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.‎