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Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's
1. | Clark Atlanta University, Department of Physics, Atlanta, GA 30314, USA |
2. | Morehouse College, Department of Physics, Atlanta, GA 30314, USA |
We investigate traveling wave (TW) solutions to modified versionsof the Burgers and Fisher PDE’s. Both equations are nonlinear parabolicPDE’s having square-root dynamics in their advection and reaction terms.Under certain assumptions, exact forms are constructed for the TW solutions.
References:
[1] |
R. Buckmire, K. McMurtry and R. E. Mickens, Numerical studies of a nonlinear heat equation with square root reaction term, Numerical Methods for Partial Differential Equations, 25 (2009), 598-609.
doi: 10.1002/num.20361. |
[2] |
L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, 1997. |
[3] |
P. M. Jordan, Finite-amplitude acoustic traveling waves in a fluid that saturates a porous media, Physics Letters A, 355 (2006), 216-221. |
[4] |
P. M. Jordan, A Note on the Lambert W-function: Applications in the mathematical and physical sciences, Contemporary Mathematics, 618 (2014), 247-263.
doi: 10.1090/conm/618/12351. |
[5] |
J. D. Logan, Nonlinear Partial Differential Equations Wiley-Interscience, New York, 1994. |
[6] |
R. E. Mickens, Exact finite difference scheme for an advection equation having square-root dynamics, Journal of Difference Equations and Applications, 14 (2008), 1149-1157.
doi: 10.1080/10236190802332209. |
[7] |
R. E. Mickens, Wave front behavior of traveling waves solutions for a PDE having square-root dynamics, Mathematics and Computers in Simulation, 82 (2012), 1271-1277.
doi: 10.1016/j.matcom.2010.08.010. |
[8] |
R. E. Mickens, Mathematical Methods for the Natural and Engineering Sciences, 2nd edition, World Scientific, London, 2017. |
[9] |
J. D. Murray, Mathematical Biology, Springer, Berlin, 1993.
doi: 10.1007/b98869. |
[10] |
S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium, Soviet Physics - Acoustic, 8 (1962), 170-175. |
[11] |
S. R. Valluri, D. J. Jeffrey and R. M. Corless, Some applications of the Lambert W function to physics, Canadian Journal of Physics, 78 (2000), 823-831. |
[12] |
G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974. |
[13] |
H. Wilhelmsson, M. Benda, B. Etlicher, R. Jancel and T. Lehner, Non-linear evolution of densities in the presence of simultaneous diffusion and reaction processes, Physica Scripta, 38 (1988), 1482-1489.
doi: 10.1103/PhysRevA.38.1482. |
show all references
References:
[1] |
R. Buckmire, K. McMurtry and R. E. Mickens, Numerical studies of a nonlinear heat equation with square root reaction term, Numerical Methods for Partial Differential Equations, 25 (2009), 598-609.
doi: 10.1002/num.20361. |
[2] |
L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, 1997. |
[3] |
P. M. Jordan, Finite-amplitude acoustic traveling waves in a fluid that saturates a porous media, Physics Letters A, 355 (2006), 216-221. |
[4] |
P. M. Jordan, A Note on the Lambert W-function: Applications in the mathematical and physical sciences, Contemporary Mathematics, 618 (2014), 247-263.
doi: 10.1090/conm/618/12351. |
[5] |
J. D. Logan, Nonlinear Partial Differential Equations Wiley-Interscience, New York, 1994. |
[6] |
R. E. Mickens, Exact finite difference scheme for an advection equation having square-root dynamics, Journal of Difference Equations and Applications, 14 (2008), 1149-1157.
doi: 10.1080/10236190802332209. |
[7] |
R. E. Mickens, Wave front behavior of traveling waves solutions for a PDE having square-root dynamics, Mathematics and Computers in Simulation, 82 (2012), 1271-1277.
doi: 10.1016/j.matcom.2010.08.010. |
[8] |
R. E. Mickens, Mathematical Methods for the Natural and Engineering Sciences, 2nd edition, World Scientific, London, 2017. |
[9] |
J. D. Murray, Mathematical Biology, Springer, Berlin, 1993.
doi: 10.1007/b98869. |
[10] |
S. I. Soluyan and R. V. Khokhlov, Finite amplitude acoustic waves in a relaxing medium, Soviet Physics - Acoustic, 8 (1962), 170-175. |
[11] |
S. R. Valluri, D. J. Jeffrey and R. M. Corless, Some applications of the Lambert W function to physics, Canadian Journal of Physics, 78 (2000), 823-831. |
[12] |
G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974. |
[13] |
H. Wilhelmsson, M. Benda, B. Etlicher, R. Jancel and T. Lehner, Non-linear evolution of densities in the presence of simultaneous diffusion and reaction processes, Physica Scripta, 38 (1988), 1482-1489.
doi: 10.1103/PhysRevA.38.1482. |
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