Geodesic
Fixed-point theorem for Caputo--Fabrizio fractional Nagumo equation with nonlinear diffusion and convection
Alqahtani, Rubayyi T.1
1 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), P. O. Box 65892, Riyadh 11566, Saudi Arabia
Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 5, p. 1991-1999.

Voir la notice de l'article provenant de la source International Scientific Research Publications

We make use of fractional derivative, recently proposed by Caputo and Fabrizio, to modify the nonlinear Nagumo diffusion and convection equation. The proposed fractional derivative has no singular kernel considered as a filter. We examine the existence of the exact solution of the modified equation using the method of fixed-point theorem. We prove the uniqueness of the exact solution and present some numerical simulations.
DOI : 10.22436/jnsa.009.05.05
Classification : 47H10, 34A08
Keywords: Nonlinear Nagumo equation, Caputo-Fabrizio derivative, fixed-point theorem, uniqueness.

Alqahtani, Rubayyi T. 1

1 Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), P. O. Box 65892, Riyadh 11566, Saudi Arabia 
@article{JNSA_2016_9_5_a4,
     author = {Alqahtani, Rubayyi T.},
     title = {Fixed-point theorem for {Caputo--Fabrizio} fractional {Nagumo} equation with nonlinear diffusion and convection},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {1991-1999},
     publisher = {mathdoc},
     volume = {9},
     number = {5},
     year = {2016},
     doi = {10.22436/jnsa.009.05.05},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.05.05/}
}
TY  - JOUR
AU  - Alqahtani, Rubayyi T.
TI  - Fixed-point theorem for Caputo--Fabrizio fractional Nagumo equation with nonlinear diffusion and convection
JO  - Journal of nonlinear sciences and its applications
PY  - 2016
SP  - 1991
EP  - 1999
VL  - 9
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.05.05/
DO  - 10.22436/jnsa.009.05.05
LA  - en
ID  - JNSA_2016_9_5_a4
ER  - 
%0 Journal Article
%A Alqahtani, Rubayyi T.
%T Fixed-point theorem for Caputo--Fabrizio fractional Nagumo equation with nonlinear diffusion and convection
%J Journal of nonlinear sciences and its applications
%D 2016
%P 1991-1999
%V 9
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.05.05/
%R 10.22436/jnsa.009.05.05
%G en
%F JNSA_2016_9_5_a4
Alqahtani, Rubayyi T. Fixed-point theorem for Caputo--Fabrizio fractional Nagumo equation with nonlinear diffusion and convection. Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 5, p. 1991-1999. doi : 10.22436/jnsa.009.05.05. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.009.05.05/

[1] D. G. Aronson The role of the diffusion in mathematical population biology: skellam revisited, Lecture Notes in Biomath, Springer, Berlin, 1985

[2] Atangana, A. On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., Volume 273 (2016), pp. 948-956

[3] Atangana, A.; Alkahtani, B. S. T. Analysis of the Keller-Segel model with a fractional derivative without singular kernel , Entropy, Volume 17 (2015), pp. 4439-4453

[4] Atangana, A.; B. S. T. Alkahtani Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel , Adv. Mech. Eng., Volume 7 (2015), pp. 1-6

[5] Atangana, A.; Noutchie, S. C. Oukouomi On the fractional Nagumo equation with nonlinear diffusion and convection, Abstr Appl. Anal., Volume 2014 (2014), pp. 1-7

[6] Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel , Progr. Fract. Differ. Appl., Volume 1 (2015), pp. 73-85

[7] Chen, Z. X.; Guo, B. Y. Analytic solutions of the Nagumo equation, IMA J. Appl. Math., Volume 48 (1992), pp. 107-115

[8] FitzHugh, R. Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys., Volume 17 (1955), pp. 257-278

[9] FitzHugh, R. Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., Volume 1 (1961), pp. 445-466

[10] FitzHugh, R. Mathematical models of excitation and propagation in nerve, McGraw-Hill, New York, 1969

[11] Grindrod, P.; Sleeman, B. D. Weak travelling fronts for population models with density dependent dispersion, Math. Methods Appl. Sci., Volume 9 (1987), pp. 576-586

[12] S. Hastings The existence of periodic solutions to Nagumo's equation, Quart. J. Math., Volume 25 (1974), pp. 369-378

[13] Li, H.; Y. Guo New exact solutions to the FitzHugh-Nagumo equation, Appl. Math. Comput., Volume 180 (2006), pp. 524-528

[14] Losada, J.; J. J. Nieto Properties of a new fractional derivative without singular kernel , Progr. Fract. Differ. Appl., Volume 1 (2015), pp. 87-92

[15] M. B. A. Mansour On the sharp front-type solution of the Nagumo equation with nonlinear diffusion and convection, Pramana, Volume 80 (2013), pp. 533-538

[16] Nagumo, J.; Arimoto, S.; S. Yoshizawa An active pulse transmission line simulating nerve axon, Proc. IRE, Volume 50 (1962), pp. 2061-2070

[17] Nucci, M. C.; Clarkson, P. A. The non-classical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation, Phys. Lett. A, Volume 164 (1992), pp. 49-56

[18] Pandir, Y.; Tandogan, Y. A. Exact solutions of the time-fractional Fitzhugh-Nagumo equation, 11th internattional conference of numerical analysis and applied mathematics, Volume 1558 (2013), pp. 1919-1922

[19] Sánchez-Garduño, F.; P. K. Maini Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations, J. Math. Biol., Volume 35 (1997), pp. 713-728

Cité par Sources :