Gevrey Expansions of Hypergeometric Integrals II

We study integral representations of the Gevrey series solutions of irregular hypergeometric systems under certain assumptions. We prove that, for such systems, any Gevrey series solution, along a coordinate hyperplane of its singular support, is the asymptotic expansion of a holomorphic solution given by a carefully chosen integral representation.

On Holomorphic Solution for Space- and Time-Fractional Telegraph Equations in Complex Domain

We consider some classes of space- and time-fractional telegraph equations in complex domain in sense of the Riemann-Liouville fractional operators for time and the Srivastava-Owa fractional operators for space. The existence and uniqueness of holomorphic solution are established. We illustrate our theoretical result by examples.

On the asymptotics of meromorphic solutions for nonlinear Riemann–Hilbert problems

This paper is devoted to a global existence theorem of meromorphic solutions of the form Z(z)=Zo(z)+R(z) of a nonlinear Riemann–Hilbert problem (RHP) for multiply connected domains Gq(q[ges ]1), where Zo(z) is the singular part of the solution, R(z) is the regular part which is a holomorphic solution of some appropriate nonlinear RHP for Gq(q[ges ]1). Under appropriate conditions on the characteristics of both the singular part Zo(z) (number of poles) and regular part (winding number) we prove the existence of meromorphic solutions Z(z) of the form Z(z)=Zo(z)+R(z). The proof is based on a special construction of the singular part Zo(z) and an adequate formulation of Newton's method for the regular part R(z).

Holomorphic solutions about an irregular singular point of an ordinary linear differential equation

It is shown that that an ordinary linear differential equation may possess a holomorphic solution in a neighbourhood of an irregular singular point even though the usual linearly independent solutions corresponding to the two roots of the indicial equation both have zero radius of convergence.