SOLUTION TO A PARAMETRIC EQUATION WITH GENERALIZED BOUNDARY CONDITION IN TRANSPORT THEORY

This paper deals with the solution of a parametric equation with generalized boundary condiiton in transport theory. It gives the distribution of parameter (so called delta-eigenvalue [1]) with which the equation has non-zero solution. A necessary and sufficient condition for the existence of; he control critical eigenvalue delta0 is established.

Induced generalized exact boundary synchronizations for a coupled system of wave equations

Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.

Existence of Positive Solutions of Generalized Sturm-Liouville Boundary Value Problems for a Singular Differential Equation

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular differential equation with a parameter. Some sufficient conditions for the existence of positive solutions are established. In the last section, an example is presented to illustrate the applications of our main results.

Navier-Stokes/Allen-Cahn System with Generalized Navier Boundary Condition

In this paper,we focus on the immiscible compressible two-phase flow described by the coupled compressible Navier-Stokes system and the modified Allen-Cahn equations.The generalized Navier boundary condition and the relaxation boundary condition are established in order to solve the problem of moving contact lines on the solid boundary by using the principle of minimum energy dissipation.The existence and uniqueness for local strong solution in three dimensional bounded domain for this type of boundary value problem is obtained by the elementary energy method and the maximum principle.

Generalized Exact Boundary Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

This paper deals with the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in the framework of weak solutions.A necessary and sufficient condition for the generalized exact boundary synchronization is obtained,and some results for its generalized exactly synchronizable states are given.

Determination of generalized exact boundary synchronization matrix for a coupled system of wave equations

For a coupled system of wave equations with Dirichlet boundary controls,this paper deals with the possible choice of its generalized synchronization matrices so that the admissible generalized exact boundary synchronizations for this system are obtained.

EXPLICIT DISSIPATIVE SCHEMES FOR BOUNDARY PROBLEMS OF GENERALIZED SCHRDINGER SYSTEMS

In this paper, the author constructs a class of explicit schemes, spanning two time levels, forthe initial--boundary-value problems of generalized nonlinear Schrodinger systems, and proves theconvergence of these schemes with a series of prior estimates. For a single Schrodinger equation, theschemes are identical with those of the article [1].

PROBABILISTIC APPROACH TO SEMILINEAR AND GENERALIZED MIXED BOUNDARY VALUE PROBLEMS

A probabilistic approach is developed to solve semilinear and generalized mixed boundaryvalue problems involving Schrodinger operators. The results obtained in this paper generalize thecorresponding results of [1] and partly generalize the result of [2] as well.

The Estimates L_(1)-L_(∞) for the Reduced Radial Equation of Schrodinger

Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.

Analysis of thermal injuries using classical Fourier and DPL models for multi-layer of skin under different boundary conditions

In this paper,the temperature distribution in the multi-layer of the skin is studied when the skin surface is subjected to most generalized boundary condition.Our skin model consists of three layers known as the epidermis,dermis,and subcutaneous layers.All layers of skin are assumed to be connected with point of interface condition and taking the barrier in between each of the two layers by symmetric flux condition and analyzing each layer separately.The classical Fourier and non-Fourier(DPL)models are extended to analyze the behavior of heat transfer in the multi-layer of the skin.The Laplace transform technique is used to derive analytical solutions for the multi-layer of skin models.The effects of the variability of different parameters such as relaxation time,layer thickness,and different types of boundary conditions on the behavior of temperature distribution in the multi-layer of skin are analyzed and discussed in detail.All the effects are shown graphically.It has been observed that during temperature distribution in the multi-layer of skin,the measurement of skin damage is less on the DPL model(rq>Tt)in comparison to the classical Fourier model.

On a Class of Non-local Operators in Conformal Geometry

In this expository article, the authors discuss the connection between the study of non-local operators on Euclidean space to the study of fractional GJMS operators in conformal geometry. The emphasis is on the study of a class of fourth order operators and their third order boundary operators. These third order operators are generalizations of the Dirichlet-to-Neumann operator.

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

In this paper,an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods.In order to efficiently solve the generated linear large-scale system,the generalized minimal residual(GMRES)algorithm is applied.For accelerating the convergence rate of the it erative,the St rang-type,Chantype and P-type preconditioners are introduced.The suggested met hod can reach higher order accuracy both in space and in time than the existing met hods.When the used boundary value method is Ak1,K2-stable,it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1.It implies that the iterative solution is convergent rapidly.Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.

Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

In this paper,we compute a phase field(diffuse interface)model of CahnHilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids.The generalized Navier boundary condition proposed by Qian et al.[1]is adopted here.We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time.We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable.Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid.We also derive the discrete energy law for the droplet displacement case,which is slightly different due to the boundary conditions.The accuracy and stability of the scheme are validated by examples,results and estimate order.