相关论文: The generalized Taylor series approach is not equi…
A simple analytic tool namely the general series expansion method is proposed to find the solutions for nonlinear differential equations. By choosing a set of suitable basis functions $\{e_n(t,t_0)\}_{n=0}^{+\infty}$ such that the solution…
The generalized Taylor expansion including a secret auxiliary parameter $h$ which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of $h$ can't be…
The explicit analytic solution of the Thomas Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao (Appl. Math. Comp. 144, (2003)). However, the base functions and the auxiliary…
This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is…
The homotopy analysis method known from its successful applications to obtain quasi-analytical approximations of solutions of ordinary and partial differential equations is applied to stochastic differential equations with Gaussian…
A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular…
In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…
In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…
We prove simultaneous Universal Approximation of a certain type of Pade Approximants and of Taylor series with the same indexes. This is a generic phenomenon in the space of holomorphic functions in any simply connected domain, as well as…
The homotopy analysis method is studied in the present paper. The question of convergence of the homotopy analysis method is resolved. It is proven that under a special constraint the homotopy analysis method does converge to the exact…
Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and…
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…
We comment on some analytical solutions to a class of Boussinesq-like equations derived recently by means of the homotopy perturbation method (HPM). We show that one may obtain exactly the same result by means of the Taylor series in the…
We analyze a recent application of homotopy perturbation method to some heat-like and wave-like models and show that its main results are merely the Taylor expansions of exponential and hyperbolic functions. Besides, the authors require…
An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the six-order boussinesq equation. We summarize the general formulas for similarity reduction solutions and similarity reduction equations of…
We prove a formula for the Taylor series coefficients of a zero of the sum of a complex-exponent polynomial and a base function which is a general holomorphic function with a simple zero. Such a Taylor series is more general than a Puiseux…
In this work, Lienard equations are considered. The limit cycles of these systems are studied by applying the homotopy analysis method. The amplitude and frequency obtained with this methodology are in good agreement with those calculated…
Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…
Based on the Taylor expansion, we propose a renormalization method for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence the mathematical essence…
In the paper we analyse the exact solutions to scalar PDEs obtained thanks to summable Taylor series provided by Adomian's decomposition method. We propose the modification of the method which makes the calculations of Taylor coefficients…