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2 2015

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

Tahara H., Yamazawa H.

Opuscula Mathematica

2015

Vol. 35, no. 5

713--738

Let q > 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 X(t, z) one can construct an actual holomorphic solution which admits X(t, z) as a q-Gevrey asymptotic expansion of order 1.

Analytic continuation of solutions of some nonlinear convolution partial differential equations

Tahara H.

Opuscula Mathematica

2015

Vol. 35, no. 5

739--773

The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summa-bility theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.