bi-singular type of a fractional-order multi-points boundary value condition problem

In this paper, a numerical method based on modified hat functions (mhfs) is investigated to find an approximate solution for a fractional-order system of stochastic integro-differential equations. the fractional and stochastic operational matrices of integration of these functions are employed to present the numerical approach. by using these operational matrices and properties of mhfs, the considered problem is transformed into a system of algebraic equations which can be easily solved by an iterative method. also, error analysis of the proposed method is discussed. at the end, the accuracy and effectiveness of this approach are studied by some numerical examples.