scholarly journals The Lagrangian, Self-Adjointness, and Conserved Quantities for a Generalized Regularized Long-Wave Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Long Wei ◽  
Yang Wang
Keyword(s):  
Wave Equation ◽  
Auxiliary Function ◽  
Order Equation ◽  
Conserved Quantities ◽  
Minimal Order ◽  
Third Order ◽  
The Third ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Differential Substitution

We consider the Lagrangian and the self-adjointness of a generalized regularized long-wave equation and its transformed equation. We show that the third-order equation has a nonlocal Lagrangian with an auxiliary function and is strictly self-adjoint; its transformed equation is nonlinearly self-adjoint and the minimal order of the differential substitution is equal to one. Then by Ibragimov’s theorem on conservation laws we obtain some conserved qualities of the generalized regularized long-wave equation.

Euler operators and conservation laws of the BBM equation

1979 ◽  
Vol 85 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Calculus Of Variations ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Formal Calculus ◽  
Euler Operators ◽  
Bbm Equation

AbstractThe BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.

scholarly journals New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

2020 ◽  
Vol 66 (3 May-Jun) ◽  
pp. 297
Author(s):  
Mehmet Senol
Keyword(s):  
Wave Equation ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Algebraic Method ◽  
Symbolic Calculation ◽  
Solution Sets ◽  
Flow Models ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

Strongly Oscillatory and Nonoscillatory Subspaces of Linear Equations

1975 ◽  
Vol 27 (1) ◽  
pp. 106-110 ◽  
Author(s):  
J. Michael Dolan ◽  
Gene A. Klaasen
Keyword(s):  
Linear Equation ◽  
Linear Space ◽  
Nontrivial Solution ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Oscillatory Solutions ◽  
The Third ◽  
All Solutions

Consider the nth order linear equationand particularly the third order equationA nontrivial solution of (1)n is said to be oscillatory or nonoscillatory depending on whether it has infinitely many or finitely many zeros on [a, ∞). Let denote respectively the set of all solutions, oscillatory solutions, nonoscillatory solutions of (1)n. is an n-dimensional linear space. A subspace is said to be nonoscillatory or strongly oscillatory respectively if every nontrivial solution of is nonoscillatory or oscillatory. If contains both oscillatory and nonoscillatory solutions then is said to be weakly oscillatory.

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