IJPAM: Volume 117, No. 3 (2017)

Title

SYMMETRIES OF THE GENERALIZED FISHER
EQUATION WITH $t-$DEPENDENT COEFFICIENT

Authors

Hamid Reza Yazdani$^1$, Mehdi Nadjafikhah$^2$
$^1$Department of Mathematics
Payame Noor University (PNU)
P.O. Box 19395-3697, Tehran, IRAN
$^2$School of Mathematics
Iran University of Science and Technology
Narmak, Tehran, IRAN

Abstract

There are many tools for analyzing PDEs. In the equivalence theory, the symmetry methods like the Lie symmetry and Fushchych methods are tools for solving PDEs. Indeed, these methods can determine classical and non-classical invariants and then by reformulating the equations according to these invariants, they can reduce the order of PDEs and convert them to ODEs.

In this paper, we consider the generalized version of FKPP equation (GFKPP) with $t$-dependent coefficient f(t)u_tt(x,t) + u_t(x,t) = u_xx(x,t) + u(x,t) - u^2(x,t), Where $f(t)$ is a smooth function of $t$. In this study, the Lie symmetry and Fushchych methods applied on the GFKPP and obtained the symmetry groups, differential invariants & invariant solutions. After that, the Fourier transform method (FTM) (as the harmonic analysis method) applied on the GFKPP, and calculated solutions. Finally, the results of the equivalence methods (Lie and Fushchych methods) compared with FTM.

History

Received: 2017-03-04
Revised: 2017-08-29
Published: January 15, 2018

AMS Classification, Key Words

AMS Subject Classification: 54H15, 42A38, 42B10, 76M60
Key Words and Phrases: Lie symmetry method, Fushchych method, non-exact symmetries, Fourier transform method, The GFKPP equation

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Bibliography

1
R.D. Benguria and M.C. Depassier, Speed of Fronts of the Reaction-Diffusion Equation, Physical Review, 1996.

2
G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations, Applied Mathematical Sciences, Springer, 1974.

3
G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, 1989.

4
A. Boggess and F. J. Narcowich, A first course in wavelets with Fourier analysis, John Wiley, 2009.

5
W. E. Boyce and R. C. Diprima, Elementary Differential Equations and Boundary Value Problems, John Wiley, 2001.

6
G. Cicogna, A Discussion on the Different Notions of Symmetry of Differential Equations, Proceedings of Institute of Mathematics of NAS of Ukraine, 2004.

7
R. Enberg, Traveling Waves and the Renormalization Group Improved Balitsky-Kovchegov Equation, Physical Review, 2007.

8
R. A. Fisher, The Genetical Theory of Natural Selection, Oxford University Press, 2000.

9
R. A. Fisher, The Wave of Advance of Advantageous Genes, Annals of Eugenics, 1937.

10
W.I. Fushchych, On symmetry and particular solutions of some multidimensional physics equations, In: Algebraic-Theoretical Methods in Mathematical Physics Problems, Institute of Mathematics of Ukrainian Academy of Sciences (1983).

11
A. Lemarchand, A. Lesne and M. Mareschal, Langevin Approach to a Chemical Wave Front: Selection of the Propagation Velocity in the Presence of Internal Noise, Physical Review, 1995.

12
M. Nadjafikhah, Lie Symmetries of Inviscid Burger's Equation, Advances in applied clifford algebras, Springer, 2010.

13
P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, 1993.

14
A.D. Polyanin and V.F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC, 2004.

15
H. R. Yazdani and M. Nadjafikhah, Symmetries of the generalized fisher equation with $x-$dependent coefficient, International Journal of Mathematics and Computation, 28, No. 4 (2017).

How to Cite?

DOI: 10.12732/ijpam.v117i3.5 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 117
Issue: 3
Pages: 401 - 413


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