Analytic continuation of Borel sum of formal solution of semilinear partial differential equation

Article type: Research Article

Authors: Yoshino, Masafumi

Affiliations: Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-hiroshima, Hiroshima 739-8526, Japan. Tel.: +81 82 424 7345; Fax: +81 82 424 0710; E-mail: [email protected]

Abstract: We study the Borel summability and the analytic behavior of the Borel sum of a formal solution of first-order semilinear system with a singular perturbative parameter. By virtue of the representation formula of the Borel sum of a formal series solution expanded in terms of a parameter, we show that the analytic continuation of the Borel sum with respect to the parameter to a regular point in a singular direction coincides with the solution of the initial value problem expanded in the space variable. We also show that a similar phenomenon occurs outside the origin of the independent variables.

Keywords: Borel sum, singular equation, singular direction

DOI: 10.3233/ASY-141270

Journal: Asymptotic Analysis, vol. 92, no. 1-2, pp. 65-84, 2015