The distribution of values of taylor series

Clarify some classical results and their proofs on the distribution of values of simple Taylor series and extend them to multiple Taylor series.     

A new non‐polynomial univariate interpolation formula of Hermite type

A new C ^∞ interpolant is presented for the univariate Hermite interpolation problem. It differs from the classical solution in that the interpolant is of non‐polynomial nature. Its basis functions are a set of simple, compact support, transcendental functions. The interpolant can be regarded as a truncated Multipoint Taylor series. It has essential singularities at the sample points, but is well behaved over the real axis and satisfies the given functional data. The interpolant converges to the underlying real‐analytic function when (i) the number of derivatives at each point tends to infinity and the number of sample points remains finite, and when (ii) the spacing between sample points tends to zero and the number of specified derivatives at each sample point remains finite. A comparison is made between the numerical results achieved with the new method and those obtained with polynomial Hermite interpolation. In contrast with the classical polynomial solution, the new interpolant does not suffer from any ill conditioning, so it is always numerically stable. In addition, it is a much more computationally efficient method than the polynomial approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

We show that in contrast to a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. We consider an approximation based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the second-order term. To escape technical difficulties related to the infinite dimensionality of the phase space for quantum mechanics, we consider finite-dimensional quantum mechanics. On one hand, this is a simple example with high pedagogical value. On the other hand, quantum information operates in a finite-dimensional state space. Therefore, our investigation can be considered a construction of a classical statistical model for quantum information. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 278–291, August, 2007.     

Unsteady draining of a fluid from a circular tank

Three-dimensional draining flow of a two-fluid system from a circular tank is considered. The two fluids are inviscid and incompressible, and are separated by a sharp interface. There is a circular hole positioned centrally in the bottom of the tank, so that the flow is axially symmetric. The mean position of the interface moves downwards as time progresses, and eventually a portion of the interface is withdrawn into the drain. For narrow drain holes of small radius, the interface above the centre of the drain is pulled down towards the hole. However, for drains of larger radius the portion of the interface above the drain edge is drawn down first, rather than the central section. Non-linear results are obtained with a novel spectral technique, and are also compared against the predictions of linearized theory. Unstable Rayleigh–Taylor type flows, in which the upper fluid is heavier than the lower one, are also discussed.     

非线性Blasius方程求解的一种新算法

通过一半无限大平板的不可压缩的两维稳定流是一个典型的工程问题,被称为边界层流问题.它是一个由三阶非线性微分方程描述的边值问题,其微分方程称为Blasius方程.首先将该边值问题转化为一对初值问题,然后用状态方程直接积分法和Taylor级数展开法对这对初值问题进行求解.与其它算法相比,具有算法简单,精度高的优点.     

Finite Element Analysis for Mass-Lumped Three-Step Taylor Galerkin Method for Time Dependent Singularly Perturbed Problems with Exponentially Fitted Splines

The motive of the current study is to derive pointwise error estimates for the three-step Taylor Galerkin finite element method for singularly perturbed problems. Pointwise error estimates have not been derived so far for the said method in the finite element framework. Singularly perturbed problems represent a class of problems containing a very sharp boundary layer in their solution. A small parameter called singular perturbation parameter is multiplied with the highest order derivative terms. When this parameter becomes smaller and smaller, a boundary layer occurs and the solution changes very abruptly in a very small portion of the domain. Because of this sudden change in the nature of the solution, it becomes very difficult for the numerical methods to capture the solution accurately specially in the boundary layer region. In the present study finite element analysis has been carried out for such one-dimensional singularly perturbed time dependent convection-diffusion equations. Exponentially fitted splines have been used for the three-step Taylor Galerkin finite element method to converge. Pointwise error estimates have been derived for the method and it is shown that the method is conditionally convergent of first order accurate in space and third order accurate in time. Numerical results have been presented for both the linear and nonlinear problems.     

两个平均值问题的注记

利用函数得幂级数展开式,证明了两个平均值定理相反问题的解均为多项式函数.     

对一道数学分析题证明方法的改进

某文献在处理一道关于高阶导数的应用问题时,反复利用Rolle定理来证明高阶导数为零．考虑到这种做法过于繁琐,遂通过对其证明方法的改进,综合使用Lagrange中值定理和Taylor公式,使该问题的解决获得简化．     

A return toward equilibrium in a 2D Rayleigh–Taylor instability for compressible fluids with a multidomain adaptive Chebyshev method

We numerically simulate a single-mode Rayleigh–Taylor instability between compressible miscible fluids with a highly accurate self-adaptive pseudospectral Chebyshev multidomain method in two two-dimensional boxes at small aspect ratios. The simulations are started from rest and pursued until the return toward mechanical equilibrium of the mixing. Four regimes—linear and weakly nonlinear, nonlinear steady bubble rise, return toward equilibrium, and finally a system of acoustic waves—can be identified. We show that this one-dimensional system of stationary acoustic waves is damped by the physical viscosity. This provides a reference solution.