In this article, we discuss the existence and uniqueness of proportional Itô–Doob stochastic fractional order systems (PIDSFOS) by using Picard iteration method. The paper provides new results integral calculus techniques. We have shown convergence solution averaged PIDSFOS to that standard in context mean square also probability. One example is given illustrate our results.

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 ...

Dynamic systems in many branches of science and industry are often perturbed by various types of environmental noise. Analysis of this class of models are very popular among researchers. In this paper, we present a method for approximating solution of fractional-order stochastic delay differential equations driven by Brownian motion. The fractional derivatives are considered in the Caputo sense...