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Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions. (English) Zbl 1463.34093

Summary: 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 \(\mathcal{D}_r\) 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 \(\mathcal{D}_r\) 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.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34B05 Linear boundary value problems for ordinary differential equations
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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