scholarly journals INTEGRAL REPRESENTATION OF SOLUTIONS OF AN ORDINARY DIFFERENTIAL EQUATION AND THE LOEWNER– KUFAREV EQUATION

2020 ◽  
pp. 28-39
Author(s):  
Olga V. ZADOROZHNAYA ◽  
◽  
Vladimir K. KOCHETKOV ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Integral Representation ◽  
Univalent Functions ◽  
Integral Form ◽  
Second Order ◽  
Partial Differential ◽  
Representation Of Solutions ◽  
Order Part

The article presents a method of integral representation of solutions of ordinary differential equations and partial differential equations with a polynomial right-hand side part, which is an alternative to the construction of solutions of differential equations in the form of different series. The method is based on the introduction of additional analytical functions establishing the equation of connection between the introduced functions and the constituent components of the original differential equation. The implementation of the coupling equations contributes to the representation of solutions of the differential equation in the integral form, which allows solving some problems of mathematics and mathematical physics. The first part of the article describes the coupling equation for an ordinary differential equation of the first order with a special polynomial part of a higher order. Here, the integral representation of the solution of a differential equation with a second-order polynomial part is indicated in detail. In the second part of the paper, we consider the integral representation of the solution of a partial differential equation with the polynomial second-order part of the Loewner–Kufarev equation, which is an equation for univalent functions.

scholarly journals On the Existence of Eigenmodes of Linear Quasi-Periodic Differential Equations and their Relation to the MHD Continuum

1982 ◽  
Vol 37 (8) ◽  
pp. 830-839 ◽  
Author(s):  
A. Salat
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Infinite Sequence ◽  
Periodic Coefficient ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Magnetic Surfaces ◽  
Toroidal Geometry

The existence of quasi-periodic eigensolutions of a linear second order ordinary differential equation with quasi-periodic coefficient f{ω1t, ω2t) is investigated numerically and graphically. For sufficiently incommensurate frequencies ω1, ω2, a doubly indexed infinite sequence of eigenvalues and eigenmodes is obtained.The equation considered is a model for the magneto-hydrodynamic “continuum” in general toroidal geometry. The result suggests that continuum modes exist at least on sufficiently ir-rational magnetic surfaces

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

XIII.—Partial Differential Equations and the Calculus of Variations

1927 ◽  
Vol 46 ◽  
pp. 126-135 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Calculus Of Variations ◽  
Order Equation ◽  
Second Order ◽  
Physical Problem ◽  
Partial Differential ◽  
Hamiltonian Principle

A partial differential equation of physics may be defined as a linear second-order equation which is derivable from a Hamiltonian Principle by means of the methods of the Calculus of Variations. This principle states that the actual course of events in a physical problem is such that it gives to a certain integral a stationary value.

scholarly journals Positive decreasing solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1995-2010 ◽  
Author(s):  
Jelena Milosevic ◽  
Jelena Manojlovic
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Ordinary Differential Equations ◽  
Regular Variation ◽  
Second Order ◽  
Precise Information ◽  
Varying Coefficients

This paper is concerned with asymptotic analysis of positive decreasing solutions of the secondorder quasilinear ordinary differential equation (E) (p(t)?(|x'(t)|))'=q(t)?(x(t)), with the regularly varying coefficients p, q, ?, ?. An application of the theory of regular variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that lim t?? x(t)=0, lim t?? p(t)?(-x'(t))=?.

A non-standard numerical approach to the solution of some second-order ordinary differential equations

2015 ◽  
Vol 08 (04) ◽  
pp. 1550076 ◽  
Author(s):  
A. Adesoji Obayomi ◽  
Michael Olufemi Oke
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Approach ◽  
Local Approximation ◽  
Second Order ◽  
Discrete Models ◽  
Order Ordinary Differential Equation ◽  
Non Local

In this paper, a set of non-standard discrete models were constructed for the solution of non-homogenous second-order ordinary differential equation. We applied the method of non-local approximation and renormalization of the discretization functions to some problems and the result shows that the schemes behave qualitatively like the original equation.

scholarly journals Generalized multitime expansions for equations with slowly varying coefficients

1984 ◽  
Vol 7 (1) ◽  
pp. 151-158
Author(s):  
L. E. Levine ◽  
W. C. Obi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Time Scales ◽  
Constant Coefficient ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Varying Coefficients

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameterϵmay be obtained without resorting to secularity conditions if the time scalesti=ϵit(i=0,1,…)are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.

Numerical Analysis of the Exact Solution of the Wave Equation of the Longitudinal Seismic Oscillations of Soil and the Construction as a Point Insertion

2018 ◽  
Vol 931 ◽  
pp. 152-157 ◽  
Author(s):  
Kamil D. Yaxubayev ◽  
Dinara D. Kochergina
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Wave Equation ◽  
Numerical Analysis ◽  
Differential Equations ◽  
Partial Differential ◽  
Seismic Oscillations

The numerical analysis of the exact solution of the system of the differential equations which includes the partial differential equation of the longitudinal seismic oscillations of the soil and the ordinary differential equation of oscillations of the construction in the form of a point rigid insertion.

Spurious roots of delay differential equations using Galerkin approximations

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1178-1184
Author(s):  
Shanti S Kandala ◽  
Surya Samukham ◽  
Thomas K Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Second Order ◽  
Partial Differential ◽  
First Order ◽  
Galerkin Approximations ◽  
Delay Differential

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

scholarly journals Focal decompositions for linear differential equations of the second order

2003 ◽  
Vol 2003 (14) ◽  
pp. 813-821 ◽  
Author(s):  
L. Birbrair ◽  
M. Sobolevsky ◽  
P. Sobolevskii
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Equivalence Relation ◽  
Complete Classification ◽  
Second Order ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Second Order Equations

Focal decomposition associated to an ordinary differential equation of the second order is a partition of the set of all two-points boundary value problems according to the number of their solutions. Two equations are called focally equivalent if there exists a homomorphism of the set of two-points problems to itself such that the image of the focal decomposition associated to the first equation is a focal decomposition associated to the second one. In this paper, we present a complete classification for linear second-order equations with respect to this equivalence relation.

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