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March  2019, 8(1): 139-147. doi: 10.3934/eect.2019008

Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's

Ronald Mickens 1, and  Kale Oyedeji 2,,

1. 

Clark Atlanta University, Department of Physics, Atlanta, GA 30314, USA
2. 

Morehouse College, Department of Physics, Atlanta, GA 30314, USA

Corresponding author: Kale Oyedeji, 470-639-0285

Received  October 2017 Revised  January 2018 Published  March 2019 Early access  January 2019

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.

Keywords: Traveling wave solutions, parabolic PDE', Burgers' equation, Fisher equations, asymptotics.
Mathematics Subject Classification: Primary: 34E05, 35K57; Secondary: 35B09, 35B40.
Citation: Ronald Mickens, Kale Oyedeji. Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's. Evolution Equations and Control Theory, 2019, 8 (1) : 139-147. doi: 10.3934/eect.2019008
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.

Figure 1.  a) $ v(z) $ vs $ z $, b) $ f(z) = v(z)^2 $ vs $ z $. See Eqs. (5.10) and (5.13).
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Figure 2.  a) $ v(z) $ vs $ z $, \quad b) $ f(z) $ vs $ z $. See Eq. (5.15).
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