scholarly journals Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
M. De la Sen ◽  
A. Ibeas ◽  
R. Nistal
Keyword(s):  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Series Expansions ◽  
Initial Condition ◽  
Shadowing Property ◽  
Sampling Points ◽  
Taylor Series Expansions ◽  
Well Posed

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling pointsti∈[t0,tJ]fori=0,1,…,Jof the solution. Two examples are provided.

scholarly journals A Shortcut to LAD Estimator Asymptotics

1991 ◽  
Vol 7 (4) ◽  
pp. 450-463 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Asymptotic Expansion ◽  
Regression Model ◽  
Taylor Series ◽  
Asymptotic Theory ◽  
Random Variables ◽  
Generalized Functions ◽  
Infinite Variance ◽  
Series Expansions ◽  
Linear Functionals ◽  
Taylor Series Expansions

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

A simple analytical approach to a nonlinear equation arising in porous catalyst

2017 ◽  
Vol 27 (4) ◽  
pp. 861-866 ◽  
Author(s):  
Chun-Hui He
Keyword(s):  
Differential Equations ◽  
Nonlinear Equation ◽  
Taylor Series ◽  
Analytical Approach ◽  
Nonlinear Differential Equations ◽  
Solution Process ◽  
Content Type ◽  
Porous Catalyst ◽  
Taylor Series Method ◽  
Heat And Fluid Flow

Purpose Analytical methods are widely used in heat and fluid flow; the purpose of this paper is to couple Taylor series method and Bubbfil algorithm to solve nonlinear differential equations. Design/methodology/approach A series solution is obtained with some unknown constants, which can be determined by incorporating boundary conditions, and the constants are calculated by the Bubbfil algorithm. Findings This paper gives an analytical approach to a nonlinear equation arising in porous catalyst using Taylor series and Bubbfil algorithm, and a high accuracy can be achieved. Research limitations/implications The coupled method of Taylor series and Bubbfil algorithm is a powerful method for nonlinear differential equations. Practical implications The proposed technology can be used in various numerical methods. Originality/value A new analytical method is proposed based on Taylor series and Bubbfil algorithm, which is a development of Newton’s iteration method and an ancient Chinese algorithm. The solution process is simple and easy to follow.

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

scholarly journals Approximate Solutions of Nonlinear Partial Differential Equations by Modifiedq-Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaheed N. Huseen ◽  
Said R. Grace
Keyword(s):  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
Power Series Solution ◽  
Analysis Method ◽  
Exactly Solvable ◽  
Homotopy Analysis

A modifiedq-homotopy analysis method (mq-HAM) was proposed for solvingnth-order nonlinear differential equations. This method improves the convergence of the series solution in thenHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting thenth-order nonlinear differential equation in the form ofnfirst-order differential equations. The solution of thesendifferential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

scholarly journals A Method for the Generation of Various Ghost Correction Algorithms—the Example of the Positivity Method and the Exponential Method

1991 ◽  
Vol 13 (4) ◽  
pp. 199-212 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Pole Figure ◽  
Taylor Series ◽  
The Other ◽  
New Method ◽  
Series Expansions ◽  
Theoretical Scheme ◽  
Taylor Series Expansions ◽  
Exponential Method ◽  
Pole Figure Data

A theoretical strategy is presented that can derive the algorithms of several existing ghost correction methods. The examples of the positivity method and the “GHOST” method are elaborated. A new method is derived as well: the “exponential” method. It can successfully replace the quadratic method as a method that yields an exactly non-negative complete C.O.D.F. from pole figure data. The theoretical scheme that can generate all these algorithms makes use of the fact, that several parameter sets can be defined in order to describe a C.O.D.F. The parameters of one set are then functions of those of the other. The algorithms are derived from Taylor series expansions of these functions.

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