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Article overview
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Well-posedness and Mittag--Leffler Euler integrator for space-time fractional SPDEs with fractionally integrated additive noise | | Xinjie Dai
; Jialin Hong
; Derui Sheng
; | | Date: |
1 Jun 2022 | | Abstract: | This paper considers the space-time fractional stochastic partial
differential equation (SPDE, for short) with fractionally integrated additive
noise, which is general and includes many (fractional) SPDEs with additive
noise. Firstly, the existence, uniqueness and temporal regularity of the mild
solution are presented. Then the Mittag--Leffler Euler integrator is proposed
as a time-stepping method to numerically solve the underlying model. Two key
ingredients are developed to overcome the difficulty caused by the interaction
between the time-fractional derivative and the fractionally integrated noise.
One is a novel decomposition way for the drift part of the mild solution, named
here the integral decomposition technique, and the other is to derive some fine
estimates associated with the solution operator by making use of the properties
of the Mittag--Leffler function. Consequently, the proposed Mittag--Leffler
Euler integrator is proved to be convergent with order $min{ frac{1}{2} +
frac{alpha}{2eta}(r+lambda-kappa), 1 }$ if $alpha + gamma = 1$,
otherwise order $min{ frac{alpha}{2eta}min{kappa,r+lambda} +
(gamma-frac{1}{2})^{+}, 1-varepsilon}$ in the sense of $L^2(Omega,H)$-norm
for the nonlinear case. In particular, the corresponding convergence order can
attain $min{ alpha + frac{alpha r}{2eta} +
(gamma-frac{1}{2})^{+}-varepsilon, alpha+gamma -varepsilon, 1 }$ for the
linear case. | | Source: | arXiv, 2206.00320 | | Services: | Forum | Review | PDF | Favorites | |
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