Numerical Solution of Fractional Control System by Haar-wavelet Operational Matrix ‎Method

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.‎