equations with Newtonian temperature feedback and multi-group of precursor delayed neutrons is presented. This model is a couple of the stiff stochastic nonlinear differential equations. The matrix formula of this stochastic nonlinear model is solved by the analytical exponential technique (AET). This proposed technique is based on the integration factor, Euler’s method and the exponential function of the coefficient matrix. This exponential function is determined via the eigenvalues and corresponding eigenvectors of the coefficient matrix. The mean neutron population of the stochastic nonlinear model in the presence Newtonian temperature feedback and six-groups of delayed neutrons is computed for various cases of the external reactivity. The numerical results of the analytical exponential technique are compared with the results of the Euler–Maruyama method and the deterministic results. This comparison confirms that the AET for stochastic nonlinear model is efficient to study the natural behavior of neutron population in the presence temperature feedback effects and multi-group of precursor delayed neutrons.     

Non-extinction of a Fleming-Viot particle model

We consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.     

Optimal boundary controls for a phase field model

The phase field model is a nonlinear system of parabolic equationswhich describes the phase transitions between two differentphases, e.g. solid and liquid. In this paper, we consider ageneral optimal boundary control problem which is governed bythis model. The existence of the solutions of the phase fieldmodel is established by a rigorous analysis of the method oflines. The existence of the optimal solutions and the necessaryconditions for optimality are proved. For a special unconstrainedboundary control problem, we also prove some results concerningthe uniqueness of the optimal solutions. For a special constrainedboundary control problem, we obtain a result concerning thebang-bang principle.     

Equilibrium of a Degenerate Gas of Nucleons and Electrons in a Strong Magnetic Field

A model of a degenerate ideal gas of nucleons and electrons in a superstrong magnetic field is used to describe the state of matter in the central region of a strongly magnetized neutron star. The influence of a constant uniform superstrong magnetic field on the equilibrium conditions and the equation of state for the degenerate gas of neutrons, protons, and electrons is investigated in the framework of this model. The contribution determined by the interaction of the anomalous magnetic moments of the fermions with the magnetic field is taken into account. The influence of the superstrong magnetic field on the process of gravitational collapse of a magnetized neutron star is discussed under the assumption that the central region of the star consists mostly of degenerate neutrons. We show that if the densities of electrons, protons, and neutrons are relatively low depending on the field strength, the fermion gases in a superstrong uniform magnetic field become totally polarized with respect to the spin. We discuss the possibility of spontaneous magnetization occurring in a system of degenerate neutrons where the exchange interaction effects are taken into account.     

Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions

The existence of global smooth solutions to the multidimensional hydrodynamic model for plasmas of electrons and positively charged ions with insulating boundary conditions is shown under the assumption that the initial densities are close to a constant. Furthermore it is proved that the particle densities converge exponentially fast to the constant steady state.     

On a multiple credit rating migration model with stochastic interest rate

In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.     

Optimal control problem of a generalized Ginzburg–Landau model equation in population problems

In this paper, we consider the problem for distributed optimal control of the generalized Ginzburg–Landu model equation in population. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved, and the optimality system is established. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal control of time discrete two-phase flow governed by a diffuse interface model

We present results on optimal control of two-phase flows. The fluid is modeled by a thermodynamically consistent diffuse interface model and allows to treat fluids of different densities and viscosities. In earlier work we proposed an energy stable time discretization for this model that we now employ to derive existence of optimal controls for a time discrete optimal control problem. The control aim is to obtain a desired distribution of the two phases in the system. For this we investigate three control actions. We use tangential Dirichlet boundary control and distributed control. We further consider the inverse problem of finding an initial distribution such that the evolution over a given time horizon starting from this value is close to a desired distribution. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of phase boundary evolution in metal solidification for new thermodynamic parameters of the metal

The problem of controlling the phase boundary evolution in the course of solidification of metals with different thermodynamic properties is studied. The underlying mathematical model of the process is based on a three-dimensional nonstationary two-phase initial–boundary value problem of the Stefan type. The control functions are determined by optimal control problems, which are solved numerically with the help of gradient optimization methods. The gradient of the cost function is exactly computed by applying the fast automatic differentiation technique. The research results are described and analyzed. Some of them are illustrated.     

Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks

We consider a dynamic linear shallow shell model, subject to nonlinear dissipation active on a portion of its boundary in physical boundary conditions. Our main result is a uniform stabilization theorem which states a uniform decay rate of the resulting solutions. Mathematically, the motion of a shell is described by a system of two coupled partial differential equations, both of hyperbolic type: (i) an elastic wave in the 2-d in-plane displacement, and (ii) a Kirchhoff plate in the scalar normal displacement. These PDEs are defined on a 2-d Riemann manifold. Solution of the uniform stabilization problem for the shell model combines a Riemann geometric approach with microlocal analysis techniques. The former provides an intrinsic, coordinate-free model, as well as a preliminary observability-type inequality. The latter yield sharp trace estimates for the elastic wave—critical for the very solution of the stabilization problem—as well as sharp trace estimates for the Kirchhoff plate—which permit the elimination of geometrical conditions on the controlled portion of the boundary.     

Optimal control in a model of the motion of a viscoelastic medium with objective derivative

In this paper we consider the Jeffreys model of the motion of a viscoelastic incompressible medium with the Yaumann derivative. Within this model we study the optimal control problem for the right-hand sides of the initial boundary value problem. We prove the existence of the optimal strong solution.     

On the multiplicity of solutions of a simplified tumor growth model with a free boundary

We consider a mathematical model of tumor growth taking into account the spatial structure and include a different phases because of the lack of the nutriments. This model is expressed as a PDE on a spherical domain describing the tumor region. The nonlinearity of this PDE is discontinuous, and our problem can be regarded as a free boundary problem. In fact, the boundary of the tumor and another part in the interior are unknown. We obtain a multiplicity result of solutions together with some properties of the associated free boundaries.