Analysis of time-fractional non-linear Kawahara Equations with power law kernel

In this article, we study time-fractional non-linear Kawahara and modified Kawahara equations with Caputos fractional derivative. A variety of integral transforms with Adomian decomposition are applied to obtain the general series solutions of the considered models. The efficiency of the proposed methods is confirmed by numerical examples under suitable initial conditions. From the numerical results, one can see that the attained series solutions converge to the exact solutions of the systems. The stability of the applied methods is investigated using the principle of Banach contraction and $S$-stable mapping. For the Kawahara Equation, it is observed that the amplitude of the system enhances as time increases for fixed values of fractional orders. Similarly, for stable temporal variables, when the fractional-order increases, the amplitude of the solitary wave solution also increases. Similarly, for the modified system, the wave amplitude is also enhanced with variations in time $\left(t\right)$. It infers that when $\left(\alpha \right)$ increases, it significantly decreases the wave amplitude. It is also observed that uniform changes take place in the wave amplitude with time $\left(t\right)$. However, non-uniform changes in wave amplitude occur for different values of $\alpha$. The absolute error between the exact and obtained series solutions is presented. It is revealed that the absolute error in the systems reduces promptly when $x$ increases at a comparatively small time $t$, whereas the increment in iterations decreases the error in the systems.