Version 1
: Received: 26 October 2023 / Approved: 27 October 2023 / Online: 27 October 2023 (12:51:38 CEST)
How to cite:
Malek, S. Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem. Preprints2023, 2023101781. https://doi.org/10.20944/preprints202310.1781.v1
Malek, S. Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem. Preprints 2023, 2023101781. https://doi.org/10.20944/preprints202310.1781.v1
Malek, S. Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem. Preprints2023, 2023101781. https://doi.org/10.20944/preprints202310.1781.v1
APA Style
Malek, S. (2023). Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem. Preprints. https://doi.org/10.20944/preprints202310.1781.v1
Chicago/Turabian Style
Malek, S. 2023 "Double-Scale Expansions for a Logarithmic Type Solution to a Q-analog of a Singular Initial Value Problem" Preprints. https://doi.org/10.20944/preprints202310.1781.v1
Abstract
We examine a linear q-difference differential equation which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation under study represents a q-analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. A sectorial holomorphic solution to the equation is constructed as a double complete Laplace transform in both time t and its complex logarithm log t and Fourier inverse integral in space. For a particular choice of the forcing term, this solution turns out to solve some specific nonlinear q-difference differential equation with polynomial coefficients in some positive rational power of t. Asymptotic expansions of the solution relatively to time t are investigated. A Gevrey type expansion is exhibited in a logarithmic scale. Furthermore, a formal asymptotic expansion in power scale is displayed, revealing a new fine structure involving remainders with both Gevrey and q-Gevrey type growth.
Keywords
Asymptotic expansion; Borel-Laplace transform; Fourier transform; initial value problem; formal power series
Subject
Computer Science and Mathematics, Analysis
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
