一类四阶两点边值问题多个正解的存在性

两端简单支撑弹性梁的形变可以用四阶常微分方程两点边值问题来描述。由于其在物理中的重要性,已有许多人研究了该类问题解的存在性,但在实际应用中该类问题正解以及多个正解的存在性更为重要。本文应用锥上的不动点定理,研究了该类四阶常微分方程两点边值问题多个正解的存在性,给出了该类问题多个正解存在的充分条件,本文结果推广和改进了一些已知结果。最后给出一例作为所获结果的应用。     

Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Summary Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: (1) the interior of a spherical triangle, and (2) the exterior of a great circle arc extending for less than radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.     

Localized wrinkling instabilities in radially stretched annular thin films

Summary We investigate a pre-stressed annular thin film subjected to a uniform displacement field along its inner boundary. This loading scenario leads to a variable stress distribution characterized by an orthoradial component that may change sign along a concentric circle within the annular domain. When the intensity of the applied field is strong enough, elastic buckling occurs circumferentially, leading to a localized wrinkling pattern near the inner edge. Using a linear non-homogeneous pre-bifurcation state, the eigenvalue problem describing this instability is cast as a singularly-perturbed fourth-order linear differential equation with variable coefficients. The dependence of the lowest eigenvalue on various non-dimensional quantities is numerically investigated using the compound matrix method. These results are complemented by a WKB analysis which suggests that the qualitative and quantitative features of the full model can be described by a simplified second-order eigenvalue problem which takes into account the finite stiffness of the system.     

The interface crack in a tension field: an eigenvalue problem for the gap

The problem of the interface crack in a tension field is reexamined with the objective of obtaining simple and explicit expressions for all quantities of physical interest. The known exact solution to this problem is complicated and difficult to analyze. Here the gap is shown to satisfy an eigenvalue problem. The condition that the gap is nonnegative implies that the smallest eigenvalue is the correct one. A simple uniform approximation for the gap is obtained by using the method of matched asymptotic expansions to take advantage of the smallness of the contact zones. As the tangential shift satisfies the same differential equation as the gap, a uniform approximation for this quantity is obtained by the same method. The tractions are also shown to satisfy a second order ordinary differential equation, and uniform approximations are given for the quantities.     

A note on the integrability of a remarkable static Euler–Bernoulli beam equation

It has recently been shown that the fourth-order static Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y ^?5/3, and the equation has the nonsolvable algebra ${sl(2, \mathbb{R})}$ . We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of ${sl(2, \mathbb{R})}$ remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y ^?5/3.     

On Some Approximate Methods for the Tensile Instabilities of Thin Annular Plates

A thin annular plate is subjected to a uniform tensile field at its inner edge which leads to compressive circumferential stresses. When the intensity of the applied field is strong enough, elastic buckling occurs circumferentially, leading to a wrinkling pattern. Using a linear non-homogeneous pre-bifurcation state, the linearised eigenvalue problem describing this instability is cast as a fourth-order linear differential equation with variable coefficients. This problem is investigated numerically and it is shown that the simple application of the Galerkin technique reported in the literature leads to gross errors in the corresponding approximations. Several novel mathematical features of the eigenvalue problem are included as well.     

Numerical solution of a fourth-order ordinary differential equation

Summary In this paper we have derived numerical methods of orderO(h ⁴) andO(h ⁶) for the solution of a fourth-order ordinary differential equation by finite differences. A method ofO(h ²) was earlier discussed by Usmani and Marsden [6]. Convergence of the fourth-order method is shown. Two examples are computed to show the superiority of our methods.     

Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical circular cylinder

Summary The transformation group theoretic approach is applied to present an analysis of the problem of unsteady free convection from the outer surface of a vertical circular cylinder. The application of two-parameter group reduces the number of independent variables by two, and consequently the system of the governing partial differential equations with the boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. Numerical results are obtained for the study of the boundary-layer characteristics. The general analysis developed in this study corresponds to the case of surface temperature that varies exponentially with time and uniform with respect to the axial coordinate, i.e., in the formT _w =ae ^bt , wherea andb are constants. The effect of Prandtl number,Pr, andb on the boundary layer characteristics and the maximum value of the vertical component of the velocity are studied.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

Natural element analysis of the Cahn–Hilliard phase-field model

We present a natural element method to treat higher-order spatial derivatives in the Cahn–Hilliard equation. The Cahn–Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase separation in binary mixtures. Standard classical C⁰{{\mathcal{C}}^0}-continuous finite element solutions are not suitable because primal variational formulations of fourth-order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally C¹{{\mathcal{C}}^1}-continuous. To ensure C¹{{\mathcal{C}}^1}-continuity, we develop a natural-element-based spatial discretization scheme. The C¹{{\mathcal{C}}^1}-continuous natural element shape functions are achieved by a transformation of the classical Farin interpolant, which is basically obtained by embedding Sibsons natural element coordinates in a Bernstein–Bézier surface representation of a cubic simplex. For the temporal discretization, we apply the (second-order accurate) trapezoidal time integration scheme supplemented with an adaptively adjustable time step size. Numerical examples are presented to demonstrate the efficiency of the computational algorithm in two dimensions. Both periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are considered. We demonstrate that the use of C¹{{\mathcal{C}}^1}-continuous natural element shape functions enables the computation of topologically correct solutions on arbitrarily shaped domains.     

On the departure of a sphere from contact with a permeable membrane

Summary A sphere in contact with a porous membrane can depart with non-zero velocity under the influence of a finite force. The flow field and the viscous force resisting the motion are evaluated by reducing the Stokes equations to a fourth-order ordinary differential equation through the use of conformal mapping. Explicit results are given for high and low membrane permeabilities by using regular and singular perturbation techniques respectively.     

Nonsteady diffusion with variable coefficients in the boundary conditions

A method of generalized integral transformations is used to solve the problem of nonsteady diffusion with time-dependent coefficients in the boundary conditions. Such an approach does not require a solution of an integral equation for the surface potential or of a time-dependent eigenvalue problem. A formal solution is obtained on the basis of an infinite system of ordinary differential equations. An example is considered and numerical results are discussed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 61, No. 5, pp. 829–837, November, 1991.     

Squeezing flow of MHD fluid between parallel disks

Squeezing flow between parallel disks is studied for the case when one disk is porous and the other is impermeable. Viable similarity transform is used to reduce the problem to a highly nonlinear ordinary differential equation. Variation of Parameters Method (VPM) is then employed to determine the solution to resulting ordinary differential equation. Numerical solution is also obtained using R-K 4 method and comparison shows an excellent agreement between both the solutions. Effects of different physical parameters on the flow are also discussed with the help of graphs coupled with comprehensive discussions.     

Analytic parametric solutions for the HRR nonlinear elastic field with low hardening exponents

Summary In this paper, we restore the already constructed approximate asymptotic solutions extracted in [10] concerning the HRR [1] strongly nonlinear fourth-order ordinary differential equation (ODE) for plane strain conditions in nonlinear elastic (plastic) fracture. It is proved that the above equation, for low strain hardening exponents (0 < N ? 1), is reduced to a strongly nonlinear ODE of the second order. The method of the total differentials is used so that the last equation is reduced to Abels' equations of the second kind of the normal form, that can be analytically solved in parametric form. In addition, the case of rigid perfect-plasticity (N=0) is extensively investigated and several important results are extracted.     

Self-similar solutions of the magnetohydrodynamic boundary layer system for a dilatable fluid

Summary A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting dilatable fluid along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness, existence, and nonexistence results for generalized normal solutions are established.     

Solution of the Pontriagin–Vitt equation for the moments of time to first passage of the randomly accelerated particle by the finite element method

The Pontriagin–Vitt equation governing the mean of the time of first passage of a randomly accelerated particles has been studied extensively by Franklin and Rodemich.¹ In their paper is presented the analytic solution for the two-sided barrier problem and solutions by several finite difference procedures. This note demonstrates solution of the problem by a Petrov–Galerkin finite element method using upstream weighting functions,² shown to give rapidly convergent results. In addition, the equation is generalized to include higher statistical moments, and solutions for the first few ordinary moments are reported.     

Solution of low p'eclet number heat transfer with viscous dissipation in the infinite axial region between parallel plates via functional analytic methods

Abstract

Functional analytic methods have been applied to the analysis of the extended Graetz problem with prescribed wall flux and viscous dissipation between parallel plates. First, the non‐self‐adjoint elliptic energy equation is decomposed into a set of first order partial differential equations to obtain a self‐adjoint formulism. Next, the induced eigenvalue problems are solved by applying an approximation method in a Hilbert space, and an algebraic characteristic equation is obtained. In addition, the expansion coefficients of the solutions on upstream and downstream regions can be explicitly obtained and unnecessary to match at the entrance.     

热过屈曲梁振动的解析解

该文导出了面内热载荷作用下, 梁在其过屈曲构形附近微幅振动的解析解。首先基于经典梁理论, 推导了控制轴向和横向变形的基本方程。然后, 将2 个非线性方程化为一个关于横向挠度的四阶非线性积分-微分方程。假设梁的振幅以及由此引起的附加应变为无限小, 另设其响应为谐振, 则该非线性积分-微分方程将化为两组耦合的微分方程:一组控制非线性静态响应;另一组就是叠加于梁屈曲构形之上的线性振动方程。直接求解这些问题, 可以得到梁热过屈曲构形以及固有频率的解析解, 这些解是外加热载荷的函数。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

On the derivation of first integrals for similarity solutions

In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equation. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.