In this paper, the second kind Chebyshev polynomials (SKCPs) basis is used to solve time-fractional diffusion-wave equations with damping. We present some notations and definitions of the fractional calculus and introduce some basic properties of the SKCPs. Bivariate shifted SKCPs are defined and the operational matrix of fractional integration and some other needed operational matrices are constructed. Our approach uses the properties of bivariate shifted SKCPs to transform the considered problem to a matrix equation without using any collocation points. The main characteristic of this technique is that only a small number of the basic functions is needed to obtain a satisfactory result. An estimation of the error is given in the sense of Sobolev norms. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.
