Abstarct: In this thesis, we solve a special class of partial differential equations called reaction-diffusion equation and present a new method to solve this class of equations. The proposed method is baxsed on the spectral- collocation method. At first, we approximate the reaction-diffusion equation with the help of interpolator polynomials depending on the spectral Legendre-Gauss-Lubato points, and then, according to the properties of this approximation, we obtain a system of algebraic equations whose unknowns are the desired approximation coefficients. In the following, we use the proposed method to solve the one-dimensional and two-dimensional delayed reaction-diffusion equation. The convergence of the proposed method is discussed and analyzed in this thesis with using module of continuity theory. To show the capability and efficiency of the method, several numerical examples are also presented at the end of each chapter.‎‎‎ ‎