On Summable Formal Power Series Solutions to Some Initial Value Problem with Infinite Order Irregular Singularity and Mahler Transforms Article Swipe

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Stéphane Malek ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.20944/preprints202412.0591.v1 · OA: W4405092424

In this paper, we examine a nonlinear partial differential equation in complex time t and complex space z combined with so-called Mahler transforms acting on time. This equation is endowed with a leading term represented by some infinite order formal differential operator of irregular type which enables the construction of a formal power series solution in t obtained by means of a Borel-Laplace procedure known as k-summability. The so-called k-sums are shown to solve some related differential functional equations involving integral transforms which stem from the analytic deceleration operators appearing in the multisummability theory for formal power series.