Korteweg-de Vries-Burgers' equation: Difference between revisions
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The ''' |
The '''Korteweg-de Vries–Burgers equation''' is a nonlinear [[partial differential equation]]: |
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:<math>u_t+u_{xxx} |
:<math>u_t+\alpha u_{xxx} + uu_x - \beta u_{xx}=0. </math> |
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The equation gives a description for nonlinear waves in dispersive-dissipative media by combining the nonlinear and [[Dispersion_relation|dispersive]] elements from the [[KdV equation]] with the [[Dissipative_system|dissipative]] element from [[Burgers' equation]].{{sfn | Polyanin | Zaitsev | 2003}} |
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The '''modified KdV-Burgers equation''' can be written as:{{sfn | Wang | 1996}} |
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:<math>u_t+ a u_{xxx} + u^2u_x - b u_{xx}=0. </math> |
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==See also== |
==See also== |
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*[[Unnormalized modified KdV equation|modified KdV equation]] |
*[[Unnormalized modified KdV equation|modified KdV equation]] |
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== |
==Notes== |
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{{reflist}} |
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<references/> |
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==References== |
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#Graham W. Griffiths William E. Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press |
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* {{cite book | last=Polyanin | first=Andrei D. | last2=Zaitsev | first2=Valentin F. | title=Handbook of Nonlinear Partial Differential Equations | publisher=Chapman and Hall/CRC | publication-place=Boca Raton, Fla | year=2003 | isbn=978-1-58488-355-5|chapter= |
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# Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 |
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9.1.7. Burgers–Korteweg–de Vries Equation and Other Equation}} |
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#Inna Shingareva, Carlos Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple Springer. |
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* {{cite journal | last=Wang | first=Mingliang | title=Exact solutions for a compound KdV-Burgers equation | journal=Physics Letters A | volume=213 | issue=5-6 | date=1996 | doi=10.1016/0375-9601(96)00103-X | pages=279–287}} |
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#Eryk Infeld and George Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge 2000 |
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#Saber Elaydi, An Introduction to Difference Equationns, Springer 2000 |
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#Dongming Wang, Elimination Practice, Imperial College Press 2004 |
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# David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 {{ISBN|9780387983004}} |
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# George Articolo Partial Differential Equations and Boundary Value Problems with Maple V Academic Press 1998 {{ISBN|9780120644759}} |
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{{DEFAULTSORT:Modified KdV-Burgers equation}} |
{{DEFAULTSORT:Modified KdV-Burgers equation}} |
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Revision as of 09:30, 14 June 2024
The Korteweg-de Vries–Burgers equation is a nonlinear partial differential equation:
The equation gives a description for nonlinear waves in dispersive-dissipative media by combining the nonlinear and dispersive elements from the KdV equation with the dissipative element from Burgers' equation.[1]
The modified KdV-Burgers equation can be written as:[2]
See also
Notes
References
- Polyanin, Andrei D.; Zaitsev, Valentin F. (2003). "9.1.7. Burgers–Korteweg–de Vries Equation and Other Equation". Handbook of Nonlinear Partial Differential Equations. Boca Raton, Fla: Chapman and Hall/CRC. ISBN 978-1-58488-355-5.
- Wang, Mingliang (1996). "Exact solutions for a compound KdV-Burgers equation". Physics Letters A. 213 (5–6): 279–287. doi:10.1016/0375-9601(96)00103-X.
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