On The Motion of Three Vortices

1949 ◽  
Vol 1 (3) ◽  
pp. 257-270 ◽  
Author(s):  
J. L. Synge
Keyword(s):  
Differential Equations ◽  
Incompressible Fluid ◽  
Linear Differential Equations ◽  
Vortex Filaments ◽  
First Order ◽  
Perfect Incompressible Fluid ◽  
Non Linear ◽  
The Mean ◽  
Linear Differential

In a perfect incompressible fluid extending to infinity, the determination of the motion of N parallel rectilinear vortex filaments involves the solution of N non-linear differential equations, each of the first order. The method of Kirchhoff provides certain constants of the motion. If we describe the positions of the vortices by their point-traces on a plane perpendicular to them, the following facts follow from the theory of Kirchhoff:(1.1) The mean centre of the system is fixed.

On reducible non-linear differential equations occurring in mechanics

1953 ◽  
Vol 217 (1130) ◽  
pp. 327-343 ◽  
Keyword(s):  
Differential Equations ◽  
Basic Problem ◽  
Order Differential Equation ◽  
Linear Differential Equations ◽  
Qualitative Information ◽  
First Order ◽  
Internal Ballistics ◽  
Integral Curves ◽  
Non Linear ◽  
Linear Differential

The purpose of this paper is to present a new method of approach to certain problems in mechanics which give rise to ordinary non-linear differential equations of the second order. The method, which is based on the topology of the integral curves of a first-order differential equation, aims at providing qualitative information which can be used, if necessary, in guiding numerical calculations of the solutions. Among the equations discussed are those of Emden and Blasius, which occur in astrophysics and in boundary-layer theory respectively; these, together with the equation of a basic problem of internal ballistics, are shown to be reducible to different forms of the same first-order equation, which is itself of a type studied originally by Poincaré in another connexion.

scholarly journals Some Important Criteria for Oscillation of Non-Linear Differential Equations with Middle Term

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 346 ◽  
Author(s):  
Saad Althobati ◽  
Omar Bazighifan ◽  
Mehmet Yavuz
Keyword(s):  
Differential Equations ◽  
Comparison Method ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Middle Term ◽  
Oscillation Criteria ◽  
First Order ◽  
Non Linear ◽  
Linear Differential ◽  
Higher Order Differential Equations

In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with first-order equations. The obtained results extend and simplify known conditions in the literature. Furthermore, examining the validity of the proposed criteria is demonstrated via particular examples.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

scholarly journals On non-homogeneous canonical third-order linear differential equations

1994 ◽  
Vol 57 (2) ◽  
pp. 138-148 ◽  
Author(s):  
N. Parhi
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Linear Differential Equations ◽  
Third Order ◽  
Order Linear ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

AbstractIn this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

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