the operational matrix of fractional derivative of the fractional-order chebyshev functions and its applications

in this paper, we introduce a family of fractional-order chebyshev functions based on the classical chebyshev polynomials. we calculate and derive the operational matrix of derivative of fractional order $gamma$ in the caputo sense using the fractional-order chebyshev functions. this matrix yields to low computational cost of numerical solution of fractional order differential equations to the solution of a system of algebraic equations. several numerical examples are given to illustrate the accuracy of our method. the results obtained, are in full agreement with the analytical solutions and numerical results presented by some previous works.