On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities
We consider a Cauchy problem for some family of linear q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables X (t, z) for given formal power series initial conditions. Under suitable conditions and by the appl...
| Authors: | ; ; |
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| Format: | Article |
| Publication date: | 2012 |
| Country: | España |
| Institution: | Universidad de Alcalá (UAH) |
| Repository: | e_Buah Biblioteca Digital Universidad de Alcalá |
| Language: | English |
| OAI Identifier: | oai:ebuah.uah.es:10017/41470 |
| Online Access: | http://hdl.handle.net/10017/41470 https://dx.doi.org/10.1016/j.jde.2012.01.038 |
| Access Level: | Open access |
| Keyword: | q-Difference-differential equations q-Laplace transform Formal power series solutions q-Gevrey asymptotic expansions Small divisors Fuchsian and irregular singularities Matemáticas Mathematics |
| Summary: | We consider a Cauchy problem for some family of linear q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables X (t, z) for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we are able to deal with the small divisors phenomenon caused by the Fuchsian singularity, and to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion in t, uniformly for z in the compact sets of , is X (t, z) . The small divisorsʼ effect is an increase in the order of q-exponential growth and the appearance of a power of the factorial in the corresponding q-Gevrey bounds in the asymptotics. |
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