Integral representation of a solution to three-dimensional elliptic equation with strong degeneration

2015 ◽  
Vol 59 (11) ◽  
pp. 62-67
Author(s):  
G. Z. Khabibullina ◽  
S. V. Makletsov ◽  
R. M. Mavlyaviev
Keyword(s):  
Elliptic Equation ◽  
Integral Representation ◽  
Three Dimensional ◽  
Strong Degeneration

A THREE-DIMENSIONAL COVARIANT APPROACH TO MONODROMY (SKEIN RELATIONS) IN CHERN-SIMONS THEORY

1990 ◽  
Vol 05 (32) ◽  
pp. 2747-2751 ◽  
Author(s):  
B. BRODA
Keyword(s):  
Integral Representation ◽  
Path Integral ◽  
Wilson Loop ◽  
Three Dimensional ◽  
Simons Theory ◽  
Monodromy Operator ◽  
Covariant Approach ◽  
Path Integral Representation ◽  
Chern Simons ◽  
Chern Simons Theory

A genuinely three-dimensional covariant approach to the monodromy operator (skein relations) in the context of Chern-Simons theory is proposed. A holomorphic path-integral representation for the holonomy operator (Wilson loop) and for the non-abelian Stokes theorem is used.

scholarly journals Time-varying formation control of multiple quad-rotors based on ellipsoid

2021 ◽  
Vol 18 (2) ◽  
pp. 172988142110109
Author(s):  
Kecai Cao ◽  
Debao Xu
Keyword(s):  
Elliptic Equation ◽  
Formation Control ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Time Varying ◽  
Tuning Parameters ◽  
Model Control ◽  
Time Varying Parameters ◽  
Position Controller ◽  
Three Dimensional Space

Time-varying formation control problem for a group of multiple quad-rotors has been considered in this article with the help of ellipsoid. Firstly, an elliptic equation with time-varying parameters has been firstly introduced to describe the desired formation patterns for multiple quad-rotors in three-dimensional space. Then position controller and attitude controller are constructed using the method of sliding model control, respectively. Through tuning parameters of the elliptic equation, time-varying formation control of multiple quad-rotors has been realized using the controllers proposed in this article where smoothing transition between rigid formations has been guaranteed. Simulation results for formation control of quad-rotors that perform translation, scaling, and rotating actions have illustrated effectiveness of the time-varying formation controller that proposed in this article.

scholarly journals BEM Solutions of Frequency Domain Gradient Elastodynamic 3-D Porblems

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
D. Polyzos ◽  
K. G. Tsepoura ◽  
D. E. Beskos
Keyword(s):  
Integral Representation ◽  
Frequency Domain ◽  
Three Dimensional ◽  
Normal Derivative ◽  
Solution Procedure ◽  
Boundary Integral ◽  
Material Behavior ◽  
Linear Elastic ◽  
Microstructural Effects ◽  
Boundary Integral Representation

A boundary element methodology is presented for the frequency domain elastodynamic analysis of three-dimensional solids characterized by a linear elastic material behavior coupled with microstructural effects taken into account with the aid of the simple gradient elastic theory of Aifantis. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. All the kernels in the integral equations are explicitly provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy. The present version of the method does not provide explicit expressions for the computation of interior stresses.

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