# American Institute of Mathematical Sciences

October  2019, 12(6): 1791-1806. doi: 10.3934/dcdss.2019118

## 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

Received  November 2017 Revised  February 2018 Published  November 2018

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.

Citation: Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
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##### References:
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)
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
Example 1, 2-point Taylor series (n = 2). Convergence with the degree p
Example 1, degree p = 4. Convergence with the number expansion points
Example 1, 2-point Taylor series, c $\pm \frac{10}{3}$, p = 6. Distribution of the residual and of the error in the interval
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
Example 2, 4-point Taylor series, expansion points located in ($\pm 0.8; \pm 1.8$). Convergence with the degree
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 1)$, $u_{max} = 4.718$
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 0.1)$, $u_{max} = 5.0788$
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 4$, $(L = 10, a = 0.1)$, $u_{max} = 5.109$
Example 3, residual with 3 subdomains, $p = 5$, $n = 4$. Case of a smooth distribution of points
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