Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

2018 ◽  
Vol 24 (2) ◽  
pp. 175-183
Author(s):  
Jean-Claude Ndogmo
Keyword(s):  
Explicit Form ◽  
Nonlinear Equations ◽  
Variational Principles ◽  
Symmetry Algebra ◽  
First Integrals ◽  
Maximal Symmetry ◽  
General Order ◽  
Linear And Nonlinear ◽  
Variational Symmetries ◽  
Variational Symmetry

Abstract Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form

2018 ◽  
Vol 24 (1) ◽  
pp. 17-26
Author(s):  
Jean-Claude Ndogmo
Keyword(s):  
Explicit Form ◽  
Canonical Form ◽  
Variational Principles ◽  
Arbitrary Dimension ◽  
Linear Equations ◽  
Symmetry Algebra ◽  
First Integrals ◽  
Maximal Symmetry ◽  
General Order

AbstractVariational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.

scholarly journals First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
J. C. Ndogmo
Keyword(s):  
Hamiltonian Systems ◽  
Linear Equations ◽  
Symmetry Algebra ◽  
First Integrals ◽  
Second Order ◽  
Maximal Symmetry ◽  
General Solutions ◽  
Linearly Independent ◽  
Complete Sets

Complete sets of linearly independent first integrals are found for the most general form of linear equations of maximal symmetry algebra of order ranging from two to eight. The corresponding Hamiltonian systems are constructed and it is shown that their general solutions can also be found by a simple superposition formula from the solutions of a scalar second-order source equation.

scholarly journals The variational iteration method for solving n-th order fuzzy differential equations

Open Physics ◽  
2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Mohammad Saeidy ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Analytical Method ◽  
Iteration Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Linear Differential Equations ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
Linear Differential

AbstractThe variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

scholarly journals On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance

1991 ◽  
Vol 32 (4) ◽  
pp. 457-468 ◽  
Author(s):  
O. P. Bhutani ◽  
K. Vijayakumar
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Lie Group ◽  
Variational Principles ◽  
First Integrals ◽  
Noether Theorem ◽  
Repeated Application ◽  
Scale Invariant ◽  
Emden Equation ◽  
Fowler Equation

AbstractAfter formulating the alternate potential principle for the nonlinear differential equation corresponding to the generalised Emden-Fowler equation, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the said equation via the Noether theorem. Further, for physical realisable forms of the parameters involved and through repeated application of invariance under the transformation obtained, a number of exact solutions are arrived at both for the Emden-Fowler equation and classical Emden equations. A comparative study with Bluman-Cole and scale-invariant techniques reveals quite a number of remarkable features of the techniques used here.

Solution of linear and nonlinear equations by optical methods

1984 ◽  
Vol 23 (18) ◽  
pp. 3144 ◽  
Author(s):  
George Lewak ◽  
Sing H. Lee ◽  
W. Thomas Cathey
Keyword(s):  
Nonlinear Equations ◽  
Optical Methods ◽  
Linear And Nonlinear
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