Numerical Solutions of Neutral Stochastic Functional Differential Equations

Numerical Solutions of Neutral Stochastic Functional Differential Equations

This paper examines the numerical solutions of neutral stochastic functional differential equations (NSFDEs) d[x(t)− u(xt)] = f(xt)dt + g(xt)dw(t), t ≥ 0. The key contribution is to establish the strong mean square convergence theory of the Euler– Maruyama approximate solution under the local Lipschitz condition, the linear growth condition and the contractive mapping. These conditions are gene...

Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

Adaptive Numerical Solutions of Stochastic Differential Equations

In this paper we present an adaptive multi-element generalized polynomial chaos (ME-gPC) method, which can achieve hp-convergence in random space. ME-gPC is based on the decomposition of random space and generalized polynomial chaos (gPC). Using proper numerical schemes to maintain the local orthogonality on-the-fly, we perform gPC locally and adaptively. The key idea is to combine the polynomi...

Existence and continuous dependence for fractional neutral functional differential equations

In this paper, we investigate the existence, uniqueness and continuous dependence of solutions of fractional neutral functional differential equations with infinite delay and the Caputo fractional derivative order, by means of the Banach's contraction principle and the Schauder's fixed point theorem.

Asymptotic stability of neutral stochastic functional integro-differential equations*

This paper is concerned with the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic delay neutral partial functional integro-differential equations. We suppose that the linear part possesses a resolvent operator in the sense given in [8], and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is used to achie...