On the utility of the homotopy analysis method for non-analytic and global solutions to nonlinear differential equations

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations.     

Insufficiency of chemical network model integration using a high-order Taylor series method

The rate of change for the concentrations of chemical substances in a set of reactions is modeled by a nonlinear dynamical system, which warrants the use of numerical integration methods for differential equations. Previous work advocates the use of a specialized high-order Taylor series method because of an observed reduction in computation time. Contrastingly, we show combinatorial and computational difficulties of the standard Taylor series method, which may dramatically increase computational time or reduce the quality of output. We provide two implementations, a naïve algorithm and an algorithm employing dynamic programming; we are able to overcome only some numerical obstacles and therefore conclude that the Taylor series approach is insufficient for large sets of reactions having many chemical substances.     

Symplectic invariants near hyperbolic-hyperbolic points

We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system.We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C. Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.      

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.     

Taylor type formula with Fréchet polynomials

We extend Taylor’s formula for functions defined on uniquely 2-divisible Abelian semigroups with values in Banach spaces. As corollaries, we give sufficient conditions for the representation of such functions as sums of generalized power series, and also a stability criterion for Fréchet’s polynomial equation which extends the known results.     

On the Range of Universal Functions

We examine the range of universal Taylor series. We prove thatevery universal Taylor series on the unit disc assumes everycomplex number, with one possible exception, infinitely often.On the other hand, we prove that on any simply connected domainthere exist universal functions that omit one value. 2000 MathematicsSubject Classification 30B10.     

A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation

A Taylor matrix method is proposed for the numerical solution of the two-space-dimensional linear hyperbolic equation. This method transforms the equation into a matrix equation and the unknown of this equation is a Taylor coefficients matrix. Solutions are easily acquired by using this matrix equation, which corresponds to a system of linear algebraic equations. As a result, the finite Taylor series approach with three variables is obtained. The accuracy of the proposed method is demonstrated with one example.     

Breaking the limits: The Taylor series method

This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge-Kutta or similar schemes; indeed, in some cases these other schemes are not able to solve such systems at all. In particular, we explore the use of the high-precision arithmetic in the Taylor series method for numerically integrating ODEs. We show how to compute the partial derivatives, how to propagate sets of initial conditions, and, finally, how to achieve the Brouwer’s Law limit in the propagation of errors in long-time simulations. The TIDES software that we use for this work is freely available from a website.     

Choosing a stepsize for Taylor series methods for solving ODE'S

Problem-dependent upper and lower bounds are given for the stepsize taken by long Taylor series methods for solving initial value problems in ordinary differential equations. Taylor series methods recursively generate the terms of the Taylor series and estimate the radius of convergence as well as the order and location of the primary singularities. A stepsize must then be chosen which is as large as possible to minimize the required number of steps, while remaining small enough to maintain the truncation error less than some tolerance.One could use any of four different measures of trunction error in an attempt to control the global error : i) absolute truncation error per step, ii) absolute trunction error per unit step, iii) relative truncation error per step, and iv) relative truncation error per unit step. For each of these measures, we give bounds for error and for the stepsize which yields a prescribed error. The bounds depend on the series length, radius of convergence, order, and location of the primary singularities. The bounds are shown to be optimal for functions with only one singularity of any order on the circle of convergence.     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

The affine connection in the normal coordinates

We propose a method to calculate the derivatives of the Christoffel symbols in the normal coordinates on a smooth manifold with symmetric affine connection. As an application, we construct an algorithm for computing the Taylor series of the double exponential map.     

Simultaneous approximation by universal series

We show that, if individual universal series exist, then we can choose a sequence of universal series performing simultaneous universal approximation with the same sequence of indices. As an application we derive the existence of universal Laurent Series on an annulus using only the existence of universal Taylor Series on discs. Our results are generic (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremal problems for numerical series

In this paper, we consider extremal problems for numerical positive series. The terms of these series are pairwise products of the elements of two sequences, one of which is fixed and the other varies within a given set of sequences. We obtain exact solutions for a number of such problems. As one of the possible applications of the results obtained, we find solutions of some extremal problems related to best n-term approximations of periodic functions.     

Boundary-variation solution of eigenvalue problems for elliptic operators

We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated far outside their radii of convergence—by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.     

Taylor series method for dynamical systems with control: Convergence and error estimates

To optimize a complicated function constructed from a solution of a system of ordinary differential equations (ODEs), it is very important to be able to approximate a solution of a system of ODEs very precisely. The precision delivered by the standard Runge-Kutta methods often is insufficient, resulting in a “noisy function” to optimize. We consider an initial-value problem for a system of ordinary differential equations having polynomial right-hand sides with respect to all dependent variables. First we show how to reduce a wide class of ODEs to such polynomial systems. Using the estimates for the Taylor series method, we construct a new “aggregative” Taylor series method and derive guaranteed a priori step-size and error estimates for Runge-Kutta methods of order r. Then we compare the 8,13-Prince-Dormand’s, Taylor series, and aggregative Taylor series methods using seven benchmark systems of equations, including van der Pol’s equations, the “brusselator,” equations of Jacobi’s elliptic functions, and linear and nonlinear stiff systems of equations. The numerical experiments show that the Taylor series method achieves the best precision, while the aggregative Taylor series method achieves the best computational time. The final section of this paper is devoted to a comparative study of the above numerical integration methods for systems of ODEs describing the optimal flight of a spacecraft from the Earth to the Moon. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Stroboscopic averaging of highly oscillatory nonlinear wave equations

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non‐oscillatory high‐order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz‐Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations

We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Padé approximants and conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE). Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they present. After having indicated that the Taylor series method almost always requires the recourse to analytical continuation procedures to be efficient, we use the complementarity of the two remaining methods (Padé and conformal mapping) to illustrate their respective advantages and limitations. We emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods, particularly for the so-called Padé-Hankel method. (We show that this latter method is equivalent to pushing a movable pole to infinity.) For each configuration, we determine the singularity distribution (in the complex plane of the independent variable) of the solution sought and show how this distribution controls the efficiency of the two methods. In general the method based on Padé approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required.     

Computing the region of convergence for power series in many real variables: A ratio-like test

We give an elementary proof that the region of convergence for a power series in many real variables is a star-convex domain but not, in general, a convex domain. In doing so, we deduce a natural higher-dimensional analog of the so-called ratio test from univariate power series. From the constructive proof of this result, we arrive at a method to approximate the region of convergence up to a desired accuracy. While most results in the literature are for rather specialized classes of multivariate power series, the method devised here is general. As far as applications are concerned, note that while theorems such as the Cauchy-Kowalevski theorem (and its generalizations to many variables) grant the existence of a region of convergence for a multivariate Taylor series to certain PDEs under appropriate restrictions, they do not give the actual region of convergence. The determination of the maximal region of convergence for such a series solution to a PDE is one application of our result.     

顶点着色随机图边数的中偏差

本文讨论了对顶点按照一定比列着色的随机图,利用泰勒展式和斯特灵公式,得到了随机图边数的中偏差和重对数律.     

A New Type of Euler Polynomials and Numbers

By defining two specific exponential generating functions, we introduce a kind of Euler polynomials and study its basic properties in detail. As an application of the introduced polynomials, we use them in computing some new series of Taylor type that contain the associated Euler numbers \(E_n(0)\) where \(E_n(x)\) is the Euler polynomial.