Analytic Solutions of Fractal and Fractional Time Derivative-
Burgers-Nagumo Equation
Abstract
The Nagumo equation describes a reaction-diffusion system in biology.
Here, it is coupled to Burgers equation, via including convection, which
is, namely; Burgers-Nagumo equation BNE. The first objective of this
work is to present a theorem to reduce the different versions of the
fractional time derivatives FTD to “non autonomous” ordinary ones,
that is ordinary derivatives with time dependent coefficients. The
second objective is to find the exact solutions of the fractal and
fractional time derivative -BNE, that is to solve BNE with time
dependent coefficient. On the other hand FTD can be transformed to BNE
with constant coefficients via similarity transformations. The unified
and extended unified method are used. Self-similar solutions are also
obtained. It is found that significant fractal effects hold for smaller
order derivatives. While significant fractional effects hold for
higher-order derivatives. The solutions obtained show solitary, wrinkle
soliton waves, with double kinks, undulated, or with spikes. Further It
is shown that wrinkle soliton wave, with double kink configuration holds
for smaller fractal order. While in the case of fractional derivative,
this holds for higher orders.