Double‐Scale Expansions for a Logarithmic Type Solution to a q‐Analog of a Singular Initial Value Problem

Stéphane Malek

Abstract and Applied Analysis, 2024, vol. 2024, issue 1

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 logt 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.

Date: 2024
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https://doi.org/10.1155/2024/8904337

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2024:y:2024:i:1:n:8904337

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