Studying of the solution's stability of a generalized non stationary stochastic differential equations system

Defining some of the essential definitions and conceptions. Stochastic matrix. Stability. Approximate stability. Approximate stability in the quadratic middle. Formula of a system of unsettled non stationary stochastic differential equations. Formula of a generalized system of unsettled non- stationary stochastic differential equations. Foundations of a system of differential equations that divines the partial moments of the second order. Foundations of a system of differential equations that divines matrices of Lyapunov's functions. The necessary and sufficient conditions formatrisses of Lyapunov's functions to assure the stability of the studied system's solution approximately in the quadratic middle.

Numerical Simulation Stochastic of Differential Equations by Using Spline Function Approximations

In this paper, spline approximations with five collocation points are used for the numerical simulation of stochastic of differential equations(SDE). First, we have modeled continuous-valued discrete wiener process, and then numerical asymptotic stochastic stability of spline method is studied when applied to SDEs. The study shows that the method when applied to linear and nonlinear SDEs are stable and convergent. Moreover, the scheme is tested on two linear and nonlinear problems to illustrate the applicability and efficiency of the purposed method. Comparisons of our results with Euler–Maruyama method, Milstein’s method and Runge-Kutta method, it reveals that the our scheme is better than others.