Modified Fractional Power Series Method for solving fractional partial differential equations

The literature revealed that the Fractional Power Series Method (FPSM), which uses the Mittag-Leffler function in one parameter, has been gainfully applied in obtaining the solutions of fractional partial differential equations (FPDEs) in one dimension. However, the solutions in the multi-dimensional space have not been explored by researchers across the globe. The solutions of the FPDEs are feasible with the involvement of parameter $\alpha$ in the Mittag-Leffler function. However, the FPSM, which uses the Mittag-Leffler function in two parameters, has not been considered by researchers. Incorporating two parameters, $\alpha$ and $\beta$, in the Mittag-Leffler function of the FPSM is beyond reasonable doubt; it provides the continuum solution of the FPDEs and also yields more consistent and fast convergence of the solution in Holder’s spaces compared to the FPSM with the Mittag-Leffler function in one parameter. The FPSM is extended by replacing the Mittag-Leffler function in one parameter with the Mittag-Leffler function in two parameters. Also, the modified FPSM is applied to obtain the solutions of both heat and telegraph equations in multi-dimensions and one-dimension respectively. The solutions obtained by the FPSM with the Mittag-Leffler function in one parameter are compared with the modified FPSM using the Mittag-Leffler function in two parameters.