Opuscula Math. 35, no. 5 (2015), 713-738
http://dx.doi.org/10.7494/OpMath.2015.35.5.713

 
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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  • 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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