TY  - JOUR
ID  - Tahara2015
LB  - Tahara2015
AU  - Tahara, Hidetoshi
AU  - Yamazawa, Hiroshi
TI  - q-analogue of summability of formal solutions of some linear q-difference-differential equations
ST  - q-analogue of summability of formal solutions of some linear q-difference-differential equations
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 5
SP  - 713
EP  - 738
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.5.713
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.5.713
AB  - 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 \(1\).
KW  - \(q\)-difference-differential equations
KW  - summability
KW  - formal power series solutions
KW  - \(q\)-Gevrey asymptotic expansions
KW  - \(q\)-Laplace transform
SN  - 1232-9274
ER  -