On exact solutions of the regularized long‐wave equation: A direct approach to partially integrable equations. I. Solitary wave and solitons

1995 ◽  
Vol 36 (7) ◽  
pp. 3498-3505 ◽  
Author(s):  
A. Parker
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Exact Solutions ◽  
Direct Approach ◽  
Integrable Equations ◽  
Long Wave ◽  
Regularized Long Wave Equation

Rosenau-KdV Equation Coupling with the Rosenau-RLW Equation: Conservation Laws and Exact Solutions

2017 ◽  
Vol 18 (6) ◽  
pp. 451-456 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Long Wave ◽  
Rlw Equation ◽  
Kudryashov Method ◽  
Regularized Long Wave Equation ◽  
De Vries

AbstractWe compute the conservation laws for the Rosenau-Kortweg de Vries equation coupling with the Regularized Long-Wave equation using Noether’s approach through a remarkable method of increasing the order of the Rosenau-KdV-RLW equation. Furthermore, exact solutions for the Rosenau- KdV-RLW equation are acquired by employing the Kudryashov method.

scholarly journals A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
E. Momoniat
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Solitary Wave ◽  
Nonlinear Term ◽  
Solitary Wave Solution ◽  
Finite Difference Schemes ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Modified Equation ◽  
Nonstandard Finite Difference

Two nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numerical schemes are shown to preserve conserved quantities when compared to an implicit scheme in which the nonlinear term is approximated in the usual way. Comparisons to the single solitary wave solution show significantly better results, measured in theL2andL∞norms, when compared to results obtained using a Petrov-Galerkin finite element method and a splitted quadratic B-spline collocation method. The growth in the error when simulating the single solitary wave solution using the two “best” nonstandard numerical schemes is shown to be linear implying the nonstandard finite difference schemes are conservative. The formation of an undular bore for both steep and shallow initial profiles is captured without the formation of numerical instabilities.

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