Fixedpoint theorem for CaputoFabrizio fractional Nagumo equation with nonlinear diffusion and convection
Authors
Rubayyi T. Alqahtani
 Department of Mathematics and Statistics, College of Science, AlImam Mohammad Ibn Saud Islamic University (IMSIU), P. O. Box 65892, Riyadh 11566, Saudi Arabia.
Abstract
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 fixedpoint theorem. We prove the uniqueness of the exact solution and present some numerical
simulations.
Keywords
 Nonlinear Nagumo equation
 CaputoFabrizio derivative
 fixedpoint theorem
 uniqueness.
MSC
References

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

[2]
A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reactiondiffusion equation, Appl. Math. Comput., 273 (2016), 948956.

[3]
A. Atangana, B. S. T. Alkahtani, Analysis of the KellerSegel model with a fractional derivative without singular kernel , Entropy, 17 (2015), 44394453.

[4]
A. Atangana, B. S. T. Alkahtani , Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel , Adv. Mech. Eng., 7 (2015), 16.

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

[6]
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel , Progr. Fract. Differ. Appl., 1 (2015), 7385.

[7]
Z. X. Chen, B. Y. Guo, Analytic solutions of the Nagumo equation, IMA J. Appl. Math., 48 (1992), 107115.

[8]
R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys., 17 (1955), 257278.

[9]
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445466.

[10]
R. FitzHugh, Mathematical models of excitation and propagation in nerve, McGrawHill, New York (1969)

[11]
P. Grindrod, B. D. Sleeman, Weak travelling fronts for population models with density dependent dispersion, Math. Methods Appl. Sci., 9 (1987), 576586.

[12]
S. Hastings , The existence of periodic solutions to Nagumo's equation, Quart. J. Math., 25 (1974), 369378.

[13]
H. Li, Y. Guo , New exact solutions to the FitzHughNagumo equation, Appl. Math. Comput., 180 (2006), 524528.

[14]
J. Losada, J. J. Nieto , Properties of a new fractional derivative without singular kernel , Progr. Fract. Differ. Appl., 1 (2015), 8792.

[15]
M. B. A. Mansour , On the sharp fronttype solution of the Nagumo equation with nonlinear diffusion and convection, Pramana, 80 (2013), 533538.

[16]
J. Nagumo, S. Arimoto, S. Yoshizawa , An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 20612070.

[17]
M. C. Nucci, P. A. Clarkson, The nonclassical method is more general than the direct method for symmetry reductions: an example of the FitzhughNagumo equation, Phys. Lett. A, 164 (1992), 4956.

[18]
Y. Pandir, Y. A. Tandogan, Exact solutions of the timefractional FitzhughNagumo equation, 11th internattional conference of numerical analysis and applied mathematics, 1558 (2013), 19191922.

[19]
F. SánchezGarduño, P. K. Maini , Travelling wave phenomena in nonlinear diffusion degenerate Nagumo equations, J. Math. Biol., 35 (1997), 713728.