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

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 given at three points of the interval: the two extreme points $x=\pm 1$ and an interior point $x=s\in(-1,1)$. We consider $\varphi(x)$, $f(x)$, $g(x)$ and $h(x)$ analytic in a Cassini disk with foci at $x=\pm 1$ and $x=s$ containing the interval $[-1,1]$. The three-point Taylor expansion of the solution $y(x)$ at the extreme points $\pm 1$ and at $x=s$ is used to give a criterion for the existence and uniqueness of the solution of the boundary value problem. This method is constructive and provides the three-point Taylor approximation of the solution when it exists. We give several examples to illustrate the application of this technique.