This paper presents an approach for solving nonlinear STOCHASTIC di ERENTIAL EQUATIONS (NSDEs) using a new basis functions (NBFs). These functions and their operational matrices are used for representing matrix form of the NBFs. With using this method in combination with the collocation method, the NSDEs are reduced a STOCHASTIC nonlinear system of 2m + 2 EQUATIONS and 2m + 2 unknowns. Then, the error analysis is proved. Finally, numerical examples illustrate applicability and accuracy of the presented method.

In this article, a new numerical method based on triangular functions for solving nonlinear STOCHASTIC di ERENTIAL EQUATIONS is presented. For this, the STOCHASTIC operational matrix of triangular functions for It^o integral are determined. Computation of presented method is very simple and attractive. In addition, convergence analysis and numerical examples that illustrate accuracy and e ciency of the method are presented.

In this paper, we investigate stability in distribution of neutral STOCHASTIC functional di, ERENTIAL EQUATIONS with in, nite delay (NSFDEwID) at the state space Cr. We drive a su, cient strong monotone condition for the existence and uniqueness of the global solutions of NSFDEwID in the state space Cr. We also address the stability of the solution map xt and illustrate the theory with an example.

Many time-varying phenomena of various  elds in science and engineering can be modeled as a STOCHASTIC di ERENTIAL EQUATIONS, so investigation of conditions for existence of solution and obtain the analytical and numerical solutions of them are important. In this paper, the Adomian decomposition method for solution of the STOCHASTIC di ERENTIAL EQUATIONS are improved. Uniqueness and convergence of their adapted solutions are reviewed. The e ciency of the method is demonstrated through the two numerical experiments.

In this paper we propose a method for computing approximations of solution of fuzzy fractional diffERENTIAL EQUATIONS using fuzzy variational iteration method. Defining a fuzzy fractional derivative, we verify the utility of the method through two illustrative examples.

By adding a suitable real function on both sides of the quadratic Riccati diffERENTIAL equation, we propose a weighted type of Adams-Bashforth rules for solving it, in which moments are used instead of the constant coefficients of Adams-Bashforth rules. Numerical results reveal that the proposed method is efficient and can be applied for other nonlinear problems.

In this paper, a class of semi-implicit two-stage STOCHASTIC Runge-Kuttamethods (SRKs) of strong global order one, with minimum principal er-ror constants are given. These methods are applied to solve It^o stochas-tic di ERENTIAL EQUATIONS (SDEs) with a Wiener process. The e ciency ofthis method with respect to explicit two-stage It^o Runge-Kutta methods(IRKs), It^o method, Milstien method, semi-implicit and implicit two-stageStratonovich Runge-Kutta methods are demonstrated by presenting somenumerical results.

In this paper, a new type of equation namely fuzzy fractional STOCHASTIC Pantograph delay diffERENTIAL system (FSPDDS) is proposed. In our previous work, a first extension of fuzzy STOCHASTIC diffERENTIAL system into fuzzy fractional STOCHASTIC diffERENTIAL system by using Granular differentiability has been established. Here we study the existence and uniqueness results for the fuzzy FSPDDS which are obtained by using generalized Granular differentiability and contraction principle with weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results.

STOCHASTIC linear combinations of some random vectors are studied where the dis-tribution of the random vectors and the joint distribution of their coe, cients have Dirichlet distributions. A method is provided for calculating the distribution of these combinations which has been studied before. Our main result is the same as but from a di, erent point of view.