Analytical Solutions for the Equal Width Equations Containing Generalized Fractional Derivative Using the Efficient Combined Method

In this paper, the efficient combined method based on the homotopy perturbation Sadik transform method  (HPSTM) is applied to solve the physical and functional equations containing the Caputo–Prabhakar fractional derivative. The mathematical model of this equation of order $\mu \in \left(\mathrm{0,1}\right]$ with $\lambda \in {\mathbb{Z}}^{+},\theta ,\sigma \in {ℝ}^{+}$ is presented as follows: ${}^C{{\displaystyle \mathfrak{D}}}_t^{\mu }u\left(x,t\right)+\theta u^{\lambda }\left(x,t\right)u_x\left(x,t\right)-\sigma u_{xxt}\left(x,t\right)=0,$ where for $\lambda =1,\theta =1,\sigma =1s$ and $\lambda =2,\theta =3,\sigma =1$ , equations are changed into the equal width and modified equal width equations, respectively. The analytical method which we have used for solving this equation is based on a combination of the homotopy perturbation method and Sadik transform. The convergence and error analysis are discussed in this article. Plots of the analytical results with three examples are presented to show the applicability of this numerical method. Comparison between the obtained absolute errors by the suggested method and other methods is demonstrated.