Approximate method for the numerical solution of singular perturbation problems

We present an approximate method for the numerical solution of linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is divided into inner and outer region problems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem. In turn, the outer region problem is also modified and the resulting problem is efficiently treated by employing the trapezoidal formula coupled with discrete invariant imbedding algorithm. The proposed method is iterative on the terminal point. Some numerical experiments have been included to demonstrate its applicability.     

Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.     

Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.     

Optimal resource allocation program in a two-sector economic model with a Cobb-Douglas production function

We consider the resource allocation problem for a two-sector economic model with a two-factor Cobb-Douglas production function on a finite time horizon with a terminal functional. The problem is reduced to some canonical form by a scaling of the phase variables and time. We prove the optimality of the extremal solution constructed on the basis of the maximum principle. The solution of the boundary value problem of the maximum principle is constructed in closed form for three cases of location of the initial plant state.     

ASYMPTOTIC BEHAVIOUR OF A DIFFUSION SYSTEM WITH TIME DELAY IN POPULATION DYNAMICS

ＡＳＹＭＰＴＯＴＩＣＢＥＨＡＶＩＯＵＲＯＦＡＤＩＦＦＵＳＩＯＮＳＹＳＴＥＭＷＩＴＨＴＩＭＥＤＥＬＡＹＩＮＰＯＰＵＬＡＴＩＯＮＤＹＮＡＭＩＣＳＤｉｎｇＣｈｏｎｇｗｅｎ(丁崇文）（ＦｕｊｉａｎＥｄｕｃａｔｉｏｎａｌＣｏｌｌｅｇｅ，福建教育学院，邮编：３５?..     

Local-in-time well-posedness of a regularized mathematical model for silicon MESFET

We prove the local-in-time well-posedness of the initial boundary value problem for a system of quasilinear equations. This system is used for finding numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). The initial boundary value problem has the following peculiarities: the quasilinear system is not a Cauchy-Kovalevskaya-type system; the boundary is a non-smooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to an integro-differential system. The local-in-time existence of a weakened generalized solution of this system is then proved by the fixed-point argument.     

Local-in-time well-posedness of a regularized mathematical model for silicon MESFET

We prove the local-in-time well-posedness of the initial boundary value problem for a system of quasilinear equations. This system is used for finding numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). The initial boundary value problem has the following peculiarities: the quasilinear system is not a Cauchy-Kovalevskaya-type system; the boundary is a non-smooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to an integro-differential system. The local-in-time existence of a weakened generalized solution of this system is then proved by the fixed-point argument.     

Numerical solution of singular perturbation problems via deviating arguments

We present a new approach to numerically solving linear, singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in the implicit form is derived. Then, the outer region problem is solved as a two point boundary value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, a new inner region problem is obtained and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Some numerical examples have been solved to demonstrate the applicability of the method.     

Application of nonclassical separation of variables to a nonlocal heat problem

We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem.     

Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes

The solution of the three-dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A ‘tensor-product’ decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence of the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence.     

A SINGULAR SINGULARLY-PERTURBED BOUNDARY VALUE PROBLEM

ＡＳＩＮＧＵＬＡＲＳＩＮＧＵＬＡＲＬＹ－ＰＥＲＴＵＲＢＥＤＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭ￥ＬｉｎＷｕｚｈｏｎｇ（林武忠）；ＷａｎｇＺｈｉｍｉｎｇ（汪志鸣）（ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，上海华东师范大学，邮编：２０００...     

Integral equations in a diffraction problem on a locally inhomogeneous medium interface

We study the diffraction of an E-polarized field on a locally inhomogeneous interface of transparent media. We prove the unique solvability of the boundary value diffraction problem and obtain integral representations of the solution. We derive a system of integral equations equivalent to the original boundary value problem and prove a solvability theorem for this system.     

Quenching Time Estimates for Semilinear Parabolic Equations Controlled by Two Absorption Sources in Control System

This paper deals with the quenching solution of the initial boundary value problem for aclass of semilinear reaction-diffusion equation controlled by two absorption sources in control system and estimate upper bound and lower bound of the quenching time. We point that the number of absorption sources influences the time of quenching phenomenon.The solution can solve some boundary value problem in control system.     

四元数广义正则函数的斜微商边值问题

该文前半部分研究四元数广义正则函数的斜微商边值问题解的存在唯一性，后半部分在闻国格教授工作[2]的基础上，研究会有四个自变数一队退化椭圆组的斜微商边值问题解的存在性．     

Singularly perturbed partially dissipative systems of equations

Theoretical and Mathematical Physics - We construct an asymptotic expansion in a small parameter of a boundary layer solution of the boundary value problem for a system of two ordinary differential...     

Homogenization in the Theory of Viscoplasticity

We study the homogenization of the quasistatic initial boundary value problem with internal variables which models the deformation behavior of viscoplastic bodies with a periodic microstructure. This problem is represented through a system of linear partial differential equations coupled with a nonlinear system of differential equations or inclusions. Recently it was shown by Alber [2] that the formally derived homogenized initial boundary value problem has a solution. From this solution we construct an asymptotic solution for the original problem and prove that the difference of the exact solution and the asymptotic solution tends to zero if the lengthscale of the microstructure goes to zero. The work is based on monotonicity properties of the differential equations or inclusions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-dimensional rotating waves in a functional-differential diffusion equation with rotation of spatial arguments and time delay

We consider the Neumann boundary value problem for a parabolic functional-differential equation in a disk. We describe spatially inhomogeneous solutions in the form of rotating waves branching from the homogeneous stationary solution in the case of an Andronov-Hopf bifurcation. By passing to a moving coordinate system and by reducing the original problem to a stationary boundary value problem for a partial differential equation with a deviating argument, we prove the existence of rotating waves appearing in the disk under the Andronov-Hopf bifurcation.     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。