American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Statistical query-based rule derivation system by backward elimination algorithm
  • DCDS-S Home
  • This Issue
  • Next Article
    Designing dynamical systems for security and defence network knowledge management. A case of study: Airport bird control falconers organizations
December  2015, 8(6): 1331-1339. doi: 10.3934/dcdss.2015.8.1331

Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term

María Rosa 1, , María de los Santos Bruzón 1, and  María de la Luz Gandarias 1,

1. 

Dpto. de Matemáticas, Universidad de Cádiz, Polígono del Río San Pedro s/n 11510 Puerto Real, Cádiz, Spain, Spain

Received  June 2015 Revised  August 2015 Published  December 2015

Memory effect in diffusion-reaction equation with finite memory transport plays an important role in physical, biological and chemical sciences. In this work we consider a Fisher equation, which has a nonlinear convection term with finite memory transport, from the point of view of Lie classical reductions. By using a direct method we obtain some travelling waves solutions. Furthermore, by using the multipliers method, we derive some nontrivial conservation laws for this equation.
Keywords: finite memory transport., Lie symmetries, conservation law, Fisher-equation.
Mathematics Subject Classification: Primary: 76M60, 92D25; Secondary: 35Q9.
Citation: María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331
References:
[1]

S. A. Anco and G. Bluman, Direct construction of conservation laws from field equations,, Physical Review letters, 78 (1997), 2869.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[2]

S. C. Anco and G. Bluman, Direct constrution method for conservation laws for partial differential equations Part II: General treatment,, Euro. J. of Applied Mathematics, 13 (2002), 567.  doi: 10.1017/S0956792501004661.  Google Scholar

[3]

M. S. Bruzón, M. L. Gandarias and N. H. Ibragimov, Self-adjoint sub-classes of generalized thin film equations,, J. Math. Anal. Appl., 357 (2009), 307.  doi: 10.1016/j.jmaa.2009.04.028.  Google Scholar

[4]

J. M. Burgers, A mathematical model illustrating the theory of turbulence,, Adv. Appl. Mech., 1 (1948), 171.   Google Scholar

[5]

C. R. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation,, Acad. Sci. Paris, (1895), 431.   Google Scholar

[6]

N. Euler and M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: two linearisable hierachies,, Journal of Nonlinear Mathematical Physics, 16 (2009), 489.  doi: 10.1142/S1402925109000509.  Google Scholar

[7]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[8]

M. L. Gandarias, Weak self-adjoint differential equations,, J. Phys. A: Math. Theor., 44 (2011).   Google Scholar

[9]

M. L. Gandarias, Weak self-adjointness and conservation laws for a porous medium equation,, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2342.  doi: 10.1016/j.cnsns.2011.10.020.  Google Scholar

[10]

M. L. Gandarias, Nonlinear self-adjointness through differential substitutions,, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3523.  doi: 10.1016/j.cnsns.2014.02.013.  Google Scholar

[11]

M. L. Gandarias, M. S. Bruzón and M. Rosa, Nonlinear self-adjointness and conservation laws for a generalized Fisher equation,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1600.  doi: 10.1016/j.cnsns.2012.11.023.  Google Scholar

[12]

N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar

[13]

N. H. Ibragimov, Quasi-self-adjoint differential equations,, Arch. ALGA, 4 (2007), 55.   Google Scholar

[14]

N. H. Ibragimov, Nonlinear self-adjointness and conservation laws,, J. Phys. A: Math.Theor., 44 (2011).  doi: 10.1088/1751-8113/44/43/432002.  Google Scholar

[15]

N. H. Ibragimov, M. Torrisi and R. Tracina, Self-adjointness and conservation laws of a generalized Burgers equation,, J. Phys. A: Math. Theor., 44 (2011).  doi: 10.1088/1751-8113/44/14/145201.  Google Scholar

[16]

S. Kar, S. K. BaniK and D. S. Ray, Exact solutions of Fisher and Burgers equations with finite transport memory,, J. Phys. A: Math. Gen., 36 (2003), 2771.  doi: 10.1088/0305-4470/36/11/308.  Google Scholar

[17]

A. Mishra and R. Kumar, Memory effects in Fisher equation with nonlinear convection term,, Physics Letters A, 376 (2012), 1833.  doi: 10.1016/j.physleta.2012.04.037.  Google Scholar

[18]

M. Torrisi and R. Tracina, Quasi self-adjointness of a class of third order nonlinear dispersive equations,, Nonlinear Analysis: Real World Applications, 14 (2013), 1496.  doi: 10.1016/j.nonrwa.2012.10.013.  Google Scholar

[19]

M. Wang, X. Li and J. Zhang, The $\frac{G'}G$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,, Phys. Lett. A, 372 (2008), 417.  doi: 10.1016/j.physleta.2007.07.051.  Google Scholar

show all references

References:
[1]

S. A. Anco and G. Bluman, Direct construction of conservation laws from field equations,, Physical Review letters, 78 (1997), 2869.  doi: 10.1103/PhysRevLett.78.2869.  Google Scholar

[2]

S. C. Anco and G. Bluman, Direct constrution method for conservation laws for partial differential equations Part II: General treatment,, Euro. J. of Applied Mathematics, 13 (2002), 567.  doi: 10.1017/S0956792501004661.  Google Scholar

[3]

M. S. Bruzón, M. L. Gandarias and N. H. Ibragimov, Self-adjoint sub-classes of generalized thin film equations,, J. Math. Anal. Appl., 357 (2009), 307.  doi: 10.1016/j.jmaa.2009.04.028.  Google Scholar

[4]

J. M. Burgers, A mathematical model illustrating the theory of turbulence,, Adv. Appl. Mech., 1 (1948), 171.   Google Scholar

[5]

C. R. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation,, Acad. Sci. Paris, (1895), 431.   Google Scholar

[6]

N. Euler and M. Euler, On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: two linearisable hierachies,, Journal of Nonlinear Mathematical Physics, 16 (2009), 489.  doi: 10.1142/S1402925109000509.  Google Scholar

[7]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 355.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[8]

M. L. Gandarias, Weak self-adjoint differential equations,, J. Phys. A: Math. Theor., 44 (2011).   Google Scholar

[9]

M. L. Gandarias, Weak self-adjointness and conservation laws for a porous medium equation,, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2342.  doi: 10.1016/j.cnsns.2011.10.020.  Google Scholar

[10]

M. L. Gandarias, Nonlinear self-adjointness through differential substitutions,, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3523.  doi: 10.1016/j.cnsns.2014.02.013.  Google Scholar

[11]

M. L. Gandarias, M. S. Bruzón and M. Rosa, Nonlinear self-adjointness and conservation laws for a generalized Fisher equation,, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1600.  doi: 10.1016/j.cnsns.2012.11.023.  Google Scholar

[12]

N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311.  doi: 10.1016/j.jmaa.2006.10.078.  Google Scholar

[13]

N. H. Ibragimov, Quasi-self-adjoint differential equations,, Arch. ALGA, 4 (2007), 55.   Google Scholar

[14]

N. H. Ibragimov, Nonlinear self-adjointness and conservation laws,, J. Phys. A: Math.Theor., 44 (2011).  doi: 10.1088/1751-8113/44/43/432002.  Google Scholar

[15]

N. H. Ibragimov, M. Torrisi and R. Tracina, Self-adjointness and conservation laws of a generalized Burgers equation,, J. Phys. A: Math. Theor., 44 (2011).  doi: 10.1088/1751-8113/44/14/145201.  Google Scholar

[16]

S. Kar, S. K. BaniK and D. S. Ray, Exact solutions of Fisher and Burgers equations with finite transport memory,, J. Phys. A: Math. Gen., 36 (2003), 2771.  doi: 10.1088/0305-4470/36/11/308.  Google Scholar

[17]

A. Mishra and R. Kumar, Memory effects in Fisher equation with nonlinear convection term,, Physics Letters A, 376 (2012), 1833.  doi: 10.1016/j.physleta.2012.04.037.  Google Scholar

[18]

M. Torrisi and R. Tracina, Quasi self-adjointness of a class of third order nonlinear dispersive equations,, Nonlinear Analysis: Real World Applications, 14 (2013), 1496.  doi: 10.1016/j.nonrwa.2012.10.013.  Google Scholar

[19]

M. Wang, X. Li and J. Zhang, The $\frac{G'}G$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,, Phys. Lett. A, 372 (2008), 417.  doi: 10.1016/j.physleta.2007.07.051.  Google Scholar

[1]

María-Santos Bruzón, Elena Recio, Tamara-María Garrido, Rafael de la Rosa. Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020222
[2]

María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038
[3]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[4]

Özlem Orhan, Teoman Özer. New conservation forms and Lie algebras of Ermakov-Pinney equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 735-746. doi: 10.3934/dcdss.2018046
[5]

Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419
[6]

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159
[7]

Gianluca Crippa, Laura V. Spinolo. An overview on some results concerning the transport equation and its applications to conservation laws. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1283-1293. doi: 10.3934/cpaa.2010.9.1283
[8]

Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020218
[9]

Giuseppe Maria Coclite, Lorenzo Di Ruvo. A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1247-1282. doi: 10.3934/dcds.2017052
[10]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2981-2990. doi: 10.3934/dcds.2016.36.2981
[11]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[12]

Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35
[13]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
[14]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099
[15]

Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520
[16]

Fabio Camilli, Raul De Maio. Memory effects in measure transport equations. Kinetic & Related Models, 2019, 12 (6) : 1229-1245. doi: 10.3934/krm.2019047
[17]

Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312
[18]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035
[19]

Raimund Bürger, Stefan Diehl, María Carmen Martí. A conservation law with multiply discontinuous flux modelling a flotation column. Networks & Heterogeneous Media, 2018, 13 (2) : 339-371. doi: 10.3934/nhm.2018015
[20]

Darko Mitrovic. Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (1) : 163-188. doi: 10.3934/nhm.2010.5.163

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences