Multi-point Taylor series to solve differential equations

Djédjé Sylvain Zézé; Michel Potier-Ferry; Yannick Tampango

doi:10.3934/dcdss.2019118

Article Contents

2019, Volume 12, Issue 6: 1791-1806. Doi: 10.3934/dcdss.2019118

This issue Previous Article Topological remarks and new examples of persistence of diversity in biological dynamics Next Article

Multi-point Taylor series to solve differential equations

1.
Laboratoire de Mathématiques-Informatique, Université Nangui Abrogoua, Unité de Formation et de Recherche en Sciences Fondamentales et Appliquées, 02 B.P. V 102 Abidjan, Côte d'Ivoire
2.
Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France
3.
EDF R & D Saclay, 7 boulevard Gaspard Monge 91120 Palaiseau, France

* Corresponding author: Zézé
* Corresponding author: Zézé

Received: October 31, 2017

Revised: January 31, 2018

Published: April 2019

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Abstract

The use of Taylor series is an effective numerical method to solve ordinary differential equations but this fails when the sought function is not analytic or when it has singularities close to the domain. These drawbacks can be partially removed by considering multi-point Taylor series, but up to now there are only few applications of the latter method in the literature and not for problems with very localized solutions. In this respect, a new numerical procedure is presented that works for an arbitrary cloud of expansion points and it is assessed from several numerical experiments.

Keywords:

Taylor meshless method,
multi-point Taylor series,
Hermite Birkhoff polynomial interpolation,
differential equations,
high degree polynomials.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. Discretization points in the case of three subdomains and two point-Taylor series. The expansion points are $x_i$ and the additional collocation points are $y_j$ (2 per subdomain)

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Figure 2. Example 1: $f(x) = g(x) = 1, L = 10$. Comparison of a one-point Taylor series, one-point Taylor series in two subdomains and two-point Taylor series

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Figure 3. Example 1, 2-point Taylor series (n = 2). Convergence with the degree p

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Figure 4. Example 1, degree p = 4. Convergence with the number expansion points

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Figure 5. Example 1, 2-point Taylor series, c $\pm \frac{10}{3}$, p = 6. Distribution of the residual and of the error in the interval

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Figure 6. Example 1, 2-point Taylor series, p = 6. Maximal value of the residual and of the error according to the location of the expansion points

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Figure 7. Example 2, 4-point Taylor series, expansion points located in ($\pm 0.8; \pm 1.8$). Convergence with the degree

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Figure 8. Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 1)$, $u_{max} = 4.718$

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Figure 9. Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 0.1)$, $u_{max} = 5.0788$

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Figure 10. Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 4$, $(L = 10, a = 0.1)$, $u_{max} = 5.109$

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Figure 11. Example 3, residual with 3 subdomains, $p = 5$, $n = 4$. Case of a smooth distribution of points

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