Opuscula Math. 35, no. 5 (2015), 713-738

Opuscula Mathematica

q-analogue of summability of formal solutions of some linear q-difference-differential equations

Hidetoshi Tahara
Hiroshi Yamazawa

Abstract. Let \(q\gt 1\). The paper considers a linear \(q\)-difference-differential equation: it is a \(q\)-difference equation in the time variable \(t\), and a partial differential equation in the space variable \(z\). Under suitable conditions and by using \(q\)-Borel and \(q\)-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution \(\hat{X}(t,z)\) one can construct an actual holomorphic solution which admits \(\hat{X}(t,z)\) as a \(q\)-Gevrey asymptotic expansion of order \(1\).

Keywords: \(q\)-difference-differential equations, summability, formal power series solutions, \(q\)-Gevrey asymptotic expansions, \(q\)-Laplace transform.

Mathematics Subject Classification: 35C10, 35C20, 39A13.

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  1. M.S. Baouendi, C. Goulaouic, Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math. 26 (1973), 455-475.
  2. L. Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press Inc., Publishers, New York, 1963.
  3. A. Lastra, S. Malek, On \(q\)-Gevrey asymptotics for singularly perturbed \(q\)-difference-differential problems with an irregular singularity, Abstr. Appl. Anal. 2012, Art. ID 860716, 35 pp.
  4. A. Lastra, S. Malek, J. Sanz, On \(q\)-asymptotics for linear \(q\)-difference-differential equations with Fuchsian and irregular singularities, J. Differential Equations 252 (2012) 10, 5185-5216.
  5. S. Malek, On complex singularity analysis for linear \(q\)-difference-differential equations, J. Dyn. Control Syst. 15 (2009) 1, 83-98.
  6. S. Malek, On singularly perturbed \(q\)-difference-differential equations with irregular singularity, Dyn. Control Syst. 17 (2011) 2, 243-271.
  7. F. Marotte, C. Zhang, Multisommabilite des series entieres solutions formelles d'une equation aux \(q\)-differences lineaire analytique, Ann. Inst. Fourier 50 (2000) 6, 1859-1890.
  8. M. Miyake, Newton polygons and formal Gevrey indices in the Cauchy-Goursat-Fuchs type equations, J. Math. Soc. Japan 43 (1991) 2, 305-330.
  9. M. Nagumo, Über das Anfangswertproblem partieller Differentialgleichungen, Japan. J. Math. 18 (1941), 41-47.
  10. S. Ouchi, Multisummability of formal solutions of some linear partial differential equations, J. Differential Equations 185 (2002) 2, 513-549.
  11. J.P. Ramis, C. Zhang, Développement asymptotique \(q\)-Gevrey et fonction thêta de Jacobi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 899-902.
  12. J.P. Ramis, J. Sauloy, C. Zhang, Developpement asymptotique et sommabilite des solutions des equations lineaires aux \(q\)-differences, C.R. Math. Acad. Sci. Paris, 342 (2006) 7, 515-518.
  13. H. Tahara, H. Yamazawa, Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations 255 (2013), 3592-3637.
  14. C. Zhang, Développements asymptotiques \(q\)-Gevrey et séries \(Gq\)-sommables, Ann. Inst. Fourier 49 (1999) 1, 227-261.
  15. C. Zhang, Une sommation discrète pour des équations aux \(q\)-différences linéaires et à coefficients analytiques: théorie générale et exemples, Differential Equations and the Stokes Phenomenon, 309-329, World Sci. Publ., River Edge, NJ, 2002.
  • Hidetoshi Tahara
  • Sophia University, Department of Information and Communication Sciences, Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan
  • Hiroshi Yamazawa
  • Shibaura Institute of Technology, College of Engineer and Design, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan
  • Communicated by P.A. Cojuhari.
  • Received: 2013-11-11.
  • Revised: 2014-07-05.
  • Accepted: 2014-07-27.
  • Published online: 2015-04-27.
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Cite this article as:
Hidetoshi Tahara, Hiroshi Yamazawa, q-analogue of summability of formal solutions of some linear q-difference-differential equations, Opuscula Math. 35, no. 5 (2015), 713-738, http://dx.doi.org/10.7494/OpMath.2015.35.5.713

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