Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches. This article was submitted by the authors in English.     

Eigenvalues of second-order difference equations with coupled boundary conditions

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.     

n阶非线性常微分方程两点及三点边值问题解的存在性的进一步结果

本文利用上、下解方法讨论了n阶非线性常微分方程y^(n)=f(t,y,y‘…,y^(n-1))满足下列非线性边界条件的边值问题解的存在性。     

Reissner厚板弹性弯曲的一般解析解

针对大型工程建设中的Reisner厚板弹性弯曲问题,本文采用复级数方法求解相应的常系数偏微分方程组的边值问题,并首次得到了任意边界条件下的一般解析解.该解形式简单,计算方便、可靠.以四边简支和三边固支一边自由两种支撑条件下厚板承受均布载荷为例进行了分析验算,与已有的计算结果相比,计算结果相当满意.同时本文还着重对解的收敛速度、正确性(合理性)及边界满足情况进行了考察.     

一类两参数非线性奇摄动边值问题解的渐近性态

本文研究了一类两参数非线性奇摄动边值问题的基本模型.利用奇摄动方法,对该问题解的结构在两个小参数相互关联的三种不同情形下作了讨论,得到了该问题的渐近解并证明了在三种情形下不同的解的结构与渐近性态.     

A posteriori error estimation and adaptive mesh-refinement techniques

We analyse three different a posteriori error estimators for elliptic partial differential equations. They are based on the evaluation of local residuals with respect to the strong form of the differential equation, on the solution of local problems with Neumann boundary conditions, and on the solution of local problems with Dirichlet boundary conditions. We prove that all three are equivalent and yield global upper and local lower bounds for the true error. Thus adaptive mesh-refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities. Some numerical examples prove the efficiency of the error estimators and the mesh-refinement techniques.     

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Vibration of non-uniform beams with different boundary conditions subjected to a moving mass is investigated. The beam is modeled using Euler–Bernoulli beam theory. Applying the method of eigenfunction expansion, equation of motion has been transformed into a number of coupled linear time-varying ordinary differential equations. In non-uniform beams, the exact vibration functions do not exist and in order to solve these equations using eigenfunction expansion method, an adequate set of functions must be selected as the assumed vibration modes. A set of polynomial functions called as beam characteristic polynomials, which is constructed by considering beam boundary conditions, have been used along with the vibration functions of the equivalent uniform beam with similar boundary conditions, as the assumed vibration functions. Orthogonal polynomials which are generated by utilizing a Gram–Schmidt process are also used, and results of their application show no advantage over the set of simple non-orthogonal polynomials. In the numerical examples, both natural frequencies and forced vibration of three different non-uniform beams with different shapes and boundary conditions are scrutinized.     

An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers

This paper consists in studying a mathematical model of solvent diffusion through the glassy polymers as a one-dimensional moving boundary problem with kinetic undercooling. We establish an iterative variable time-step method based on a nonstandard finite difference (NSFD) scheme to solve the considered moving boundary problem. The monotonicity and positivity of the numerical solution are proved. The numerical approach is investigated for three test problems composed of constant and inconstant diffusion coefficients for different values of parameters to demonstrate the validity and ability of the method.     

Hessian型方程Neumann边值问题的梯度估计

通过构造辅助函数,利用基本对称函数的性质以及函数在极大值点的性质,得到Hessian型方程S_k(D~2u-A(x,u,Du))=B(x,u)的梯度内估计,构造不同的辅助函数,分近边、边界和内部3种情形讨论该方程Neumann边值问题,进而得到全局梯度估计.     

Immobilization of smooth convex figures

We establish a curvature criterion to decide whether three points immobilize a plane convex figure with smooth boundary. Then we use it to prove in the affirmative the convex case of Kuperberg's Conjecture. Namely, we prove that any convex figure with smooth boundary, different from a circular disk, can be immobilized with three points.     

Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.     

Numerical solution of 2D elastostatic problems formulated by potential functions

2D linear elastostatic problems formulated in Cartesian coordinates by potential functions are numerically solved by network simulation method which allows an easy implementation of the complex boundary conditions inherent to this type of formulation. Four potential solutions are studied as governing equations: the general Papkovich–Neuber formulation, which is defined by a scalar potential plus a vector potential of two components, and the three simplified derived formulations obtained by deleting one of the three original functions (the scalar or one of the vector components). Application of this method to a rectangular plate subjected to mixed boundary conditions is presented. To prove the reliability and accurate of the proposed numerical method, as well as to demonstrate the suitability of the different potential formulations, numerical solutions are compared with those coming from the classical Navier formulation.     

On the numerical solution of the flow between a rotating and a stationary disk

The flow between two co-axial, infinite disks, one rotating with constant angular velocity and one stationary is treated in this paper. The problem is reduced to that of finding the solution of a two-point boundary value for a sixth order nonlinear ordinary differential equation and three boundary conditions at each of a finite interval. The numerical solutions are obtained by using a fourth order Runge-Kutta integration scheme in modification due to Gill and in conjunction with a modified shooting method to correct the initial guesses at one boundary. The numerical calculations for different Reynolds numbers are carried out. The results obtained by this method are compared with available results. The comparison shows excellent agreement.     

On the Cegrell classes

The Cegrell classes with zero boundary data are defined by certain decreasing approximating sequences of functions with different properties depending on the class in question. It is different for Cegrell classes which are given by a continuous function f, these classes are defined by an inequality. It is proved in this article that it is possible to define the Cegrell classes which are given by f in a similar manner as those classes with zero boundary data. An existence result for the Dirichlet problem for certain singular measures is proved. The article ends with three applications. Results connected to convergence in capacity, subextension of plurisubharmonic functions and integrability are proved.     

On exact solutions for thin film flows of a micropolar fluid

The present paper contains the exact solutions for thin film flows of a micropolar fluid. Thin films are considered for three different geometries namely: (i) flow down an inclined plane, (ii) flow on a moving belt and (iii) flow down a vertical cylinder. The exact solution is obtained for the velocity and micro-rotation in first two cases for both weak and strong concentrations of the microelements. However, the exact expression for the micro-rotation is obtained for the third case and a numerical solution is presented for the velocity. The micro-rotation and velocity fields are sketched and discussed for different values of the material parameters. It is observed that the rotation of the microelements at the boundary increases the velocity when compared with the case when there is no rotation at the boundary.     

Quenching rates for heat equations with coupled singular nonlinear boundary flux

We study finite time quenching for heat equations coupled via singular nonlinear bound-ary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.     

Vibration of a single-walled carbon nanotube embedded in an elastic medium under a moving internal nanoparticle

Vibration of single-walled carbon nanotube embedded in an elastic medium under excitation of a moving nanoparticle is analyzed in this paper. Based on the Winkler spring model and the Euler–Bernoulli beam model, the time-domain responses of the single-walled carbon nanotube subjected to the moving transverse load with three different boundary conditions are computed by using the Newmark method. The effects of velocity and the excitation frequency of the moving internal nanoparticle and boundary conditions on the dynamic deflections of the single-walled carbon nanotube are discussed.     

Buckling analyses of three characteristic-lengths featured size-dependent gradient-beam with variational consistent higher order boundary conditions

In this work, a three characteristic-lengths featured size-dependent gradient-beam is constructed by adopting the modified nonlocal model, resulting in much more general constitutive equation with stress gradient up to four-order and strain gradient to two-order. The six-order differential governing equation for transverse displacement is formulated. All boundary conditions especially variational consistent higher order boundary conditions of the present model are derived with the aid of weighted residual approach. The closed-form solutions to critical buckling loads under different sets of boundary conditions are systematically formulated with higher order boundary conditions incorporated. The numerical results show that both nonlocal parameters have significant effect on the buckling behaviors. Meanwhile, if two nonlocal parameters are taken as same, the present results cannot always reduce to that from Eringen's nonlocal model. Due to its clear physical meaning, the present model is expected to be widely adopted in mechanical analyses of nano-structures.     

Two perturbation formulations of the nonlinear dynamics of a cable excited by a boundary motion

A nonlinear cable excited by an inclined boundary motion, termed as cable's moving boundary problem, is attacked by two different perturbation approaches, i.e., the boundary modulation formulation and the quasi-static drift formulation. The former transforms the boundary motion into a weak modulation on cable's high-order dynamics, while the latter introduces a hybrid mode expansion using an empirical drift shape function. In both formulations, the inclined boundary motion induces three different excitation effects, i.e., longitudinal direct, vertical boundary kinematic, and high-order parametric, all of which being characterized by the parametric modulation factors. Detailed comparative studies indicate that the modulation factors in the two formulations are exactly equivalent to each other only if a new drift shape function, well defined in the boundary modulation formulation, is used for the quasi-static drift formulation. In contrast, the empirical shape functions lead only to an approximate equivalence for intermediate/large boundary motion inclinations. Moreover, for small inclinations, the two formulations induce possible quantitative and qualitative differences. The approximate analytical framework is validated and shown to be computationally efficient, by comparison with the finite difference method.     

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.