scholarly journals First Integrals of Differential Operators from SL(2,ℝ) Symmetries

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2167
Author(s):  
Paola Morando ◽  
Concepción Muriel ◽  
Adrián Ruiz
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Explicit Expression ◽  
Differential Operators ◽  
Symmetry Algebra ◽  
First Integral ◽  
First Integrals ◽  
Integrable Distribution ◽  
Algebraic Operations ◽  
Symmetry Generators

The construction of first integrals for SL(2,R)-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl(2,R). Firstly, we provide for n=2 an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of sl(2,R), and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for n>2, provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of sl(2,R) is known. Several examples illustrate the procedures.

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

scholarly journals PSEUDO-DIFFERENTIAL OPERATORS AND EQUATIONS IN A DISCRETE HALF-SPACE

2018 ◽  
Vol 23 (3) ◽  
pp. 492-506 ◽  
Author(s):  
Vladimir B. Vasilyev ◽  
Alexander V. Vasilyev
Keyword(s):  
Differential Operator ◽  
General Solution ◽  
Differential Equations ◽  
Half Space ◽  
Differential Operators ◽  
Pseudo Differential Operator ◽  
Pseudo Differential Operators ◽  
Special Case

We introduce a digital pseudo-differential operator acting in discrete Sobolev--Slobodetskii spaces and consider pseudo-differential equations with such operators in a discrete half-space. The theorem on a general solution of such equations is proved for a special case.

scholarly journals New Exact Solutions of Some Nonlinear Systems of Partial Differential Equations Using the First Integral Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shoukry Ibrahim Atia El-Ganaini
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
First Integrals ◽  
Partial Differential ◽  
First Integral Method ◽  
New Exact Solutions

The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE), (2 + 1)-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.

scholarly journals Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Oksana Bihun ◽  
Clark Mourning
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Hermite Polynomials ◽  
Pseudospectral Method ◽  
Matrix Representations ◽  
The Family ◽  
Linear Differential

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x)=qν(x)pν(x), where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

scholarly journals Newton polygons and gevrey indices for linear partial differential operators

1992 ◽  
Vol 128 ◽  
pp. 15-47 ◽  
Author(s):  
Masatake Miyake ◽  
Yoshiaki Hashimoto
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Convergent Power Series ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators

This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations.In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.

scholarly journals Symmetric Conformable Fractional Derivative of Complex Variables

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 363 ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Rafida M. Elobaid ◽  
Suzan J. Obaiys
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Unit Disk ◽  
Analytic Functions ◽  
Differential Operators ◽  
Function Theory ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Symmetric Differential Operators

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

scholarly journals On the Differential Operators with Periodic Matrix Coefficients

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
O. A. Veliev
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
The Self ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Periodic Matrix ◽  
Matrix Coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.

scholarly journals The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
X. Huang ◽  
X. Lu
Keyword(s):  
Differential Equations ◽  
Asymptotic Solution ◽  
Explicit Expression ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Differential Operators ◽  
Positive Real ◽  
B Splines ◽  
Fractional Differential ◽  
Linear Differential

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

scholarly journals Symmetries, Associated First Integrals, and Double Reduction of Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
L. Ndlovu ◽  
M. Folly-Gbetoula ◽  
A. H. Kara ◽  
A. Love
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
First Integral ◽  
First Integrals ◽  
Double Reduction ◽  
Invariance Condition ◽  
Symmetry Generator ◽  
The Difference ◽  
Symmetry Generators

We determine the symmetry generators of some ordinary difference equations and proceeded to find the first integral and reduce the order of the difference equations. We show that, in some cases, the symmetry generator and first integral areassociatedvia the “invariance condition.” That is, the first integral may be invariant under the symmetry of the original difference equation. When this condition is satisfied, we may proceed to double reduction of the difference equation.

scholarly journals Analytic integrability of quadratic–linear polynomial differential systems

2009 ◽  
Vol 31 (1) ◽  
pp. 245-258 ◽  
Author(s):  
JAUME LLIBRE ◽  
CLÀUDIA VALLS
Keyword(s):  
Singular Point ◽  
Explicit Expression ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Analytic Integrability ◽  
Linear Polynomial ◽  
Analytic First Integral

AbstractFor the quadratic–linear polynomial differential systems with a finite singular point, we classify the ones which have a global analytic first integral, and provide the explicit expression of their first integrals.

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