The Exact Solutions of the Fractional Partial Differential Equations Using the Fractional Prabhakar Power Series Method

The Fractional Power Series Method (FPSM) is a method which provides systematic procedure to obtain exact solution of the Fractional Partial Differential Equations (FPDEs). Recently, the FPSM has been applied in science and engineering to address physical problems in heat conduction, fluid dynamics, quantum mechanics, viscoelastic and so on. Interestingly, applying the FPSM to look for the solution of the FPDE contains the Mittag–Leffler function. The solution is feasible due to the involvement of the Mittag–Leffler function with one parameter. However, the Prabhakar function which generalizes the Mittag–Leffler function has been overlooked by the researchers across the globe. This Prabhakar function contains an additional parameter, the Pochhammer symbol, which when incorporated in the FPSM yields not only exact solution but also a continuum solution of the FPDE in a functional space. In this paper, a modified version, known as the Fractional Prabhakar Power Series Method (FPPSM), is introduced to find the solutions to both the fractional heat and telegraph equations. Thus, this improved method incorporates the Prabhakar function instead of the Mittag–Leffler function. The additional parameter in the Prabhakar function provides the fast convergent solution in terms of the number of steps involved in obtaining its solution. The FPPSM, when applied to find for the solution of FPDE, yields a series which converges to the exact solution of the FPDE. The FPPSM is applied to obtain the solutions of the fractional heat equations in multidimensions: two and three dimensions and the fractional telegraph equation in one dimension. The series obtained using the FPPSM is derived to be in Sobolev spaces, ensuring the existence of a unique solution on the grounds that the Sobolev spaces are complete. Also, a unique stable solution with respect to small perturbation in the initial conditions of the fractional heat and telegraph equation is established in this paper, verifying that both the fractional heat and telegraph equations are well posed. The comparative analysis among the FPPSM and other existing methods such as the HAM, the VIM and the ADM is provided to ensure the efficacy of the FPPSM.