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# Multi-point Taylor series to solve differential equations

• * Corresponding author: Zézé
• 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.

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

 Citation: • • 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)

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

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

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

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

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

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

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

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

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

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

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