Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions

Abstract

We consider the second-order linear differential equation $(x^2-1)y''+f(x)y'+g(x)y=h(x)$ in the interval $(-1,1)$ with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions $f$, $g$ and $h$ are analytic in a Cassini disk ${\cal D}_r$ with foci at $x=\pm 1$ containing the interval $[-1,1]$. Then, the two end points of the interval may be regular singular points of the differential equation. The two-point Taylor expansion of the solution $y(x)$ at the end points $\pm 1$ is used to study the space of analytic solutions in ${\cal D}_r$ of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.

Keywords

• second-order linear differential equations
• regular singular point
• boundary value problem
• frobenius method
• two-point taylor expansions