基于泰勒级数展开的一点超前差分公式的推导

传统的数值微分公式有前向差分、后向差分和中心差分公式.所谓一点超前差分公式,就是后向差分公式在形式上"前移"一点来计算一阶导数的公式.该公式有效地弥补了传统差分公式的不足之处.不同于以前研究中使用拉格朗日公式来推导一点超前公式的做法,给出了基于泰勒级数展开的对该组公式及其截断误差的推导,从另一个角度验证了一点超前公式,使其更为完善.     

A Taylor series method for the numerical solution of two-point boundary value problems

Summary For the numerical solution of two-point boundary value problems a shooting algorithm based on a Taylor series method is developed. Series coefficients are generated automatically by recurrence formulas. The performance of the algorithm is demonstrated by solving six problems arising in nonlinear shell theory, chemistry and superconductivity.     

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN Cn

Some previous results on convergence of Taylor series in Cn [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in Cn are constructed and the Taylor series expansion is deduced.     

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Abstract

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.     

Non-vanishing of Taylor coefficients and Poincaré series

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan’s Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a “generic” point, all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.     

Numerical solution of the one-dimensional time-independent Schrödinger’s equation by recursive evaluation of derivatives

We develop a simple numerical method for solving the one-dimensional time-independent Schrödinger’s equation. Our method computes the desired solutions as Taylor series expansions of arbitrarily large orders. Instead of using approximations such as difference quotients for the derivatives needed in the Taylor series expansions, we use recursive formulas obtained using the governing differential equation itself to calculate exact derivatives. Since our approach does not use difference formulas or symbolic manipulation, it requires much less computational effort when compared to the techniques previously reported in the literature. We illustrate the effectiveness of our method by obtaining numerical solutions of the one-dimensional harmonic oscillator, the hydrogen atom, and the one-dimensional double-well anharmonic oscillator.     

The construction of high order convergent look-ahead finite difference formulas for Zhang Neural Networks

ABSTRACT

Zhang Neural Networks rely on convergent 1-step ahead finite difference formulas of which very few are known. Those which are known have been constructed in ad-hoc ways and suffer from low truncation error orders. This paper develops a constructive method to find convergent look-ahead finite difference schemes of higher truncation error orders. The method consists of seeding the free variables of a linear system comprised of Taylor expansion coefficients followed by a minimization algorithm for the maximal magnitude root of the formula's characteristic polynomial. This helps us find new convergent 1-step ahead finite difference formulas of any truncation error order. Once a polynomial has been found with roots inside the complex unit circle and no repeated roots on it, the associated look-ahead ZNN discretization formula is convergent and can be used for solving any discretized ZNN based model. Our method recreates and validates the few known convergent formulas, all of which have truncation error orders at most 4. It also creates new convergent 1-step ahead difference formulas with truncation error orders 5 through 8.     

On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.     

On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.     

与中心数相关的一类级数求和公式

This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.     

Gaussian quadrature in Ramanujan’s Second Notebook

Ramanujan’s notebooks contain many approximations, usually without explanations. Some of his approximations to series are explained as quadrature formulas, usually of Gaussian type. Dedicated to the memory of Professor K G Ramanathan     

Some remarks on holomorphic functions and taylor series in C

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.     

Comparison of Taylor finite difference and window finite difference and their application in FDTD

The finite difference time domain (FDTD) method is an important tool in numerical electromagnetic simulation. There are many ways to construct a finite difference approximation such as the Taylor series expansion theorem, the filtering theory, etc. This paper aims to provide the comparison between the Taylor finite difference (TFD) scheme based on the Taylor series expansion theorem and the window finite difference (WFD) scheme based on the filtering theory. Their properties have been examined in detail, separately. In addition, the formula of the generalized finite difference (GFD) scheme is presented, which can include both the TFD scheme and the WFD scheme. Furthermore, their application in the numerical solution of Maxwell's equations is presented. The formulas for the stability criterion and the numerical dispersion relation are derived and analyzed. In order to evaluate their performance more accurately, a new definition of error is presented. Upon it, the effect of several factors including the grid resolution, the Courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined.     

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism,implementation and rigorous validation

We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.     

Numerical solution of the Burgers’ equation by automatic differentiation

We compute the solution of the one-dimensional Burgers’ equation by marching the solution in time using a Taylor series expansion. Our approach does not require symbolic manipulation and does not involve the solution of a system of linear or non-linear algebraic equations. Instead, we use recursive formulas obtained from the differential equation to calculate exact values of the derivatives needed in the Taylor series. We illustrate the effectiveness of our method by solving four test problems with known exact solutions. The numerical solutions we obtain are in excellent agreement with the exact solutions, while being superior to other previously reported numerical solutions.     

Taylor series for the Askey-Wilson operator and classical summation formulas

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As an application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context.

     

二元混合连分式展开的混合差商极限方法

For a univariate function given by its Taylor series expansion,a continuedfraction expansion can be obtained with the Viscovatov's algorithm,as the limitingvalue of a Thiele interpolating continued fraction or by means of the determinantalformulas for inverse and reciprocal differences with coincident data points.In thispaper,both Viscovatov-like algorithms and Taylor-like expansions are incorporatedto yield bivariate blending continued expansions which are computed as the limitingvalue of bivariate blending rational interpolants,which are constructed based on sym-metric blending differences.Numerical examples are given to show the effectivenessof our methods.     

Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative

We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method.     

Closed Formula for Options with Discrete Dividends and Its Derivatives

Abstract

We present a closed pricing formula for European options under the Black–Scholes model as well as formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and a proposition that relates expectations of partial derivatives with partial derivatives themselves. The closed formulas are attained assuming the dividends are paid in any state of the world. The results are readily extensible to time-dependent volatility models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis, covering calls and puts, together with results on their partial derivatives. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.     

A modified variable time step method for solving ice melting problem

A modified numerical method was used by authors for solving 1D Stefan problem. In this paper a modified method is proposed with difference formulae and different methods of calculating the variable time step, which are deduced from Taylor series expansions of different conditions at the boundary. Also an extrapolation formula for the solution at the first point at the right of the computational domain is proposed. The numerical results are compared with those obtained from other methods.