scholarly journals Complete solutions of certain Lebesgue--Ramanujan--Nagell type equations

2020 ◽  
Vol 97 (3-4) ◽  
pp. 339-352
Author(s):  
Kalyan Chakraborty ◽  
Azizul Hoque ◽  
Richa Sharma
Keyword(s):  
Complete Solutions

Thin structure of heterogeneous and multiphase fluid flows

2012 ◽  
Vol 9 (1) ◽  
pp. 175-180
Author(s):  
Yu.D. Chashechkin
Keyword(s):  
Large Scale ◽  
Fundamental System ◽  
Fluid Flows ◽  
Regular Solutions ◽  
Flow Models ◽  
Group Methods ◽  
Wide Range ◽  
Thin Structure ◽  
Multiphase Fluid ◽  
Complete Solutions

According to the results of visualization of streams, the existence of structures in a wide range of scales is noted: from galactic to micron. The use of a fundamental system of equations is substantiated based on the results of comparing symmetries of various flow models with the usage of theoretical group methods. Complete solutions of the system are found by the methods of the singular perturbations theory with a condition of compatibility, which determines the characteristic equation. A comparison of complete solutions with experimental data shows that regular solutions characterize large-scale components of the flow, a rich family of singular solutions describes formation of the thin media structure. Examples of calculations and observations of stratified, rotating and multiphase media are given. The requirements for the technique of an adequate experiment are discussed.

General Features of Plastic-Elastic Problems as Exemplified by Some Particular Solutions

1949 ◽  
Vol 16 (3) ◽  
pp. 295-300
Author(s):  
Rodney Hill
Keyword(s):  
Spherical Shell ◽  
Infinite Medium ◽  
Cylindrical Hole ◽  
Particular Solutions ◽  
Complete Solutions ◽  
Complex Cases

Abstract New complete solutions based upon the Reuss equations are obtained for various plastic-elastic problems. These include the expansion of a spherical shell and of a cylindrical hole in an infinite medium. The solutions are used to exemplify certain features common to all plastic-elastic problems, with a view to introducing valid approximations in more complex cases.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

Complete solutions of three-dimensional problems in transversely isotropic media

2018 ◽  
Vol 32 (3) ◽  
pp. 775-802 ◽  
Author(s):  
Francesco Marmo ◽  
Salvatore Sessa ◽  
Nicoló Vaiana ◽  
Daniela De Gregorio ◽  
Luciano Rosati
Keyword(s):  
Three Dimensional ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Complete Solutions

Complete Solutions to Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With M-Homogenization

1992 ◽  
Author(s):  
Anoop K. Dhingra ◽  
Jyun-Cheng Cheng ◽  
Dilip Kohli
Keyword(s):  
Group Theory ◽  
Matrix Method ◽  
Numerical Results ◽  
Path Generation ◽  
Link Function ◽  
Homotopy Methods ◽  
Motion Generation ◽  
The Matrix ◽  
Function Generators ◽  
Complete Solutions

Abstract This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.

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