An efficient computational scheme to solve a class of fractional stochastic systems with mixed delays

Providing effective numerical methods to approximate the solution of fractional order stochastic differential equations is of great importance, since the exact solution of this type of equations is not available in many cases. In this paper, a stepwise collocation method for solving a system of nonlinear stochastic fractional differential equations (NSFDEs) with mixed delays is presented. First, an approximation of the white noise term is considered and the convergence of the solution of the problem with this approximated white noise term to the solution of the main problem is proved. Then, a combination of a stepwise scheme and a Legendre collocation technique is introduced to solve the stochastic system. In each step, the problem is studied in a subdomain and the proposed method transforms the NSFDE with delays into a system of nonlinear algebraic equations. Moreover, the convergence analysis of the proposed numerical method is described. Two numerical test examples are provided to verify the efficiency of the numerical technique. Finally, a practical epidemic model is examined to show the applicability of this scheme. •A system of fractional stochastic differential equations with multiple delays is investigated.•An effective scheme is employed to approximate the white noise terms.•A novel stepwise scheme based on the shifted Legendre collocation method is proposed.•A complete discussion about the convergence of the presented method is surveyed.•The efficiency of the proposed stepwise approach is analyzed in some test examples.