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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.

Parametric Borel summability for some semilinear system of partial differential equations

Yamazawa H., Yoshino M.

Opuscula Mathematica

2015

Vol. 35, no. 5

825--845

In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with n independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.