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

Abstract

Abstract. We consider second-order linear differential equations ϕ(x)y + f (x)y + g(x)y = h(x) in the interval (−1, 1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider ϕ(x), f (x), g(x) and h(x) analytic in a Cassini disk with foci at x = ±1 containing the interval (−1, 1). The two-point Taylor expansion of the solution y(x) at the extreme points ±1 is used to give a criterion for the existence and uniqueness of solution of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the solution(s) when it exists.

A three-point Taylor algorithm for three-point boundary value problems

“…Because we will use them later, denote by R i, j , i = 1, 2, j = 1, 2, 3, 4, 5, 6, the entries of this matrix R. For simplicity in the exposition, we do not write them explicitly here, just observe that they are data of the problem because the boundary matrix B is a datum of the problem. On the other hand (and as it happens in the standard Frobenius method for initial value problems), by introducing (4), (5) and (7) into the differential equation in (11), we find that the coefficients a n , b n and c n of the three-point Taylor expansion (4) of the solution y(x) of the differential equation in (11), satisfy a system of recursions of the form:…”

“…Since, in this case, we have a system of three recurrence relations instead of only one recursion, the computation of the coefficients a n , b n , c n for n 2 requires the initial seed a 0 , a 1 , b 0 , b 1 , c 0 and c 1 . This does not mean that the linear space of solutions of the differential equation in (11) has dimension six, this space has of course dimension two. It is happening here that, apart from the two-dimensional linear space S of (true) solutions of the differential equation in (11), there is a bigger space of formal solutions W defined by:…”

“…In [11] we have considered the same problem, but with data given only at the two extreme points of the interval x = ±1. Using a two-point Taylor expansion [10] of the solution, we have given in [11] a simple algebraic criterion for the existence and uniqueness of the solution of the boundary value problem considered there.…”

“…Using a two-point Taylor expansion [10] of the solution, we have given in [11] a simple algebraic criterion for the existence and uniqueness of the solution of the boundary value problem considered there. The purpose of this paper is to investigate the possible extension of the theory developed in [11] to the problem (1) defined by a boundary condition given at three points. A two-point Taylor expansion is not suitable for these kind of problems because the boundary data are given at three points.…”

A new spectral method for a class of linear boundary value problems

“…Also, they can be differentiated and integrated easily and can approximate a great variety of functions to any accuracy desired. For these reasons, for the numerical solution of problem (1)-(2), many methods have been developed which uses polynomial basis, such as spline functions ( [1,13,18,28,29]), Bernstein polynomials ( [3]), Hermite polynomials ( [30]), classical Taylor polynomials ( [19,31]), two-point Taylor polynomials ( [9,26]), Lagrange polynomials ( [8]), Lidstone polynomials ( [10]). In [33] the author uses Chebyshev polynomials, along with fast Fourier transform and baricentric interpolation, in order to construct a powerful tool, Chebfun ( [12]), for the solution of differential equations.…”