L 1 andL ∞ Uniform convergence of a difference scheme for a semilinear singular perturbation problem

Summary A nonlinear difference scheme is given for solving a semilinear singularly perturbed two-point boundary value problem. Without any restriction on turning points, the solution of the scheme is shown to be first order accurate in the discreteL ¹ norm, uniformly in the perturbation parameter. When turning points are excluded, the scheme is first order accurate in the discreteL norm, uniformly in the perturbation parameter.Partly supported by the Arts Faculty Research Fund of University College, Cork     

A uniformly convergent difference scheme for a semilinear singular perturbation problem

Summary We present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to that of the continuous problem.     

An error analysis for a difference scheme of exponential type applied to a nonlinear singular perturbation problem without turning points

A difference scheme of exponential type given in [5] is analyzed again for solving a nonlinear singular perturbation problem without turning points. It is shown that a solution of the scheme converges, uniformly in a perturbation parameter, to an exact solution of the continuous problem with order one in L¹ sense.     

Numerical computation of nonsimple turning points and cusps

Summary A method is developed for the numerical computation of a double turning point corresponding to a cusp catastrophe of a nonlinear operator equation depending on two parameters. An augmented system containing the original equation is introduced, for which the cusp point is an isolated solution. An efficient implementation of Newton's method in the finite-dimensional case is presented. Results are given for some chemical engineering problems and this direct method is compared with some other techniques to locate cusp points.     

Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven

Summary A numerical technique is described for computing turning points of a space curveL implicitly defined by a nonlinear system ofn equations inn+1 variables. The basic idea is a local parametrization ofL where the parameter that gives the next iterate is determined by applying one step of the well-known method for minimizing a real function using cubic Hermite interpolation with two nodes. The method is shown to convergeQ-super-linearly and withR-order of at least two. A numerical example concerning the analysis of nonlinear resistive circuits shows the algorithm to work effectively on real life problems.

     

Finite element methods on piecewise equidistant meshes for interior turning point problems

We consider linear second order singularly perturbed two-point boundary value problems with interior turning points. Piecewise linear Galerkin finite element methods are constructed on various piecewise equidistant meshes designed for such problems. These methods are proved to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usualL ² norm. Supporting numerical results are presented.     

The Adomian decomposition method in turning point problems

In this paper, the decomposition method is applied to boundary-value problems of ordinary differential equations with a parameter exhibiting turning points.     

An efficient algorithm for the determination of certain bifurcation points

Many practical problems require information about a branch of solutions of a system of nonlinear equations dependent upon a scalar parameter. We discuss some techniques for following such a branch through a turning point and describe an efficient method, with second order convergence, for finding the turning point. We also show that, if extra information is available about the solution branch, the method can be successfully applied to finding simple bifurcation points.     

论扩充系统——处理分歧问题的一种有效方法

§1.引言考察非线性算子方程: g(x,λ)=0,g:X×R→Y, (1.1)其中X,Y是Banach空间,gC~3,λ是参数。我们要处理的是(1.1)的奇异点的情形,故设g(x_0,λ_0)=0,g_x~0奇异,且称(x_0,λ_0)是(1.1)的简单奇异点。如果(1.2),(1.3)成立。由于g_x~0是指标为0的Fredholm算子,故有     

Asymptotic Methods for Bifurcation and Stability Problems

A method of asymptotically determining the bifurcating solutions of a nonlinear eigenvalue problem is described. The method is based on the smallness of a parameter which is different from the usual parameter used in the LyapunovSchmidt procedure. A discussion of the stability and evolution of bifurcating solutions is included. It is shown how the method may be useful for determining secondary bifurcations and turning points on bifurcation curves.     

The layer-resolving transformation and mesh generation for quasilinear singular perturbation problems

The relationship is analyzed between layer-resolving transformations and mesh-generating functions for numerical solution of singularly perturbed boundary-value problems. The analysis is carried out for one-dimensional quasilinear problems without turning points, which are discretized by first-order finite-difference schemes. It is proved that if a general layer-resolving function is used to generate the discretization mesh, then the numerical solution converges uniformly in the perturbation parameter.     

Uniform asymptotic expansion of $J_\nu(\nu a)$ via a difference equation

Summary. There are two ways of deriving the asymptotic expansion of , as , which holds uniformly for . One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation . Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations. Received December 15, 2000 / Published online September 19, 2001     

具有三个转向点方程渐近解的完全表达式

本文研究二阶线性常微分方程,λ为大参数,即具有三个转向点的方程。而。本文使用JL函数得到方程在转向点附近形式一致有效渐近解的完全表达式。     

Uniform methods for semilinear problems with attractive boundary turning points

An upwind finite‐difference scheme for the numerical solution of semilinear convection‐diffusion problems with attractive boundary turning points is considered. We show that the maximum nodal error is bounded by a special weighted ℓ₁‐type norm of the truncation error. This result is used to establish uniform convergence with respect to the perturbation parameter on Shishkin meshes.     

求解奇异摄动问题的一个耦合差分格式

本文考察奇异摄动问题(1.1).在一特殊的非均匀网格上,将不稳定、二阶精度的中心差格式和稳定、一阶精度的Abrahamsson-Keller-Kreiss箱子格式相耦合,得到了一个二阶一致收敛的差分格式.最后给出了数值结果.     

打靶法在常微分方程边界层型奇异摄动问题中的应用

用自适应步长积分格式结合打靶技巧,可以有效地求解比较困难的常微分方程边界层型奇异摄动问题.本文给出了若干计算实例,说明了这种方法应用于线性问题时的一次收敛性,以及应用于单端、双端边界层、转向点和多个边界层时的效果,特别是能方便地求出多解.最后并与习用的差分方法作了比较.     

A Direct Method for Pitchfork Bifurcation Points

To overcome the difficulty caused by the singularity at the pitchfork bifurcation points, we introduce the homotopy parameter so that the problem of computing the pitchfork bifurcation points can be transferred to that of computing the fold points of degree 3 with respect to the homotopy parameter. An extended system for pitchfork bifurcation points is given. The regularity of the extended system is proved. Finally, the numerical examples show the effectiveness of our method.     

一类具转向点椭圆型方程奇异摄动问题的数值解法

本文考虑如下一类具转向点椭圆型方程奇异摄动问题的数值解法文［１］已经研究了该问题的渐近解．本文在得到渐近解余项的更好估计式后，证明了所构造的差分格式关于小参数ε的一致收敛结果．误差估计达到α阶，其中．     

On parametrization of interpolating curves

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.