Stability of Displacement to the Second Fundamental Problem in Plane Elasticity

In this article, by using the stability of Cauchy type integral when the smooth perturbation for integral curve and the Sobolev type perturbation for kernel density happen, we discuss the stability of the second fundamental problem in plane elasticity when the smooth perturbation for the boundary of the elastic domain (unit disk) and the Sobolev type perturbation for the displacement happen. And the error estimate of the displacement between the second fundamental problem and its perturbed problem is obtained.     

RIEMANN BOUNDARY VALUE PROBLEM WITH RESPECT TO THE PERTURBATION OF BOUNDARY CURVE TO BE AN OPEN ARC

By dint of the stability of Cauchy-type integral with kernel density of class H＊ for an open arc, this paper discusses the stability of the solution of Riemann boundary value problem with respect to the perturbation of boundary curve to be an open arc.     

STABILITY OF DISPLACEMENT TO THE SECOND FUNDAMENTAL PROBLEM IN PLANE ELASTICITY

In this article, by using the stability of Cauchy type integral when the smooth perturbation for integral curve and the Sobolev type perturbation for kernel density happen, we discuss the stability of the second fundamental problem in plane elasticity when the smooth perturbation for the boundary of the elastic domain （unit disk） and the Sobolev type perturbation for the displacement happen. And the error estimate of the displacement between the second fundamental problem and its perturbed problem is obtained.     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

STABILITY OF A KIND OF COMPOUND BOUNDARY VALUE PROBLEM WITH RESPECT TO THE PERTURBATION OF BOUNDARY CURVE

In this paper, we discuss the stability of general compound boundary value prob-lems combining Riemann boundary value problem for an open arc and Hilbert bound-ary value problem for a unit circle with respect to the perturbation of boundary curve.     

New estimates of the stability of one-dimensional plane-parallel flows of a viscous incompressible fluid

The stability of a number of one-dimensional plane-parallel steady flows of a viscous incompressible fluid is investigated analytically using the method of integral relations. The mathematical formulation is reduced to eigenvalue problems for the Orr–Sommerfeld equation. One of three versions is chosen as the boundary conditions: all the components of the velocity perturbation are equal to zero on both boundaries of the layer (in this case we have the classical Orr–Sommerfeld problem), all the components of the velocity perturbation on one of the boundaries are equal to zero and the perturbations of the shear component of the stress vector and of the normal component of the velocity are equal to zero on the other, and all the components of the velocity perturbation are equal to zero on one boundary and the other boundary should be free. The boundary conditions derived in the latter case, are characterized by the occurrence of a spectral parameter in them. For kinematic conditions the lower estimates of the critical Reynolds number – the Joseph–Yih estimates, are improved. In the remaining cases the technique of the integral-relations method is developed, leading to new estimates of the stability. Analogs of Squire's theorem are derived for the boundary conditions of all the types mentioned above. Upper estimates of the increment of the increase in perturbations in eigenvalue problems for the Rayleigh equation with two types of boundary conditions are given.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Boundary value problem for abstract first order differential equation with integral condition

The present paper is devoted to the study of a boundary value problem for abstract first order linear differential equation with integral boundary conditions. We obtain necessary and sufficient conditions for the unique solvability and well-posedness. We also study the Fredholm solvability. Finally, we obtain a result of the stability of solution with respect to small perturbation.     

Invariant Integrals for the Equilibrium Problem for a Plate with a Crack

We consider the equilibrium problem for a plate with a crack. The equilibrium of a plate is described by the biharmonic equation. Stress free boundary conditions are given on the crack faces. We introduce a perturbation of the domain in order to obtain an invariant Cherepanov–Rice-type integral which gives the energy release rate upon the quasistatic growth of a crack. We obtain a formula for the derivative of the energy functional with respect to the perturbation parameter which is useful in forecasting the development of a crack (for example, in study of local stability of a crack). The derivative of the energy functional is representable as an invariant integral along a sufficiently smooth closed contour. We construct some invariant integrals for the particular perturbations of a domain: translation of the whole cut and local translation along the cut.     

On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition

We consider the spectral problem for the Schrödinger operator with an integral perturbation in the periodic boundary conditions. The unperturbed problem is assumed to have multiple eigenvalues and a system of eigenfunctions forming a Riesz basis in L ₂(0, 1). We show that the basis property of systems of root functions of the problem can change under arbitrarily small changes in the kernel of the integral perturbation.     

Generalized impedance boundary conditions for the maxwell equations as singular perturbations problems

In this paper the Maxwell equations in an exterior domain with generalaized impedance boundary conditions of Engquist-Nédélec are considered. The particular form of the assumed boundary conditions can be considered to be a singular perturbation of the Dirichlet boundary conditions. The convergence of the solution of the Maxwell equations with these generalized impedance boundary conditions to that of the corresponding Dirichlet problem is proven. The proof uses a new integral equations method combined with results dealing with singular perturbation problems of a class of pseudo-differential operators.     

Triple-deck theory in transonic flows and boundary layer stability

An analysis of the lower branch of the neutral curve for the Blasius boundary layer leads to a perturbed velocity field with a triple-deck structure, which is a rather unexpected result. It is the asymptotic treatment of the stability problem that has a rational basis, since it is in the limit of high Reynolds numbers that the basic flow has the form of a boundary layer. The principles for constructing a boundary layer stability theory based on the triple-deck theory are proposed. Although most attention is focused on transonic outer flows, a comparative analysis with the asymptotic theory of boundary layer stability in subsonic flows is given. The parameters of internal waves near the lower branch of the neutral curve are associated with a certain perturbation field pattern. These parameters satisfy dispersion relations derived by solving eigenvalue problems. The dispersion relations are investigated in complex planes.     

Numerical analysis of a non‐singular boundary integral method: Part I. The circular case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non‐singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright © 2001 John Wiley & Sons, Ltd.     

New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions

This paper investigates the forced Duffing equation with integral boundary conditions. Its approximate solution is developed by combining the homotopy perturbation method (HPM) and the reproducing kernel Hilbert space method (RKHSM). HPM is based on the use of the traditional perturbation method and the homotopy technique. The HPM can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully linear boundary value problems. Therefore, the forced Duffing equation with integral boundary conditions can be solved using advantages of these two methods. Two numerical examples are presented to illustrate the strength of the method.     

双参数非线性积分微分方程组奇摄动边值问题

研究含分算子并伴有边界振动的双参数非线性系统奇摄动边值问题,在适当的假设下证明了解的存在性,并得到了关于双参数的一致有效的渐近展开式。     

伴有边界摄动二阶非线性系统的奇摄动

讨论了伴有边界摄动的含积分算子的二阶非线性微分方程组边值问题的奇摄动.在适当的假设条件下,通过对角化技巧,证明了解的存在,并估计了余项.     

Continuous Subsonic-Sonic Flows in a Convergent Nozzle

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C^1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.     

Clifford分析中一类高阶奇异积分算子关于区域摄动的稳定性

定义了Clifford分析中一类高阶奇异Teodorescu算子,利用几个重要的不等式研究了这类算子关于积分区域的边界摄动的稳定性.     

Continuous dependence on the time and spatial geometry for the equations of thermoelasticity

In this paper we establish the stability of the solution of the standard initial-boundary value problem of linear anisotropic thermoelasticity under perturbations of the initial time geometry and of the spatial geometry. This is done by deriving appropriate explicit a priori inequalities which permit us to bound in particular the L₂ integral of the perturbation in terms of some well defined measure of the perturbation in the geometry.