Second order differential equations with an irregular singularity at the origin and a large parameter: convergent and asymptotic expansions

• Published in: Carpathian Journal of Mathematics 2016-01, Vol.32 (1), p.63-70
• Main Authors: Ferreira, Chelo, López, José L., Sinusía, Ester Pérez
• Format: Article
• Language: English
• Subjects: Approximation; Asymptotic methods; Bessel functions; Cauchy problem; Differential equations; Esters; Greens function; Mathematical independent variables; Second order differential equations
• ISSN: 1584-2851; 1843-4401
• DOI: 10.37193/CJM.2016.01.06

We consider the second order linear differential equation $y = \left[ {\frac{{{ \wedge ^2}}}{{{t^\alpha }}}+ g\left( t \right)} \right]y,$ where Λ is a large complex parameter and g is a continuous function. In previous works we have considered the case α ϵ (—∞, 2] and designed a convergent and asymptotic method for the solution of the corresponding initial value problem with data at t = 0. In this paper we complete the research initiated in those works and analyze the remaining case α ϵ (2, ∞). We use here the same fixed point technique; the main difference is that for α ϵ (2, ∞) the convergence of the method requires that the initial datum is given at a point different from the origin; for convenience we choose the point at the infinity. We obtain a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. generalization to non-linear problems is straightforward. An application to a quantum mechanical problem is given as an illustration.