Abstract
The analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain are constructed and studied from an asymptotic point of view with respect to the perturbation parameter. Two types of holomorphic solutions, the so-called inner and outer solutions, are considered. Each of them holds a particular asymptotic relation with the formal ones in terms of asymptotic expansions in the perturbation parameter. The growth rate in the asymptotics leans on the − 1 -branch of Lambert W function, which turns out to be crucial.
Highlights
The intent of this study is to provide analytic solutions and their parametric asymptotic expansions to a family of singularly perturbed q-difference-differential equations in the complex domain, in which two time variables act, for some fixed q > 1
After a brief summary on q-asymptotic expansions in the first part of Section 6, and the description of Ramis–Sibuya type theorems, we provide formal power series expansions in the perturbation parameter of the analytic solutions and relate them asymptotically in adequate domains
We provide information about the difference of two solutions in consecutive sectors of the good covering, which will be crucial to determine the asymptotic behavior of the analytic solutions
Summary
The intent of this study is to provide analytic solutions and their parametric asymptotic expansions to a family of singularly perturbed q-difference-differential equations in the complex domain, in which two time variables act, for some fixed q > 1. After a brief summary on q-asymptotic expansions in the first part of Section 6, and the description of Ramis–Sibuya type theorems (see Theorems 4 and 5), we provide formal power series expansions in the perturbation parameter of the analytic solutions and relate them asymptotically in adequate domains. These results are attained in Theorems 6 and 7. The work concludes with a brief section of conclusions and two technical sections, Sections 8 and 9, left to the end of the work in order to not interfere with our reasonings
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