Eigenvalues and Eigenfunctions of the Laplace Operator on an Equilateral Triangle for the Discrete Case

A discretized boundary value problem for the Laplace equation with the Dirichlet and Neumann boundary conditions on an equilateral triangle with a triangular mesh is transformed into a problem of the same type on a rectangle. Explicit formulae for all eigenvalues and all eigenfunctions are given.     

Reconstruction of norm by geometry of minimal networks

An inverse problem to the Steiner minimal tree searching problem in a normed space is studied. Namely, let a normed space be given and let all Steiner minimal trees be known in this space. The problem is to describe all norms with the same Steiner minimal trees for all finite boundary sets as in the given space. The paper presents a review of known results on the problem and announces the uniqueness of the set of Steiner minimal trees for any two-dimensional space with a strongly convex and differentiable norm.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Recovery of the C ∞ jet from the boundary distance function

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the ??boundary rigidity problem??. If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the Taylor series of the metric tensor (C ^?? jet) at the boundary (see Lassas et?al. (Math Ann 325:767?C793, 2003), Uhlmann et?al. (Adv Appl Math 31:379?C387, 2003)). In this paper we give a new reconstruction procedure for the C ^?? jet at non-concave points on the boundary using the localized boundary distance function. A closely related problem is the ??lens rigidity problem??, which asks whether the lens data determine metric structure uniquely. Lens data include information of boundary distance function, the lengths of all geodesics, and the locations and directions where geodesics hit the boundary. We give the first examples that show that lens data cannot uniquely determine the C ^?? jet. The example include two manifolds with the same boundary and the same lens data, but different C ^?? jets. With some additional careful work, we can find examples with different C ¹ jets, which means the boundaries in the two lens-equivalent manifolds have different second fundamental forms.     

Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Differential-Operator Equation with Spectral Parameter Quadratically Occurring in the Boundary Condition

The asymptotic behavior of eigenvalues of a boundary value problem for a secondorder differential-operator equation in a separable Hilbert space on a finite interval is studied for the case in which the same spectral parameter occurs linearly in the equation and quadratically in one of the boundary conditions. We prove that the problem has a sequence of eigenvalues converging to zero.     

Integral formulae in the coupled problem of the thermoelasticity of an inhomogeneous body. Application in the mechanics of composite materials

A coupled unsteady problem of thermoelasticity for an inhomogeneous body, described by a system of four second-order partial differential equations with coefficients that vary depending on the coordinates, is considered, and the same problem for a homogeneous body of the same shape (the concomitant problem) is examined together with this original problem. Integral formulae are obtained that allow one to express the displacements and temperature in the original problem in terms of the displacements and temperature in the concomitant problem. Integral formulae are used to represent the solution of the original problem in the form of series over all possible derivatives of the solution of the concomitant problem. A system of recurrence problems is written for the coefficients of these series. Expressions are found for the coefficients of the concomitant problem (effective coefficients) and special boundary value problems are formulated, from the solution of which specific expressions are found for the effective thermoelasticity coefficients. A theorem concerning the fact that the effective coefficients satisfy the physicomechanical constraints imposed on the thermoelastic constants of real bodies is proved. The case of a layer that is inhomogeneous in its thickness is considered and explicit analytical expressions for all the thermoelasticity coefficients are obtained for it. The case when the thermoelasticity coefficients depend periodically on the coordinates is examined in detail.     

Analytical solution for the evolution of exit point along the sloping seepage face

This paper developed an analytical solution for the problem of exit point evolution on the seepage face in the unconfined aquifer with sloping interface. A theoretical model for the groundwater drawdown problem in a half‐infinite aquifer with a sloping boundary is built in accordance with the linearized one‐dimensional Boussinesq equation and the Neumann boundary condition at the seepage point. The homotopy analysis method is then adopted for solving this dynamic boundary problem. By constructing two continuous deformations, the original problem could be converted into a group of subproblems with the same physical essence and similar mathematical solutions. To compare this analytical solution, a numerical model based on the finite volume method is developed, which employs adaptive grids to settle the dynamic boundary condition. The comparisons show that the analytical solution agrees with the numerical model well. The results are useful for the quantification of various hydrological problems. The methodology applied in this study is referential for other dynamic boundary problems as well.     

带谱参数边界条件的四阶边值问题的矩阵表示

研究了一类具有有限谱的带有谱参数边界条件的四阶微分方程边值问题及其矩阵表示,证明了对任意正整数m,所考虑的问题至多有2m+6个特征值,进一步给出这类带有谱参数边条件的四阶边值问题与一类矩阵特征值问题之间在具有相同特征值的意义下是等价的.     

Exact boundary controllability in problems of transmission for the system of electromagneto‐elasticity

This paper considers transmission problem for the system of electromagneto‐elasticity having piecewise constant coefficients in a bounded domain. The result on exact boundary controllability is obtained provided the interfaces, where the coefficients have a jump discontinuity, are all star‐shaped with respect to one and the same point and the coefficients satisfy a certain monotonicity conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

On the uniqueness of a solution of a two-phase free boundary problem

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this free boundary problem by constructing counter-examples in various cases in any dimension. In one-dimension, a bifurcation phenomenon presents and the uniqueness problem has been completely analyzed. In fact, the critical case signifies the change from uniqueness to non-uniqueness of a solution of the free boundary problem. Non-uniqueness of a solution of the free boundary problem suggests different physical stationary states caused by different processes, such as melting of ice or solidification of water, even with the same prescribed boundary data. However, we prove that a uniqueness theorem is true for the initial-boundary value problem of an ε-evolutionary problem which is the smoothed two-phase parabolic free boundary problem.     

About the projection of kinematical quantities within the FE2-approach

Within the FE²-approach strain–like quantities are projected to the boundary of the attached microstructure yielding Dirichlet boundary conditions for the microscopic boundary value problem. By use of the divergence theorem one can show that the projection of gradient information to the boundary of the attached microstructure fulfills the condition that the volumetrical average of the strain–like quantities in the inner of the microstructure is exactly the same as the projected quantity, provided that the microstructure is continous. However, within the Cosserat continuum theory [1] the strain tensor depends directly on the extra rotation and not on the gradient of it. Thus, the projection can not be carried out in the same way as for gradient information. The present approach shows how to circumvent this problem by giving the extra–rotation within the Cosserat continuum theory a strong kinematical interpretation in the form of a displacement field, which shows the same effects as the extra–rotation itself. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comment on “Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet”

The article named above appeared recently in Applied Mathematics Letters and investigated a boundary value problem governing viscous flow over a nonlinearly stretching sheet. The authors of the work assert existence and (under certain restrictions) uniqueness of a solution to the problem for all relevant values of the parameter governing the stretching rate of the sheet. Unfortunately, several proofs presented in the article are incorrect. We will prove that for a range of parameter space the solution to the BVP is not unique. For these parameter values there are infinitely many solutions to the problem. The same incorrect analysis is reproduced in several other papers (see the references). Some of the claims of these papers are contradicted by established results on, for example, the Falkner–Skan problem.     

Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.     

Boundaries for Banach spaces determine weak compactness

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James’ sup-characterization as a corollary.     

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.     

具有边梁加固的板弯曲问题的变分-差分方法研究

具有边梁加固的板的弯曲问题,其平衡方程模型为四阶椭圆型偏微分方程的边值问题,其中的自然边界条件涉及到了沿板边的切线和法线方向的高阶导数,对于非均匀、变厚度的板,该问题还具有"变系数"的特点.由问题的变分模型入手,应用变分-差分方法构造了该边值问题的一个差分格式.由于该方法能够结合平衡方程模型中的边界条件以消除沿板边的高阶导数项,因而,所得差分算子仅仅依赖于板面网格结点,并且保持了差分算子的对称、正定性质.同时,将已得算法在计算机上进行了数值模拟,并与现有文献进行了对比计算.结果显示本文所给出的算法具有较高的精确度,该算法将可用于定量地揭示板与边梁之间相互作用的规律,为工程设计提供参考依据.     

不适定的部分Dirichlet问题广义解的右逆表示

对于曹伟平,马吉溥给出的部分Dirichlet问题的广义解的定义进行进一步研究,证得该问题的广义解等价于同一偏微分方程的齐次Dirichlet-Neumann混杂问题的弱解,并用算子右逆给出该广义解的表示.     

A study on preconditioning iterative methods and relativity of pressure in the numerical calculation of fluid flow

This article presents the effect of preconditioning iterative methods on boundary conditions of the pressure‐correction in the numerical computation of fluid flow with known velocity components on all boundaries using the SIMPLE algorithm. In such computation, a set of solutions of the pressure‐correction is indefinite, because only the Neumann condition is imposed on all boundaries. However, solutions become unique if the value of pressure‐correction is fixed at least on one boundary point, and the Dirichlet condition is additionally imposed. Though both conditions must give exactly the same velocity and temperature fields, this problem arises from the relativity of the pressure. The mathematical illustration for this problem is provided using the numerical computation of the natural convection in an enclosure. It is concluded that the preconditioner adopted and the condition that only the Neumann condition on all boundaries is given are effective to reduce the number of iterations in solving the linear system of equations of the pressure‐correction at the computation of the natural convection. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Nonlinear Transmission Problem for Wave Equation with?Boundary Condition of Memory Type

In this paper we consider a transmission problem with a boundary damping condition of memory type, that is, the wave propagation over bodies consisting of two physically different types of materials. One component is clamped, while the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. We will study the global existence of solutions for the transmission problem, and moreover we show that if the relaxation function decays exponentially or polynomially, then the solutions for the problem have the same decay rates.