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.     

Generalized solution of a one-dimensional quasilinear boundary value problem of the hydriding type with nonlinear boundary conditions and state evolution

We consider a quasilinear parabolic boundary value problem of the third kind on an interval. The coefficients of the partial differential equation and the right-hand sides in the boundary conditions and the evolution equation for the state vector nonlinearly depend on time, the point, the state vector, and the values of the solution at the endpoints. This problem generalizes a number of models of formation and decomposition of metal hydrides. For the simplest finite-difference scheme, we prove the uniform convergence to a continuous generalized solution of the boundary value problem. A sample model is given.     

Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval

We investigate the nonhomogeneous initial boundary value problem for the Camassa-Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.     

Numerical Solution of Time-Dependent Problems with a Fractional-Power Elliptic Operator

A time-dependent problem in a bounded domain for a fractional diffusion equation is considered. The first-order evolution equation involves a fractional-power second-order elliptic operator with Robin boundary conditions. A finite-element spatial approximation with an additive approximation of the operator of the problem is used. The time approximation is based on a vector scheme. The transition to a new time level is ensured by solving a sequence of standard elliptic boundary value problems. Numerical results obtained for a two-dimensional model problem are presented.     

Time step selection for the numerical solution of boundary value problems for parabolic equations

An algorithm is proposed for selecting a time step for the numerical solution of boundary value problems for parabolic equations. The solution is found by applying unconditionally stable implicit schemes, while the time step is selected using the solution produced by an explicit scheme. Explicit computational formulas are based on truncation error estimation at a new time level. Numerical results for a model parabolic boundary value problem are presented, which demonstrate the performance of the time step selection algorithm.     

Existence of positive solutions for Sturm-Liouville-like boundary value problem with p-Laplacian on time scales

We consider the existence of positive solutions for a class of Sturm-Liouville-like boundary value problem with p-Laplacian on time scales. Using the well known Krasnosel’ski’s fixed point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, two examples are given to illustrate the main results.     

The construction of solutions to pseudoparabolic equations in noncylindrical domains

We consider the first initial boundary value problem for pseudoparabolic equations in a domain with time dependent boundaries. Sufficient conditions on the coefficients of the equation and the boundary of the region are given in order that this initial boundary value problem has a unique solution.     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

Existence and Uniqueness of Solutions to Singular Higher Order Two-point BVPs on Time Scales

This paper investigates the existence and uniqueness of solutions for singular higher order boundary value problems on time scales by using mixed monotone method.The theorems obtained are very general.For the different time scale,the problem may be the corresponding continuous or discrete boundary value problem.     

Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation

An initial–boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial–boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time.     

Optimization of a nonlocal boundary control of vibrations of a string with a fixed end for an arbitrary time interval that is a multiple of 2l

We consider the optimization problem for a nonlocal boundary control of vibrations of an elastic string with fixed right end for an arbitrary sufficiently large time interval that is a multiple of twice the string length. We prove the existence and uniqueness of a generalized solution of the first boundary value problem and indicate an explicit expression for the solution. The optimal control is found in closed form.     

Stefan problem for the laplace equation with regard for the curvature of the free boundary

We consider a nonstationary problem with free boundary for an elliptic equation in the case where the value of the required function on an unknown boundary is proportional to the curvature of this boundary. We prove the existence of a solution in the small with respect to time in the spaces of smooth functions.     

Third order boundary value problem with finite spectrum on time scales

The eigenvalues of a class of third order boundary value problem on time scales is investigated. It is shown that this kind of third order boundary value problem has finite number of eigenvalues, and the same results on time scales are previously known only for even order cases. It can be illustrated that the number of eigenvalues depend on the partition of the time scale and the order of the equation.     

Motion of a rigid body in a viscous fluid

We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.     

Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

On the existence and uniqueness of a generalized solution of the mixed problem for the wave equation with the second and third boundary conditions

We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy.     

Global existence of solutions of initial boundary value problem for nonlocal parabolic equation with nonlocal boundary condition

We prove the global existence and blow-up of solutions of an initial boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.     

Lower bounds for the eigenvalues of first-order nonlinear Hamiltonian systems on time scales

We study first-order nonlinear planar Hamiltonian boundary value problems on time scales. Estimates on lower bounds for the eigenvalues of the problems are established by way of the Lyapunov inequality method. Our results are interpreted to nonlinear differential and difference planar Hamiltonian boundary value problems. As a special case, an estimate on lower bounds for eigenvalues of half-linear dynamic equations is obtained which generalizes and improves the existing ones to nonlinear Hamiltonian systems. Based on the main results, we establish existence and uniqueness of solutions of a related linear boundary value problem.     

时间测度上奇异高阶两点边值问题解的存在性与唯一性（英文）

This paper investigates the existence and uniqueness of solutions for singular higher order boundary value problems on time scales by using mixed monotone method.The theorems obtained are very general. For the different time scale, the problem may be the corresponding continuous or discrete boundary value problem.