On the summability of divergent power series solutions for certain first-order linear PDES

• Autorzy: Hibino M.

Identyfikatory

DOI

http://dx.doi.Org/10.7494/OpMath.2015.35.5.595

• Języki publikacji: EN

Abstrakty

EN

This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.

Słowa kluczowe

EN

partial differential equation; divergent power series; summability; asymptotic expansion; analytic continuation; integro-differential equation; integral equation

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2015
• Tom: Vol. 35, no. 5
• Strony: 595--624
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Hibino M.

• Meijo University Department of Mathematics Nagoya, Japan

Bibliografia

• [1] W. Balser, From Divergent Power Series to Analytic Functions. Theory and Application of Multisummable Power Series, Lecture Notes in Mathematics, 1582, Springer-Verlag, 1994.
• [2] W. Balser, Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schafke, Pacific J. Math. 188 (1999), 53-63.
• [3] W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Universitext, Springer-Verlag, 2000.
• [4] W. Balser, Multisummability of formal power series solutions of partial differential equations with constant coefficients, J. Differential Equations 201 (2004), 63-74.
• [5] W. Balser, M. Miyake, Summability of formal solutions of certain partial differential equations, Acta Sci. Math. (Szeged) 65 (1999), 543-552.
• [6] M. Hibino, Divergence property of formal solutions for singular first order linear partial differential equations, Publ. Res. fnst. Math. Sci. 35 (1999), 893-919.
• [7] M. Hibino, Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. I, J. Differential Equations 227 (2006), 499-533.
• [8] M. Hibino, Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients. II, J. Differential Equations 227 (2006), 534-563.
• [9] M. Hibino, Summability of formal solutions for singular first-order linear PDEs with holomorphic coefficients, Differential Equations and Exact WKB Analysis, 47-62, RIMS Kokyuroku Bessatsu BIO, Res. Inst. Math. Sci., 2008.
• [10] M. Hibino, Summability of formal solutions for singular first-order partial differential equations with holomorphic coefficients. I, in preparation.
• [If] M. Hibino, Summability of formal solutions for singular first-order partial differential equations with holomorphic coefficients. II, in preparation.
• [12] D.A. Lutz, M. Miyake, R. Schafke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J. 154 (1999), 1-29.
• [13] B. Malgrange, Sommation des series divergentes, Exposition. Math. 13 (1995), 163-222.
• [14] M. Miyake, Borel summability of divergent solutions of the Cauchy problem to non-Kowalevskian equations, Partial Differential Equations and Their Applications (Wuhan, 1999), 225-239, World Sci. Publ., 1999.
• [15] S. Ouchi, Multisummability of formal solutions of some linear partial differential equa­tions, J. Differential Equations 185 (2002), 513-549.