Fixed point indexes and its applications to nonlinear integral equations modelling infectious diseases

In this paper the fixed point index problem for a class of positive operators with boundary control conditions is discussed, and some sufficient conditions for the fixed point index to be equal to 1 or 0 are given. Moreover, a general fixed point theorem of expansions and compressions for cone is obtained, which generalizes and improves the corresponding results of [3,8,9]. As an application, we utilize the results presented above to study the existence conditions of positive solutions of nonlinear integral equations modelling infectious diseases.Project supported by National Natural Science Foundation of China.     

均布载荷作用下具有光滑中心波纹膜片的非线性分析

采用轴对称旋转壳体的简化Reissner方程，研究了在均布载荷作用下具有光滑中心波纹膜片的非线性弯曲问题。应用格林函数方法，波纹膜片的非线性边值问题化为了非线性积分方程的求解。为了求解积分方程并防止发散，一个插值参数被引入到迭代格式中。计算表明，当载荷很小时，任何插值参数值均能保证迭代的收敛性，取插值参数值接近或等于1获得较快的收敛速度，而当载荷较大时，插值参数值不能取得过大。绘出了波纹膜片的特征曲线，得到的特征曲线可供设计参考。可以断言，当载荷不大时，特征曲线是近似线性的，随着载荷的增大，特征曲线开始向上弯曲，明显偏离线性。本文中提出的解决方法适应于任意轴向截面的波纹壳体。     

Singular perturbation of second-order nonlinear system with boundary perturbation

Ｗｅｃｏｎｓｉｄｅｒｉｎｔｈｉｓｐａｐｅｒｔｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｓｅｃｏｎｄ＿ｏｒｄｅｒｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｉｎｖｏｌｖｉｎｇｉｎｔｅｒｇｒａｌｏｐｅｒａｔｏｒεｙ″=ｆ(ｔ,ｙ,Ｔｙ,ε)ｙ′ ｇ(ｔ,ｙ,Ｔｙ,ε),(1)ｗｉｔｈｂｏｕｎｄａｒｙｐｅｒｔｕｒｂａｔｉｏｎｙ(ｔ,ε)｜ｔ=φ(ε)=α(ε),ｙ(ｔ,ε)｜ｔ=1 ψ(ε)=β(ε),(2)ｗｈｅｒｅε>0ｉｓａｓｍａｌｌｐａｒａｍｅｔｅｒ,ａｎｄφ(ε),ψ(ε)ａｒｅｂｏｔｈ,ｗｉｔｈｒｅｓｐｅｃｔｔｏε,ｓｕｆｆｉｃｉｅｎｔｌｙｓｍｏ…     

ON EXACT SOLUTION OF KÁRMÁN’S EQUATIONS OF RIGID CLAMPED CIRCULAR PLATE AND SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

It is extremely difficult to obtain an exact solution of von Karman’s equations because the equations are nonlinear and coupled. So far many approximate methods have been used to solve the large deflection problems except that only a few exact solutions have been investigated but no strict proof on convergence is presented yet. In this paper, first of all, we reduce the von KÁrmÁn’s equations to equivalent integral equations which are nonlinear, coupled and singular. Secondly the sequences of continuous function with general form are constructed using iterative technique. Based on the sequences to be uniformly convergent, we obtain analytical formula of exact solutions to von Karman’s equations related to large deflection problems of circular plate and shallow spherical shell with clamped boundary subjected to a concentrated load at the centre.     

Nonlinear bending of corrugated diaphragm with large boundary corrugation under compound load

IntroductionCorrugateddiaphragmisatypeofelasticthinshells .Itsdesignisverycomplicatedbecauseoftoomanyparametersthatinfluenceeachother.Inanumberofinstrumentsmeasuringdisplacements,corrugateddiaphragmissubjectedtoelasticdisplacementthatisatleastthesameorderasitsthickness,sothatitisnecessarytousegeometricalnonlineartheoryofthinshellstoanalyze.Sofarasweknow ,inmostcases,investigatorsdiscussedonlytheproblemofcorrugateddiaphragmwithuniformanddensecorrugationsundertheactionofaunique(uniformlyorconcen…     

动压推力轴承的边界元分析

本文利用Green第二公式,将Reynolds方程转化为沿边界的积分方程,并将非线性项作为自由项的一部分处理,采用常单元离散边界Γ,用迭代技术求出油膜压力分布,与有限差分法和有限元法比较,边界元法的结果更接近解析解.     

复杂载荷三维裂纹分析双重边界元法

提出可用于高温、高转速状态下的热动力机械三维含裂构件热弹性分析方法——双重边界元法.首先建立了考虑温度及离心载荷的双重边界积分方程组,并对边界积分方程组的选取及适用范围进行了讨论。然后提出角非快调元模型离散技术。接着提出超奇异积分方程分析去除奇异性方法及数值积分技术.数值算例表明计算结果与有关权函数解十分吻合,说明了用双重边界元法计算复杂载荷条件下三维应力强度因子的有效性.还讨论了有关热应力强度因子权函数解的适用范围.     

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method.The invariance of some sets under the flow of these problems and the vac- uum isolation of solutions are obtained by introducing a family of potential wells.Then the threshold result of global existence and nonexistence of solutions are given.Finally, the problem with critical initial conditions are discussed.     

METHOD OF GREEN＇S FUNCTION OF CORRUGATED SHELLS

By using the fundamental equations of axisymmetric shallow shells of revolution, the nonlinear bending of a shallow corrugated shell with taper under arbitrary load has been investigated. The nonlinear boundary value problem of the corrugated shell was reduced to the nonlinear integral equations by using the method of Green＇s function. To solve the integral equations, expansion method was used to obtain Green＇s function. Then the integral equations were reduced to the form with degenerate core by expanding Green＇s function as series of characteristic function. Therefore, the integral equations become nonlinear algebraic equations. Newton＇ s iterative method was utilized to solve the nonlinear algebraic equations. To guarantee the convergence of the iterative method, deflection at center was taken as control parameter. Corresponding loads were obtained by increasing deflection one by one. As a numerical example,elastic characteristic of shallow corrugated shells with spherical taper was studied.Calculation results show that characteristic of corrugated shells changes remarkably. The snapping instability which is analogous to shallow spherical shells occurs with increasing load if the taper is relatively large. The solution is close to the experimental results.     

Solution of the elastic boundary value problems for a layer with tunnel stress raisers

A novel procedure for solving three-dimensional problems for elastic layer weakened by through-thickness tunnel cracks has been developed and is presented in this paper. This procedure reduces the given boundary value problem to an infinite system of one-dimensional singular integral equations and is based on a system of homogeneous solutions for a layer. Integral representations of single- and double-layer potentials are used for metaharmonic and harmonic functions entering in the singular integral equations. These representations provide a continuous extendibility of the stress vector while allowing a jump in the displacement vector in the transition through the cut.Expanding the potential and biharmonic solutions in the Fourier series over the thickness coordinate yields the integral representations of the displacement vector and stress tensor. The problem of reducing a denumerable set of the integral equations of the given boundary value problem to one-to-one correspondence with the set of unknown densities appearing in the Fourier’s coefficient representations has been settled efficiently. Numerical investigations show a rapid convergence of the proposed reduction procedure as applied to the solution of the infinite system of one-dimensional integral equations. Numerical examples illustrate the proposed method and demonstrate its advantages.     

The method of mixed boundary condition for a kion of linear and nonlinear composite structure

This paper deals with the axisymmetrical deformation of shallow shells in large deflection which are in conjunction with linear elastic structures at the boundary: A method of mixed boundary condition for this problem is introduced, then the problem of a composite structure is transformed into a problem of a single structure and the integral equations are given. The perturbation method is used to obtain the solutions and an example of composite structure consisting of a shallow spherical and a cylindrical shell is presented.Communicated by Yeh Kai-yuan     

Third-order nonlinear singularly perturbed boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｎｃｅ 1 958,Ｌｅｖｓｉｏｎａｎｄｓｅｖｅｒａｌｏｔｈｅｒａｕｔｈｏｒｓ[1]- [4 ]ｈａｖｅｍａｄｅａｓｕｂｓｔａｎｔｉａｌｓｔｕｄｙｆｏｒｅｘｉｓｔｅｎｃｅａｎｄｅｓｔｉｍａｔｅｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｔｈｉｒｄ＿ｏｒｄｅｒｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｗｉｔｈｔｗｏｐｏｉｎｔｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｘ( 0 ) =Ａ ,ｘ′( 0 ) =Ｂ ,ｘ′( 1 ) =Ｃ ,ｏｒｓｌｉｇｈｔｌｙｅｘｔｅｎｓｉｖｅｌｙｌｉｎｅａｒｂｏ…     

平面弹性力学充要的随机边界元

将平面弹性力学确定性的充分必要的边界积分方程推广到含材料常数随机的不确定问题中去，给出了位移的均值以及偏差的充分必要的边界积分方程。数值计算结果表明，和确定性的积分方程一样，习用的随机边界积分方程在退化尺度附近，无论是均值还是偏差都存在巨大的误差，而充要的随机边界积分方程则始终保持良好的精度     

非线性大变形问题边界元分析的自校正方法

针对用增量法求解非线性方程解的漂移问题，在非线性问题边界元法计算中建立了自我校正方法，对在拖带坐标上建立的增量形式的基本方程，引入Ｌａｎｇｒａｎｇｅ校正因子，以全量形式的基本方程作为其辅助方程，在此基础上导出含校正项的边界积分方程，边界元自我校正方法的建立有效地保证了在非线性问题的计算中最终收敛在其解附近，提高了计算精度和运算效率。     

Solution of acoustic scattering problems by means of second kind integral equations

In the present paper, the problem of scattering of acoustic waves from a fluid inclusion in two dimensions is solved numerically. The boundary conditions are those of continuous pressure and normal displacement. First, the problem in the frequency domain is reduced to a pair of second kind Fredholm integral equations on the boundary of the scatterer. Then, the integral equations are discretized by means of the Nyström algorithm, and the resulting linear systems are solved iteratively. Finally, the time domain solution is obtained from a sequence of frequency domain values. The integral equations of the present paper possess a very simple physical interpretation which guarantees the stability of their numerical solution and rapid convergence of the iterative solver. The resulting algorithm is an efficient tool for solving relatively large scale two-dimensional scattering problems.     

BOUNDARY INTEGRAL EQUATIONS FOR BENDING PROBLEM OF REISSNER'S PLATES ONTWO-PARAMETER FOUNDATION

I.IntroductionThickplatesonelastict'oundationarewidelyusedinengineering,suchasthebottomplatesofoffShorestructures,surfaceplatesonrunwayofairportsandfoundationsofhigh-risebuildingsandthelike.Itisextremelydifficulttoobtainanalyticalsolutiontarathickplatewithcomplicatedshapeorcomplicatedboundaryconditiononelasticfoundation.Inrecentyears,theboundaryelementmethod(BEM)hasbeensuccessl'ullyusedtoanalyzethebendingproblemofplatesoneverykindofelasticfoundation(Ref.[l,2.3]).Butthereareonlyfewreferences…     

Boundary integral equations of unique solutions in elasticity

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｂｏｕｎｄａｒｙｅｌｅｍｅｎｔｍｅｔｈｏｄ（ＢＥＭ）ｐｒｏｖｉｄｅｓａｎａｔｔｒａｃｔｉｖｅａｌｔｅｒｎａｔｉｖｅｆｏｒｔｈｅａｎａｌｙｓｉｓｏｆｅｎｇｉｎｅｅｒｉｎｇｐｒｏｂｌｅｍｓ．Ｉｔｓｍａｉｎａｄｖａｎｔａｇｅｓａｒｅｅｃｏｎｏｍｉｃａｌａｎｄｐａｒｔｉｃｕｌａｒｌｙｃｏｎｖｅｎｉｅｎｔｆｏｒｕｎｂｏｕｎｄｅｄｄｏｍａｉｎａｎｄｓｔｒｅｓｓｃｏｎｃｅｎｔｒａｔｉｏｎｐｒｏｂｌｅｍｓ．Ｔｈｅｂｏｕｎｄａｒｙｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎ（ＢＩＥ）ｉｓｔｈｅ…     

Self-similar solutions of nonstationary equations of a plane laminar magnetohydrodynamic boundary layer

Self-similar solutions of nonstationary equations of the boundary layer in ordinary hydrodynamics are discussed in [1, 2]. In this paper self-similar solutions of nonstationary equations of a plane magnetohydrodynarnic boundary layer are sought. In this case, a transformation to curvilinear coordinates of a certain special form is employed. Its choice is determined by the requirements essential to reducing the equations of the boundary layer to a system of ordinary equations. H. Weyl's iterative method is used to solve the equations describing the flow over a plate suddenly set in motion.     

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

The traditional Saint-Venant problem of three-dimensional viscoelasticity is discussed under the Hamiltonia system with the use of the Laplace integral transformation, and the original problem is transformed into finding eigenvalues and eigenvectors of the Hamiltonia operator matrix. Since local effect near the boundary is usually neglected, all solutions of Saint-Venant problems can be obtained directly by the combinations of zero eigenvectors. Moreover, the adjoint relationships of the symplectic orthogonality of zero eigenvectors in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigenvector space of the time domain, and the iterative application of Laplace transformation is not needed. Simply by applying the adjoint relationships of the symplectic orthogonality, an effective method for boundary condition is given. Based on this method, some typical examples are discussed, in which the whole character of total creep and relaxation of viscoelasticity is clearly revealed.     

处理一种非线性问题迭代法的计算机方法

本文依非线性问题的积分方程组,按迭代法得出一般迭代表达式.由此,任意次迭代解的所有待定系数可用计算机计算,使研究复杂的非路性问题成为可能.