In this paper, the numerical solution of the initial-value problem involving the time-fractional diffusion problem in the Caputo sense can be express as a series of the shifted Vieta-Fibonacci polynomials with unknown coefficients. Next, by making use of the collocation points and the relations between their coefficients via the boundary conditions, the recent problem is reduced to a system of fractional ordinary differential equations (SFODEs) with initial conditions. Then , utilizing the residual power series method (RPSM) on SFODEs, the analytic approximate solution can be achieved. To illustrate the simplicity and accuracy of the proposed method, some numerical examples are considered
