On the determination of approximate periodic solutions of some non-linear ODE

We propose a numerical-symbolic method for the approximation of periodic solutions of a type of non-linear ODE. The efficiency of our method is contrasted with the harmonic balance method and with another one which combines the differential transformation method with Padé approximants on a non trivial example: the relativistic oscillator. It is shown that our method is computationally more reliable.     

Defects and the Convergence of Padé Approximants

A selective survey is given of convergence results for sequences of Padé approximants. Various approaches for dealing with the convergence problems due to `defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the `smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Padé approximants to functions in the Baker–Gammel–Wills conjecture function class is presented.     

Generic approximation of functions by their Padé approximants

In the present article we obtain generic approximations, under sharp conditions, of holomorphic functions on arbitrary open sets by sequences of their Padé approximants. Similar results hold for functions smooth on the boundary of their domain of definition. In addition, the approximation is valid simultaneously with respect to all centers of expansion.     

Multivariate Padé approximants revisited

Several definitions of multivariate Padé approximants have been introduced during the last decade. We will here consider all types of definitions based on the choice that the coefficients in numerator and denominator of the multivariate Padé approximant are defined by means of a linear system of equations. In this case a determinant representation for the multivariate Padé approximant exists. We will show that a general recursive algorithm can be formulated to compute a multivariate Padé approximant given by any definition of this type. Here intermediate results in the recursive computation scheme will also be multivariate Padé approximants. Up to now such a recursive computation of multivariate Padé approximants only seemed possible in some special cases.     

On the convergence of type I Hermite–Padé approximants

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions.     

Anti-Gaussian Padé Approximants

This paper gives a synthesis of Padé approximants and anti-Gaussian quadratures. New rational approximants for Stieltjes series have been constructed. In addition, a three term recurrence relation is given for the numerator and denominator, which is useful when the given functional is not defin ite positive.We give the different algebraic properties of these new polynomials, which are similar to those obtained with the Gaussian quadrature formula. We find an easy definition and several relations with Padé approximants. Finally, some numerical results are given in the last section.     

A split-step finite difference method for nonparaxial nonlinear Schrödinger equation at critical dimension

The critical nonlinear Schrödinger equation (NLS) is the model equation for propagation of laser beam in bulk Kerr medium. One of the final stages in the derivation of NLS from the nonlinear Helmholtz equation (NLH) is to apply paraxial approximation. However, there is numerical evidence suggesting nonparaxiality prevents singularity formation in the solutions of NLS. Therefore, it is important to develop numerical methods for solving nonparaxial NLS. Split-step methods are widely used for finding numerical solutions of NLS equation. Nevertheless, these methods cannot be applied to nonparaxial NLS directly. In this study, we extend the applicability of split-step methods to nonparaxial NLS by using Padé approximant operators. In particular, split-step Crank-Nicolson (SSCN) method is used in conjunction with Padé approximants to provide examples of numerical solutions of nonparaxial NLS.     

Formal Orthogonal Polynomials and Newton–Padé Approximants

We proved in [8] that the denominators of Newton–Padé approximants for a formal Newton series are formal orthogonal with respect to linear functionals. The same functional is used along an antidiagonal of the Newton–Padé denominator table. The two linear functionals, corresponding to two adjacent antidiagonals, are linked with a very simple relation. Recurrence relations between denominators are given along an antidiagonal or two adjacent antidiagonals in the normal and non-normal case. The same recurrence relations are also satisfied by the Newton–Padé numerators, which implies another formal orthogonality.     

Computer extension and analytic continuation of problems of porous thrust bearings

Summary In this paper we propose new type of series solution, the semi-analytical, semi-numerical technique for the steady flow of a viscous fluid between two parallel disks in which the fluid is injected through the lower porous disk. We develop a double series expansion of the solution function and sufficiently large number of terms (30 terms-universal coefficients) in the expansion are obtained by delegating routine complex algebra to computer. The expression obtained in the form of power series for the normalized lift is analysed by means of Padé approximants. The change of roles of dependent and independent variables and the use of bilinear Euler transformation increases the region of validity of the corresponding power series. The results obtained from double series expansion method for small as well as moderately large values of cross-flow Reynolds number,R are more accurate and the computing time required in this method is negligible compared with pure numerical methods [1]. Besides this, we find that the method proposed by Phan-Thien and Bush [2] has to be implemented for each value ofR separately, whereas the one proposed by us has advantage of yielding, at a stretch, the results for larger range ofR which agree with exact values and at the same time requires less computer time. It is of interest to note that the pure numerical results in respect of normalized lift coefficient agree closely with the analytic continuation of our findings. In addition various Padé approximants are found to bracket the numerical results.     

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain.     

On two-point Padé-Type and two-Point Padé approximants

Summary Two-point Padé-type approximants are introduced in the case of a non-commutative algebra on a commutative field. Algebraic properties are given and a study of the error of approximation is done. From the relation of the error and some additional properties, two-point Padé approximants are found. Algebraic properties and recurrence relations are proved. The means to compute these approximants in following any way in the table of the approximants are given. The mixed table is introduced, as well as the normality and some results of convergence of two-point Padé-type and Padé approximants.     

A type of matrix Padé approximant inspired by scalar component models

In this paper we define a type of matrix Padé approximant inspired by the identification stage of multivariate time series models considering scalar component models. Of course, the formalization of certain properties in the matrix Padé approximation framework can be applied to time series models and in other fields. Specifically, we want to study matrix Padé approximants as follows: to find rational representations (or rational approximations) of a matrix formal power series, with both matrix polynomials, numerator and denominator, satisfying three conditions: (a) minimum row degrees for the numerator and denominator, (b) an invertible denominator at the origin, and (c) canonical representation (without free parameters).     

Some convergence results for the generalized Padé-type approximants

The aim of this paper is to give some convergence results for some sequences of generalized Padé-type approximants. We will consider two types of interpolatory functionals: one corresponding to Langrange and Hermite interpolation and the other corresponding to orthogonal expansions. For these two cases we will give sufficient conditions on the generating functionG(x, t) and on the linear functionalc in order to obtain the convergence of the corresponding sequence of generalized Padé-type approximants. Some examples are given.     

A Rational Approximant for the Digamma Function

Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use Padé approximants or other rational functions constructed from sequence transformations instead. However, neither Padé approximants nor sequence transformation utilize the information which is avaliable in the case of a special function – all power series coefficients as well as the truncation errors are explicitly known – in an optimal way. Thus, alternative rational approximants, which can profit from additional information of that kind, would be desirable. It is shown that in this way a rational approximant for the digamma function can be constructed which possesses a transformation error given by an explicitly known series expansion.     

A direct approach to convergence of multivariate,nonhomogeneous, Padé approximants

We present a direct approach for proving convergence in measure/product capacity of multivariate, nonhomogeneous, Padé approximants. Previous approaches have involved projection onto Padé-type approximation in one variable, and only yielded convergence in (Lebesgue) measure.     

Convergence Rates of Padé and Padé-Type Approximants

A comparison is made between Padé and Padé-type approximants. LetQ_nbe thenth orthonormal polynomial with respect to a positive measureμwith compact support inC. We show that for functions of the form[formula]wherewis an analytic function on the support ofμ, Padé-type approximants with denominatorQ_ngive a successful and, in general, better approximation procedure than Padé approximation.     

Two-point Padé approximants for the expansions of Stieltjes functions in real domain

The fundamental inequalities for the sequences of subdiagonal and diagonal one-point Padé approximants to Stieltjes function has been extended to the case of certain two-point Padé approximants. The results can be applied to the theory of inhomogeneous media for calculating the bounds for the effective transport coefficients of two-components heterogeneous materials.     

Numerical approximations and Padé approximants for a fractional population growth model

This paper presents an efficient numerical algorithm for approximate solutions of a fractional population growth model in a closed system. The time-fractional derivative is considered in the Caputo sense. The algorithm is based on Adomian’s decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components. Then the Padé approximants are effectively used in the analysis to capture the essential behavior of the population u(t) of identical individuals.     

Multivariate Two-Point Padé-Type and Two-Point Padé Approximants

In his paper the notions of two-point Padé-type and two-point Padé approximants are generalized for multivariate functions, with a generating denominator polynomial of general form. The multivariate two-point Padé approximant can be expressed as a ratio of two determinants and computed recursively using the E-algorithm. A comparison is made with previous definitions by other authors using particular generating denominator polynomials. The last section contains some convergence results.