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

López, José; Sinusía, Ester

doi:10.1090/S0025-5718-10-02370-7

Two-point Taylor expansions and one-dimensional boundary value problems
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by José L. López and Ester Pérez Sinusía PDF

Math. Comp. 79 (2010), 2103-2115 Request permission

Abstract:

We consider second-order linear differential equations $\varphi (x)y”+f(x)y’+g(x)y=h(x)$ in the interval $(-1,1)$ with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider $\varphi (x)$, $f(x)$, $g(x)$ and $h(x)$ analytic in a Cassini disk with foci at $x=\pm 1$ containing the interval $(-1,1)$. The two-point Taylor expansion of the solution $y(x)$ at the extreme points $\pm 1$ 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.

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