Abstract
•In this work, we have used the new extended direct algebraic method to construct the new exact traveling wave solutions in the form of rational, hyperbolic and trigonometric functions of the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation arising in Mathematical biology.•The obtained solutions are verified for the aforesaid equations with the help of symbolic software MATHEMATICS.•Some produced results are graphically illustrated using 3D surface graphs and 2D line plots, which provide useful information about the dynamical behavior.•It worth to be noticed that, compare to existent works; our results obtained in this paper are presented for the first time.•Using bifurcation theory of planar dynamical system, we have successfully studies the bifurcation behavior of some nonlinear traveling waves of the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation through numerical simulation.•It is observed that proposed equations supports nonlinear solitary waves, periodic waves and most important supernonlinear periodic waves.
In the present work, we study the dynamical behavior of nonlinear traveling waves for the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation. Novel exact solutions in the form of rational, hyperbolic and trigonometric functions are obtained using the new extended direct algebraic method. The obtained solutions are also verified for the aforesaid equations with the help of symbolic soft computations. Furthermore, some produced results are graphically illustrated using 3D surface graphs and 2D line plots, which provide useful information about the dynamical behavior. Moreover, we assurance that all the obtained solutions are accurate, efficient and an excellent contribution in the literature of solitary wave theory. Bifurcation behavior of some nonlinear traveling waves of the proposed equations is also studied with the help of bifurcation theory of planar dynamical systems through numerical simulation. It is observed that proposed equations supports nonlinear solitary waves, periodic waves and most important supernonlinear periodic waves.