Mittag--Leffler Euler Integrator for a Stochastic Fractional Order Equation with Additive Noise

Abstract

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here as the Mittag-Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.

Well-posedness and Mittag--Leffler Euler integrator for space-time fractional SPDEs with fractionally integrated additive noise

“…Existence and uniqueness of the mild solution. The following assumption is common in the theoretical analysis of semilinear fractional SPDEs; see e.g., [12,14]. Assumption 3.1.…”

“…When α ∈ ( 1 2 , 1), β ∈ ( 1 2 , 1] and γ ∈ [0, 1], the model (1.1) is solved by the Galerkin finite element method combined with the Wong-Zakai approximation in [12], which mainly focuses on the spatial discretization. Without being too exhaustive, we also refer to [6,7,14,15,18] for other numerical methods of related models. To our best knowledge, however, there seems no work on the time-stepping method and the corresponding numerical analysis for the model (1.1) with general α ∈ (0, 1], β ∈ (0, 1] and γ ∈ [0, 1].…”

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

“…When α ∈ (0, 1), equation ( 1) becomes a fractional stochastic heat equation (U 1 = 0 is this case), while for α ∈ (1, 2), a fractional stochastic wave equation. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory [36], while fractional stochastic wave type equations may be used to model random forcing effects in viscoelastic materials which exhibit a simple power-law creep [10,32]. For analytic results for stochastic Volterra-type integrodifferential equations and, in particular for fractional order differential, equations we refer to [2,3,4,8,9,5,6,7,11,10,15,16,17,18,19,22,24,25,26,27,33,36,39,38] which is, admittedly, a rather incomplete list.…”

“…We demonstrate by two examples how to obtain such estimates for fractional order equations both for spectral Galerkin and for a standard continuous finite element method. In general, when S i are resolvent families for certain parabolic integrodifferential problems arising, for example, in viscoelasticity, [2,10,32,37], these nonsmooth data estimates are direct consequences of the smoothing property of the resolvent family of the linear deterministic problem, at least for the spectral Galerkin method. For similar works in the setting of abstract evolution equations (without memory kernel) we refer to [13,14].…”

“…To our knowledge all existing work on the numerical analysis of stochastic fractional order differential equations are considering a much weaker error measure; that is, the sup t∈[0,T ] E U (t) − U n (t) p (mainly for p = 2), see, for example, [20,21,23,30,32,41], or the weak error [1,23,29,31].…”