Abstract
This paper presents an accurate numerical algorithm to solve the space fractional-order Fisher's equation where the derivative operator is described in the Caputo derivative sense. In the presented discretization process, first, we use the compact finite difference (CFD) for a semidiscrete occurrence in time derivative and implement the Chebyshev spectral collocation method (CSCM) of the third-kind to discretize the spatial fractional derivative. The presented method converts the problem understudy to be a system of algebraic equations which can be easily solved. To study the convergence and stability analysis, some theorems are given with their proofs. A numerical simulation is outputted to test the accuracy and applicability of our presented algorithm.