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

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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

Abstract
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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.
[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.
[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.
[4]	J. M. Burgers, A mathematical model illustrating the theory of turbulence,, Adv. Appl. Mech., 1 (1948), 171.
[5]	C. R. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation,, Acad. Sci. Paris, (1895), 431.
[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.
[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.
[8]	M. L. Gandarias, Weak self-adjoint differential equations,, J. Phys. A: Math. Theor., 44 (2011).
[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.
[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.
[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.
[12]	N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311. doi: 10.1016/j.jmaa.2006.10.078.
[13]	N. H. Ibragimov, Quasi-self-adjoint differential equations,, Arch. ALGA, 4 (2007), 55.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[4]	J. M. Burgers, A mathematical model illustrating the theory of turbulence,, Adv. Appl. Mech., 1 (1948), 171.
[5]	C. R. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation,, Acad. Sci. Paris, (1895), 431.
[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.
[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.
[8]	M. L. Gandarias, Weak self-adjoint differential equations,, J. Phys. A: Math. Theor., 44 (2011).
[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.
[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.
[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.
[12]	N. H. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333 (2007), 311. doi: 10.1016/j.jmaa.2006.10.078.
[13]	N. H. Ibragimov, Quasi-self-adjoint differential equations,, Arch. ALGA, 4 (2007), 55.
[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.
[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.
[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.
[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.
[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.
[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.

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