On Complex Singularity Analysis for Some Linear Partial Differential Equations in

A. Lastra, S. Malek and C. Stenger

Abstract and Applied Analysis, 2013, vol. 2013, 1-30

Abstract:

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.

Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:394564

DOI: 10.1155/2013/394564

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