The General Solutions to Some Systems of Adjointable Operator Equations

2018 ◽  
pp. 109-118
Author(s):  
Nan-Bin Cao ◽  
Yu-Ping Zhang
Keyword(s):  
Operator Equations ◽  
General Solutions

On Homogenization of Controlled Objects Described by Operator Equations of Hammerstein Type

2003 ◽  
Vol 35 (11) ◽  
pp. 19-27
Author(s):  
Victor N. Mizernyi ◽  
Peter I. Kogut ◽  
Tatyana N. Rudyanova
Keyword(s):  
Operator Equations

scholarly journals Some general solutions for linear Bragg–Hawthorne equation

2021 ◽  
Vol 33 (7) ◽  
pp. 077113
Author(s):  
Ting Yi
Keyword(s):  
General Solutions

Cylindrically Anisotropic Tubes Containing a Line Dislocation

2006 ◽  
Author(s):  
Chung-Hao Wang
Keyword(s):  
State Space ◽  
Dislocation Line ◽  
Longitudinal Axis ◽  
The State ◽  
Fourier Series Expansion ◽  
General Solutions ◽  
Mixed Dislocation ◽  
Line Dislocation ◽  
Space Formulation ◽  
Cylindrically Anisotropic

An analytical solution of the problem of a cylindrically anisotropic tube which contains a line dislocation is presented in this study. The state space formulation in conjunction with the eigenstrain theory is proved to be a feasible and systematic methodology to analyze a tube with the existence of dislocations. The state space formulation which expediently groups the displacements and the cylindrical surface traction can construct a governing differential matrix equation. By using Fourier series expansion and the well developed theory of matrix algebra, the asymmetrical solutions are not only explicit but also compact in form. The dislocation considered in this study is a kind of mixed dislocation which is the combination of edge dislocations and a screw dislocation and the dislocation line is parallel to the longitudinal axis of the tube. The degeneracy of the eigen relation and the technique to determine the inverse of a singular matrix are thoroughly discussed, so that the general solutions can be applied to the case of isotropic tubes, which is one of the novel features of this research. The results of isotropic problems, which are belong to the general solutions, are compared with the well-established expressions in the literature. The satisfied correspondences of these comparisons indicate the validness of this study. A cylindrically orthotropic tube is also investigated as an example and the numerical results for the displacements and tangential stress on the outer surface are displayed. The effects on surface stresses due to the existence of a dislocation appear to have a characteristic of localized phenomenon.

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