Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
In recent years, there has been greater attempt to find numerical solutions of differential equations using wavelet's methods. The following method is based on vector forms of Haar-wavelet functions. In this paper, we will introduce one dimensional Haar-wavelet functions and the Haar-wavelet operational matrices of the fractional order integration. Also the Haar-wavelet operational matrices of the fractional order differentiation are obtained. Then we propose the Haar-wavelet operational matrix method to achieve the Haar-wavelet time response output solution of fractional order linear systems where a fractional derivative is defined in the Caputo sense. Using collocation points, we have a Sylvester equation which can be solve by Block Krylov subspace methods. So we have analyzed the errors. The method has been tested by a numerical example. Since wavelet representations of a vector function can be more accurate and take less computer time, they are often more useful.