Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem
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 differen...
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| Format: | Article |
| Language: | English |
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Wiley
2024-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2024/8904337 |
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| _version_ | 1832543464320925696 |
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| author | Stéphane Malek |
| author_facet | Stéphane Malek |
| author_sort | Stéphane Malek |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f0077a041d60465fa0375a83639f8e60 |
| institution | Kabale University |
| issn | 1687-0409 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-f0077a041d60465fa0375a83639f8e602025-02-03T11:43:23ZengWileyAbstract and Applied Analysis1687-04092024-01-01202410.1155/2024/8904337Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value ProblemStéphane Malek0Laboratoire Paul PainlevéWe 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.http://dx.doi.org/10.1155/2024/8904337 |
| spellingShingle | Stéphane Malek Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem Abstract and Applied Analysis |
| title | Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem |
| title_full | Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem |
| title_fullStr | Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem |
| title_full_unstemmed | Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem |
| title_short | Double-Scale Expansions for a Logarithmic Type Solution to a q-Analog of a Singular Initial Value Problem |
| title_sort | double scale expansions for a logarithmic type solution to a q analog of a singular initial value problem |
| url | http://dx.doi.org/10.1155/2024/8904337 |
| work_keys_str_mv | AT stephanemalek doublescaleexpansionsforalogarithmictypesolutiontoaqanalogofasingularinitialvalueproblem |