scholarly journals Some Results on Equivalence Groups

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. C. Ndogmo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Lie Algebra ◽  
Infinitesimal Generator ◽  
Constant Factor ◽  
The Other ◽  
Invariant Functions ◽  
Levi Factor

The comparison of two common types of equivalence groups of differential equations is discussed, and it is shown that one type can be identified with a subgroup of the other type, and a case where the two groups are isomorphic is exhibited. A result on the determination of the finite transformations of the infinitesimal generator of the larger group, which is useful for the determination of the invariant functions of the differential equation, is also given. In addition, the Levidecomposition of the Lie algebra associated with the larger group is found; the Levi factor of which is shown to be equal, up to a constant factor, to the Lie algebra associated with the smaller group.

On the theory of the perturbations of the planets

1837 ◽  
Vol 3 ◽  
pp. 98-98
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
The Other ◽  
The Sun ◽  
Constant Force ◽  
First Case ◽  
Complete Determination ◽  
Analytical Formulae ◽  
Sole Action

The methods hitherto employed by mathematicians for determining the variations which the elements of the orbit of a planet undergo in consequence of perturbation, and for expressing these variations analytically in the manner best adapted for computation, are found to depend upon a theory in mechanics, of considerable intricacy, known by the name of the Variation of the Arbitrary Constants . In seeking the means for abridging the severe labour of the calculations, we must separate the general principles on which they are founded from the analytical processes by which they are carried into effect; and in some important problems great advantage is obtained by adapting the investigation to the particular circumstance of the case, and attending solely to the principles of the method in deducing the solution. The author suggests the possibility of simplifying physical astronomy by calling in the aid of only the usual principles of Dynamics, and by setting aside every formula or equation not absolutely necessary for arriving at the final results. The present paper contains a complete determination of the variable elements of the elliptic orbit of a disturbed planet, deduced from three differential equations, that follow readily from the mechanical conditions of the problem. In applying these equations the author observes, the procedure is the same whether a planet is urged by the sole action of the constant force of the sun, or is besides disturbed by the attraction of other bodies revolving round the luminary; the only difference being that, in the first case, the elements of the orbit are all constant, whereas in the other case they are all variable. The success of the method followed by the author is derived from a new differential equation between the time and the area described by the planet in its momentary plane, which greatly shortens the investigation by rendering it unnecessary to consider the projection of the orbit. But the solution given in the present paper, although it makes no reference to the analytical formulæ of the theory of the Variations of the Arbitrary Constants , is no less an application of that method and an example of its utility, and of the necessity of employing it in very complicated problems.

Determination of Stresses in Cemented Lap Joints

1953 ◽  
Vol 20 (3) ◽  
pp. 355-364
Author(s):  
R. W. Cornell
Keyword(s):  
Differential Equations ◽  
Infinite Number ◽  
Lap Joints ◽  
The Other ◽  
Complete Analysis ◽  
Stress Distributions ◽  
Lap Joint ◽  
Analogy Method ◽  
Set Up

Abstract A variation and extension of Goland and Reissner’s (1) method of approach is presented for determining the stresses in cemented lap joints by assuming that the two lap-joint plates act like simple beams and the more elastic cement layer is an infinite number of shear and tension springs. Differential equations are set up which describe the transfer of the load in one beam through the springs to the other beam. From the solution of these differential equations a fairly complete analysis of the stresses in the lap joint is obtained. The spring-beam analogy method is applied to a particular type of lap joint, and an analysis of the stresses at the discontinuity, stress distributions, and the effects of variables on these stresses are presented. In order to check the analytical results, they are compared to photoelastic and brittle lacquer experimental results. The spring-beam analogy solution was found to give a fairly accurate presentation of the stresses in the lap joint investigated and should be useful in analyzing other cemented lap-joint structures.

scholarly journals Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190564
Author(s):  
Zhi-Yong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Algebra ◽  
Infinitesimal Generator ◽  
Linear Time ◽  
Lie Symmetry ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Pde ◽  
Leading Position

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

THE CENTER PROBLEM FOR DISCONTINUOUS LIÉNARD DIFFERENTIAL EQUATION

1999 ◽  
Vol 09 (09) ◽  
pp. 1751-1761 ◽  
Author(s):  
B. COLL ◽  
R. PROHENS ◽  
A. GASULL
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Differential Equations ◽  
General Expression ◽  
The Other ◽  
Center Problem ◽  
Homogeneous Polynomials ◽  
Lyapunov Constants

We prove that the Lyapunov constants for differential equations given by a vector field with a line of discontinuities are quasi-homogeneous polynomials. This property is strongly used to derive the general expression of the Lyapunov constants for two families of discontinuous Liénard differential equations, modulus some unknown coefficients. In one of the families these coefficients are determined and the center problem is solved. In the other family the center problem is just solved by assuming that the coefficients which appear in these expressions are nonzero. This assumption on the coefficients is supported by their computation (analytical and numerical) for several cases.

ON STOCHASTICITY OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH A SMALL DELAY

2005 ◽  
Vol 05 (03) ◽  
pp. 475-486 ◽  
Author(s):  
MARK FREIDLIN ◽  
MATTHIAS WEBER
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Differential Equations ◽  
Intrinsic Property ◽  
Interior Vertex ◽  
The Other ◽  
Small Delay

We show that solutions of a class of differential equations with a small delay can be approximated, in a sense, by a stochastic process on a graph associated with the equation. This process moves as a deterministic motion inside any edge of the graph, but, after reaching an interior vertex of the graph, the process chooses one of the other adjacent edges to proceed there with a certain probability. These probabilities are calculated explicitly. The stochasticity is an intrinsic property of the differential equation with small delay.

Infinitesimal Kinematics Methods in the Mobility Determination of Kinematic Chains

2009 ◽  
Author(s):  
J. M. Rico ◽  
J. J. Cervantes ◽  
A. Tadeo ◽  
J. Gallardo ◽  
L. D. Aguilera ◽  
...  
Keyword(s):  
Lie Algebra ◽  
Good Deal ◽  
Higher Order ◽  
The Other ◽  
Velocity Analysis ◽  
Euclidean Group ◽  
Kinematic Chains ◽  
Order Analysis ◽  
The One

In recent years, there has been a good deal of controversy about the application of infinitesimal kinematics to the mobility determination of kinematic chains. On the one hand, there has been several publications that promote the use of the velocity analysis, without any additional results, for the determination of the mobility of kinematic chains. On the other hand, the authors of this contribution have received several reviews of researchers who have the strong belief that no infinitesimal method can be used to correctly determine the mobility of kinematic chains. In this contributions, it is attempted to show that velocity analysis by itself can not correctly determine the mobility of kinematic chains. However, velocity and higher order analysis coupled with some recent results about the Lie algebra, se(3), of the Euclidean group, SE(3), can correctly determine the mobility of kinematic chains.

scholarly journals Resummation methods for Master Integrals

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Dhimiter D. Canko ◽  
Nikolaos Syrrakos
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Canonical Basis ◽  
Proof Of Concept

Abstract We present in detail two resummation methods emerging from the application of the Simplified Differential Equations approach to a canonical basis of master integrals. The first one is a method which allows for an easy determination of the boundary conditions, since it finds relations between the boundaries of the basis elements and the second one indicates how using the x → 1 limit to the solutions of a canonical basis, one can obtain the solutions to a canonical basis for the same problem with one mass less. Both methods utilise the residue matrices for the letters {0, 1} of the canonical differential equation. As proof of concept, we apply these methods to a canonical basis for the three-loop ladder-box with one external mass off-shell, obtaining subsequently a canonical basis for the massless three-loop ladder-box as well as its solution.

Akbari-Ganji's Method

2017 ◽  
pp. 246-273
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Chemical Engineering ◽  
Solid Mechanics ◽  
Nonlinear Partial Differential Equation ◽  
The Other ◽  
Partial Differential ◽  
New Approach

The nonlinear partial differential equation governing on the mentioned system has been investigated by a simple and innovative method which we have named it Akbari-Ganji's Method or AGM. It is notable that this method has been compounded by Laplace transform theorem in order to covert the partial differential equation governing on the afore-mentioned system to an ODE and then the yielded equation has been solved conveniently by this new approach (AGM). One of the most important reasons of selecting the mentioned method for solving differential equations in a wide variety of fields not only in heat transfer science but also in different fields of study such as solid mechanics, fluid mechanics, chemical engineering, etc. in comparison with the other methods is as follows: Obviously, according to the order of differential equations, we need boundary conditions so in the case of the number of boundary conditions is less than the order of the differential equation, this method can create additional new boundary conditions in regard to the own differential equation and its derivatives. Therefore, a solution with high precision will be acquired. With regard to the afore-mentioned explanations, the process of solving nonlinear equation(s) will be very easy and convenient in comparison with the other methods.

scholarly journals An application of the spread relation to differential equations

1993 ◽  
Vol 36 (2) ◽  
pp. 211-229 ◽  
Author(s):  
Gary G. Gundersen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Meromorphic Functions ◽  
Meromorphic Solution ◽  
Finite Union ◽  
The Other

If a differential equation with meromorphic coefficients has a certain form where the growth of one of the coefficients dominates the growth of the other coefficients in a finite union of angles, then we show that this puts restrictions on the deficiencies of any meromorphic solution of the equation. We use the spread relation in the proofs. Examples are given which show that our results are sharp in several ways. Most of these examples are constructed from the quotients of solutions of w″ + G(z)w = 0 for certain polynomials G(z) and from meromorphic functions which are extremal for the spread relation.

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