On the Existence and Uniqueness of the Formal Solution to a Singular Partial Differential Equation

Abstract

Abstract. According to the theory used in researching certain type of doubly singular differential equations, by computing directly, we find out that a singular partial differential equation in two complex variables has a unique formal power series solution which is in a special form and whose coefficients can be worked out explicitly. By estimating coefficients of the solution, we prove this formal solution is 1-Gevrey in a monomial. This gives a way to study Gevrey order in a monomial for formal solutions to certain classes of differential equations (or systems).