Analytic Solutions of Fractal and Fractional Time Derivative- Burgers-Nagumo Equation

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.