Solving Shallow Water Wave Equation with HLL Scheme Based on Moving Grid

2021 ◽  
Vol 10 (10) ◽  
pp. 3317-3324
Author(s):  
霄 李
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Moving Grid ◽  
Shallow Water Wave

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

scholarly journals On a shallow water wave equation

Nonlinearity ◽  
1994 ◽  
Vol 7 (3) ◽  
pp. 975-1000 ◽  
Author(s):  
P A Clarkson ◽  
E L Mansfield
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Shallow Water Wave

scholarly journals The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ying Wang ◽  
YunXi Guo
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Sufficient Conditions ◽  
Global Weak Solution ◽  
Dissipative Term ◽  
Shallow Water Wave ◽  
Global Strong Solution ◽  
Special Cases ◽  
The Cauchy Problem

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that(1-∂x2)u0∈M+(R),u0∈H1(R),andu0∈L1(R), the existence and uniqueness of the global weak solution to the equation are shown to be true.

scholarly journals Diversity of Interaction Solutions of a Shallow Water Wave Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Jian-Ping Yu ◽  
Wen-Xiu Ma ◽  
Bo Ren ◽  
Yong-Li Sun ◽  
Chaudry Masood Khalique
Keyword(s):  
Fluid Dynamics ◽  
Wave Equation ◽  
General Theory ◽  
Shallow Water ◽  
Water Wave ◽  
Direct Method ◽  
Soliton Solutions ◽  
Interaction Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions

In this paper, we study the diversity of interaction solutions of a shallow water wave equation, the generalized Hirota–Satsuma–Ito (gHSI) equation. Using the Hirota direct method, we establish a general theory for the diversity of interaction solutions, which can be applied to generate many important solutions, such as lumps and lump-soliton solutions. This is an interesting feature of this research. In addition, we prove this new model is integrable in Painlevé sense. Finally, the diversity of interactive wave solutions of the gHSI is graphically displayed by selecting specific parameters. All the obtained results can be applied to the research of fluid dynamics.

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