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

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 given at three points of the interval: the two extreme points x=±1 and an interior point x=s∈(−1,1). We consider φ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x=±1 and x=s containing the interval [−1,1]. The three-point Taylor expansion of the solution y(x) at the extreme points ±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.