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.

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.

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.

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.

CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z5-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELDS OF DEGREE 5

2009 ◽  
Vol 19 (06) ◽  
pp. 2115-2121 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Vector Fields ◽  
Center Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed System ◽  
Lyapunov Constants

This paper proves that a Z5-equivariant planar polynomial vector field of degree 5 has at least five symmetric centers, if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z5-equivariant perturbations, the conclusion that the perturbed system has at least 25 limit cycles with the scheme 〈5 ∐ 5 ∐ 5 ∐ 5 ∐ 5〉 is rigorously proved.

CENTER PROBLEM AND MULTIPLE HOPF BIFURCATION FOR THE Z6-EQUIVARIANT PLANAR POLYNOMIAL VECTOR, FIELDS OF DEGREE 5

2009 ◽  
Vol 19 (05) ◽  
pp. 1741-1749 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Vector Fields ◽  
Hamiltonian Vector Field ◽  
Center Problem ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed System ◽  
Lyapunov Constants

This paper proves that a Z6-equivariant planar polynomial vector field of degree 5 has at least six symmetric centers, if and only if it is a Hamiltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z6-equivariant perturbations, the conclusion that the perturbed system has at least 24 limit cycles with the scheme 〈 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4 ∐ 4〉 is rigorously proved. Two schemes of distributions of limit cycles are given.

scholarly journals On the Growth of Solutions of Second Order Linear Complex Differential Equations whose Coefficients Satisfy Certain Conditions

2020 ◽  
Vol 17 (2) ◽  
pp. 0530
Author(s):  
Ayad Alkhalidy ◽  
Eman Hussein
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Infinite Order ◽  
Second Order ◽  
The Other ◽  
Nontrivial Solutions ◽  
Linear Complex ◽  
Order Linear ◽  
Complex Differential Equations ◽  
Growth Of Solutions

In this paper, we study the growth of solutions of the second order linear complex differential equations  insuring that any nontrivial solutions are of infinite order. It is assumed that the coefficients satisfy the extremal condition for Yang’s inequality and the extremal condition for Denjoy’s conjecture. The other condition is that one of the coefficients itself is a solution of the differential equation .

scholarly journals Discontinuous Galerkin Spectral Element Method for Solving Population Balance Differential Equations

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Discontinuous Galerkin ◽  
Legendre Polynomials ◽  
Spectral Element Method ◽  
Population Balance ◽  
Spectral Element ◽  
The Other ◽  
Test Functions ◽  
Element Method

In this paper, we study the discontinuous Galerkin spectral element method for solving the population balance differential equation. We use the Legendre polynomials of order k as test functions on each element. Calculate the matrices used by Gauss Quadrature integration and then compare the numerical results with the other methods.

Finite Difference, Finite Element and Boundary Element Formulae as a Language of Applied Physics

1991 ◽  
Vol 02 (01) ◽  
pp. 383-386
Author(s):  
JIŘÍ KAFKA ◽  
NGUYEN VAN NHAC
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Integral Form ◽  
The Other ◽  
Applied Physics ◽  
Partial Differential

When deducing the finite difference formulae, one has to discretize partial differential equations. On the other hand, those equations have been previously derived having started from laws of physics in their integral form. So, a question arises, why not avoid the approach to the limit (necessary to deduce the partial differential equation) and why not deduce the finite difference formulae directly on the base of laws of physics in their integral form.

Sign in / Sign up

Export Citation Format

Share Document