Multi-point Taylor approximations in one-dimensional linear boundary value problems

Abstract

ABSTRACT. We consider second order linear differential equations in a real interval I with mixed Dirichlet and Neumann boundary data. We consider a representation of its solution by a multi-point Taylor expansion. The number and location of the base points of that expansion are conveniently chosen to guarantee that the expansion is uniformly convergent ∀ x ∈ I. We propose several algorithms to approximate the multipoint Taylor polynomials of the solution based on the power series method for initial value problems.

Two-point Taylor expansions and one-dimensional boundary value problems

“…On the other hand, as it happens in the standard Frobenius method for initial value problems, the coefficients a n and b n of the two-point Taylor expansion (2.1) of the solution y(x) of the differential equation in (2.5) satisfy a system of recurrences of the form [5] …”

“…Then, we have that any solution of the differential equation in (1.1) can be represented in the form of a two-point Taylor expansion at the base points x = ±1 [5] (2.1)…”

“…In [5] we have shown that, when ϕ, f , g and h are analytic in a region containing the interval [−1, 1], a two-point Taylor expansion of the solution y(x) at the two extreme points of the interval ±1 is useful to approximate the solution of the boundary value problem. The convergence region for that two-point Taylor expansion is a Cassini disk (see Figure 1) that avoids the possible singularities of the coefficient functions more efficiently than the standard Taylor disk [5].…”

“…In [5] we have assumed that problem (1.1) has a unique solution and then we have proposed several algorithms to approximate that solution. The purpose of this paper is different.…”

New series expansions of the Gauss hypergeometric function

“…As has been pointed out in [6] (in a different context), the use of a multi-point Taylor expansion [4], [5] with base points in the interval (0, 1) is preferable to using a standard Taylor expansion. With a multi-point Taylor expansion we can avoid the singularity t = 1/z of f (t) in its domain of convergence in a better way, and, at the same time, include the whole interval (0, 1) in its interior (see Fig.…”