Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 76M60, 92D25; Secondary: 35Q91.

 Citation:

•  [1] S. A. Anco and G. Bluman, Direct construction of conservation laws from field equations, Physical Review letters, 78 (1997), 2869-2873.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), 567-585.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), 307-313.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), 171-199. [5] C. R. Cattaneo, On a form of heat equation which eliminates the paradox of instantaneous propagation, Acad. Sci. Paris, (1895), 431-433. [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), 489-504.doi: 10.1142/S1402925109000509. [7] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.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), 262001. [9] M. L. Gandarias, Weak self-adjointness and conservation laws for a porous medium equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2342-2349.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), 3523-3528.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), 1600-1606.doi: 10.1016/j.cnsns.2012.11.023. [12] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328.doi: 10.1016/j.jmaa.2006.10.078. [13] N. H. Ibragimov, Quasi-self-adjoint differential equations, Arch. ALGA, 4 (2007), 55-60. [14] N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math.Theor., 44 (2011), 432002.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), 145201, 5pp.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), 2771-2780.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), 1833-1835.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), 1496-1502.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), 417-423.doi: 10.1016/j.physleta.2007.07.051.