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Continuation of holomorphic solutions to convolution equations in complex domains

100%

Ishimura R. , Okada J.-i. , Okada Y.

2000

tom 74

nr 1

105-115

For an analytic functional $S$ on $ℂ^n$, we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in $ℂ^n$. We determine the directions in which every solution can be continued analytically, by using the characteristic set.

Analytic continuation of solutions of some nonlinear convolution partial differential equations

88%

Tahara H.

Opuscula Mathematica

2015

tom 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.

A screw dislocation located outside, inside or on the interface of a parabolic elastic inhomogeneity

88%

Wang X. , Schiavone P.

Archives of Mechanics

2021

tom Vol. 73, nr 3

219--235

Using conformal mapping techniques, superposition and analytic continuation, we derive analytic solutions to the problem of a screw dislocation interacting with a parabolic elastic inhomogeneity. The screw dislocation can be located anywhere either in the surrounding matrix or in the parabolic inhomogeneity or simply on the parabolic interface itself. We obtain explicit expressions for the two analytic functions in the image plane characterizing the elastic fields describing displacement and stresses in the two-phase composite. Using the Peach-Koehler formula, we also obtain the image force acting on the screw dislocation. The analytic function defined in the parabolic inhomogeneity in the physical plane can be interpreted in terms of real and image screw dislocations for any location of the real screw dislocation.

Interaction between an edge dislocation and a rigid hypotrochoidal inhomogeneity

75%

Wang X. , Schiavone P.

Archives of Mechanics

2024

tom Vol. 76, nr 1-2

143--153

We study the plane elasticity problem associated with a rigid hypotrochoidal inhomogeneity embedded in an infinite isotropic elastic matrix subjected to an edge dislocation located at an arbitrary position. A closed-form solution to the problem is derived primarily with the aid of conformal mapping and analytic continuation. All of the unknown complex constants appearing in the pair of analytic functions characterizing the elastic field in the matrix are determined in an analytical manner. In addition, a simple method distinct from that by Santare and Keer (1986) is proposed to determine the rigid body rotation of the rigid inhomogeneity.

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

75%

Hibino M.

Opuscula Mathematica

2015

tom Vol. 35, no. 5

595--624

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.