scholarly journals THE NONLINEAR SUPERPOSITION PRINCIPLE AND THE WEI-NORMAN METHOD

1998 ◽  
Vol 13 (21) ◽  
pp. 3601-3627 ◽  
Author(s):  
J. F. CARIÑENA ◽  
G. MARMO ◽  
J. NASARRE
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Order Differential Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
First Order ◽  
Second Order Differential Equations

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied to obtain the associated differential equation in the group SL(2, ℝ). The superposition principle for first order differential equation systems and Lie–Scheffers theorem are also analyzed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrödinger equation.

scholarly journals INTEGRABILITY OF THE RICCATI EQUATION FROM A GROUP-THEORETICAL VIEWPOINT

1999 ◽  
Vol 14 (12) ◽  
pp. 1935-1951 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
ARTURO RAMOS
Keyword(s):  
Riccati Equation ◽  
Superposition Principle ◽  
Theoretical Perspective ◽  
Nonlinear Superposition ◽  
Theoretical Methods ◽  
Integrability Conditions

In this paper we develop some group-theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation, and we discuss some of its integrability conditions from a group-theoretical perspective. The nonlinear superposition principle also arises in a simple way.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM

2020 ◽  
Vol 69 (1) ◽  
pp. 7-11
Author(s):  
A.K. Abirov ◽  
◽  
N.K. Shazhdekeeva ◽  
T.N. Akhmurzina ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Inhomogeneous System ◽  
Homogeneous Solutions ◽  
First Order ◽  
Zero Divisors ◽  
Hypercomplex System ◽  
Independent Variable ◽  
The Right

The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

scholarly journals A CONVENIENT COORDINATIZATION OF SIEGEL–JACOBI DOMAINS

2012 ◽  
Vol 24 (10) ◽  
pp. 1250024 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Half Plane ◽  
Order Differential Equation ◽  
Quantum Evolution ◽  
Classical Motion ◽  
Matrix Riccati Equation ◽  
First Order ◽  
Upper Half Plane ◽  
Linear Differential

We determine the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi ball [Formula: see text] as the sum of the Kähler two-form on ℂn and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a hermitian linear Hamiltonian in the generators of the Jacobi group [Formula: see text] are described by a matrix Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂn, with coefficients depending also on [Formula: see text]. Hn denotes the (2n+1)-dimensional Heisenberg group. The system of linear differential equations attached to the matrix Riccati equation is a linear Hamiltonian system on [Formula: see text]. When the transform FC : (η, W) → (z, W) is applied, the first-order differential equation in the variable [Formula: see text] becomes decoupled from the motion on the Siegel ball. Similar considerations are presented for the Siegel–Jacobi upper half plane [Formula: see text], where [Formula: see text] denotes the Siegel upper half plane.

scholarly journals Oscillation of even-order neutral differential equations via comparison principles

2014 ◽  
Vol 30 (3) ◽  
pp. 293-300
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Comparison Principles ◽  
Neutral Differential Equations ◽  
First Order ◽  
Even Order ◽  
Delay Differential

In the paper we offer oscillation criteria for even-order neutral differential equations, where z(t) = x(t) + p(t)x(τ(t)). Establishing a generalization of Philos and Staikos lemma, we introduce new comparison principles for reducing the examination of the properties of the higher order differential equation onto oscillation of the first order delay differential equations. The results obtained are easily verifiable.

scholarly journals Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 718 ◽  
Author(s):  
Emad R. Attia ◽  
Hassan A. El-Morshedy ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Oscillation Criteria ◽  
Lower Upper ◽  
First Order ◽  
Upper Limit ◽  
First Order Differential Equations

New sufficient criteria are obtained for the oscillation of a non-autonomous first order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability and strength of our results.

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

1982 ◽  
Vol 25 (3) ◽  
pp. 291-295 ◽  
Author(s):  
Lance L. Littlejohn ◽  
Samuel D. Shore
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Image Position

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

An Existence Result for Stepanoff Almost-Periodic Differential Equations

1971 ◽  
Vol 14 (4) ◽  
pp. 551-554 ◽  
Author(s):  
S. Zaidman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Existence Result ◽  
Order Differential Equation ◽  
Short Paper ◽  
Almost Periodic ◽  
First Order ◽  
Right Hand

In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.

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