Numerical approximation of stochastic time-fractional diffusion

Jin, Bangti; Yan, Yubin; Zhou, Zhi

doi:10.1051/m2an/2019025

Cited by 40 publications

(26 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For simplicity, we only consider the experimentally determined temporal convergence rates of the proposed numerical methods. More numerical simulations for the stochastic time fractional diffusions can be found in Jin et al [21].…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise

Yan

2020

Applied Numerical Mathematics

Self Cite

View full text Add to dashboard Cite

We consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated space-time white noise. The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich's first order convolution quadrature formula. We use the Euler method to approximate the noise in time and use the truncated series to approximate the noise in space. The spatial variable is discretized by using the linear finite element method. Applying the idea in Gunzburger et al. (Math. Comp. 88(2019), pp. 1715-1741), we express the approximate solutions of the fully discrete scheme by the convolution of the piecewise constant function and the inverse Laplace transform of the resolvent related function. Based on such convolution expressions of the approximate solutions, we obtain the optimal convergence orders of the fully discrete scheme in spatial multi-dimensional cases by using the Laplace transform method and the corresponding resolvent estimates.

show abstract

Section: Numerical Experimentsmentioning

confidence: 99%

“…Li et al [24] considered the finite element method for solving stochastic super-diffusion equation. Jin et al [21] considered the numerical methods for stochastic time fractional partial differential equation driven by integrated noise.…”

Section: Introductionmentioning

confidence: 99%

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise

Yan

2020

Applied Numerical Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let Q be a positive bounded linear operator on L 2 ( D ) defined by Qe n = μ n e n ,

n \in ℤ^{+}

, with finite trace

T r false(Q false) = \sum_{n \in ℤ^{+}} μ_{n} < \infty

. The Wiener process

scriptW false(\cdot, t false)

with covariance Q (see Jin et al 34 and Zou and Wang 35 ) can be defined as

W (\cdot, t) = \sum_{n \in ℤ^{+}} B_{n} (t) Q^{1 / 2} e_{n} = \sum_{n \in ℤ^{+}} μ_{n}^{1 / 2} B_{n} (t) e_{n},

where B n ( t ) is a one‐dimensional Brownian motion, for

n \in ℤ^{+}

. One can refer to Gikhman and Skorokhod 36 (on page 229) a similar representation for Wiener process, which is called Karhunen–LoÃv˙e series expansion.…”

Section: Preliminariesmentioning

confidence: 99%

Regularized solution of a Cauchy problem for stochastic elliptic equation

Tuan

Hoan²,

Zhou

et al. 2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…For the recent development of the corrections of numerical methods for (1), we refer the readers to the survey paper [12], see also [37]. For other numerical methods for solving time fractional diffusion equation, we refer to [3,[5][6][7][8]11,15,[18][19][20][21][22]24,25,28,32,[38][39][40], etc.…”

Section: Introductionmentioning

confidence: 99%

Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

Wang

Yan

et al. 2020

J Sci Comput

Self Cite

View full text Add to dashboard Cite

Two higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Grünwald–Letnikov formulae introduced in Tian et al. (Math Comput 84:2703–2727, 2015). After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders $$O(k^2)$$O(k2) and $$ O(k^3)$$O(k3), respectively for any fixed time t for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.

show abstract

Numerical approximation of stochastic time-fractional diffusion

Cited by 40 publications

References 41 publications

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise

Regularized solution of a Cauchy problem for stochastic elliptic equation

Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

Contact Info

Product

Resources

About