Singular Cauchy problem for a certain linear and 2nd order equation

Abstract

In this paper we consider the structure, in particular the singularities, of solutions of singular Cauchy problem for the following operator $L$ with holomorphic coefficients in the neighbourhood of the origin of $C^{2}$ under some conditions \[ L = D_{t}^{2} - (x +bt^{2})D_{x}^{2} - a(t, x)D_{t} - c(t, x)D_{x} - d (t , x). \] We construct its solution by so-called asymptotic expansion method and study its structure by the monodromy theory of the hypergeometric function.