Double- and Simple-Layer Potentials for a Three-Dimensional Elliptic Equation with a Singular Coefficient and Their Applications

2021 ◽  
Vol 65 (1) ◽  
pp. 72-86
Author(s):  
T. G. Ergashev
Keyword(s):  
Elliptic Equation ◽  
Three Dimensional ◽  
Layer Potentials ◽  
Singular Coefficient

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 About Three Dimensional Jump Boundary Value Problems for the Laplacian

2019 ◽  
pp. 51-54
Author(s):  
Olexandr Polishchuk
Keyword(s):  
Hilbert Space ◽  
Integral Equation ◽  
Boundary Value Problems ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Equation System ◽  
Normal Derivative ◽  
Layer Potentials ◽  
Jump Problem ◽  
Well Posed

The conditions of well-posed solvability of searched function and its normal derivative three dimensional jump problem for the Laplacian and equivalent to them integral equation system for the sum of the simple and double layer potentials are determined in the Hilbert space, element of which as well as their normal derivatives have the jump through boundary surface.

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