scholarly journals Alternative integration procedure for scale-invariant ordinary differential equations

1979 ◽  
Vol 2 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Gerald Rosen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Parametric Form ◽  
Scale Invariant ◽  
Invariant Variable ◽  
Parameter Group ◽  
Independent Variables ◽  
Scale Transformations

For an ordinary differential equation invariant under a one-parameter group of scale transformationsx→λx,y→λαy,y′→λα−1y′,y″→λα−2y″, etc., it is shown by example that an explicit analytical general solution may be obtainable in parametric form in terms of the scale-invariant variableξ=∫xy−1/αdx. This alternative integration may go through, as it does for the example equationy″=kxy−2y′, in cases for which the customary dependent and independent variables(x−αy)and(ℓnx)do not yield an analytically integrable transformed equation.

scholarly journals info geting lost in plates 7u,8,9,11 and 13 - I. On the normal series satisfying linear differential equations

1906 ◽  
Vol 205 (387-401) ◽  
pp. 1-35 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
London Mathematical Society ◽  
Linear Ordinary Differential Equation ◽  
Linear Differential Equations ◽  
Independent Variables ◽  
Normal Series ◽  
The Matrix ◽  
Linear Differential

1. The present paper is suggested by that of Dr. H. F. Baker in the ‘Proceedings of the London Mathematical Society,’ vol. xxxv., p. 333, “On the Integration of Linear Differential Equations.” In that paper a linear ordinary differential equation of order n is considered as derived from a system of n linear simultaneous differential equations dx i / dt = u i1 x +.....+ u i n x n ( i = 1... n ), or, in abbreviated notation, dx / dt = ux , where u is a square matrix of n rows and columns whose elements are functions of t , and x denotes a column of n independent variables. A symbolic solution of this system is there given and denoted by the symbol Ω( u ). This is a matrix of n rows and columns formed from u as follows :—Q ( ϕ ) is the matrix of which each element is the t -integral from t 0 to t of the corresponding element of ϕ , ϕ being any matrix of n rows and columns; then Ω( u ) = 1+Q u +Q u Q u +Q u Q u Q u ..... ad inf ., where the operator Q affects the whole of the part following it in any term.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

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.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

scholarly journals On the Polynomial Solutions and Limit Cycles of Some Generalized Polynomial Ordinary Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1139
Author(s):  
Claudia Valls
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Limit Cycles ◽  
Polynomial Solutions ◽  
Generalized Polynomial

We study equations of the form y d y / d x = P ( x , y ) where P ( x , y ) ∈ R [ x , y ] with degree n in the y-variable. We prove that this ordinary differential equation has at most n polynomial solutions (not necessarily constant but coprime among each other) and this bound is sharp. We also consider polynomial limit cycles and their multiplicity.

scholarly journals Nonindependent Session Recommendation Based on Ordinary Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zhenyu Yang ◽  
Mingge Zhang ◽  
Guojing Liu ◽  
Mingyu Li
Keyword(s):  
Neural Network ◽  
Differential Equation ◽  
Neural Networks ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Deep Neural Networks ◽  
Semantic Information ◽  
User Behavior ◽  
User Behaviors

The recommendation method based on user sessions is mainly to model sessions as sequences in the assumption that user behaviors are independent and identically distributed, and then to use deep semantic information mining through Deep Neural Networks. Nevertheless, user behaviors may be a nonindependent intention at irregular points in time. For example, users may buy painkillers, books, or clothes for different reasons at different times. However, this has not been taken seriously in previous studies. Therefore, we propose a session recommendation method based on Neural Differential Equations in an attempt to predict user behavior forward or backward from any point in time. We used Ordinary Differential Equations to train the Graph Neural Network and could predict forward or backward at any point in time to model the user's nonindependent sessions. We tested for four real datasets and found that our model achieved the expected results and was superior to the existing session-based recommendations.

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