The Fractional Power Series Method (FPSM) is an effective and efficient method that offers an analytic method to find exact solution for Fractional Partial Differential Equations (FPDEs) in a functional space. In recent time, the FPSM has been applied in various science and engineering fields to solve physical problems in areas such as fluid dynamics, quantum mechanics, viscoelasticity, and heat conduction. This paper introduces a modification of the FPSM called the Fractional Gauss Hypergeometric Power Series Method (FGHPSM) which employs the so‐called Gauss Hypergeometric Function (GHF) to replace the Mittag–Leffler function in the FPSM on the grounds that the GHF generalizes the Mittag–Leffler function. This GHF when integrated into the FPSM provides not only exact solution but a generalized solution as compared with solution of the same equation using the FPSM. The FGHPSM is applied to solve fractional heat equation in two and three dimensions as well as fractional telegraph equation in a single dimension. The series obtained by the FGHPSM is derived to be in Sobolev spaces ensuring the existence of a unique solution. Also, a unique stable solution with respect to small perturbation in the initial conditions of the fractional heat and telegraph equations is established in this paper.MSC2020 Classification: 35J10