Free Access
Issue
ESAIM: M2AN
Volume 53, Number 4, July-August 2019
Page(s) 1245 - 1268
DOI https://doi.org/10.1051/m2an/2019025
Published online 09 July 2019
  1. E.J. Allen, S.J. Novosel and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Stoch. Rep. 64 (1998) 117–142. [CrossRef] [Google Scholar]
  2. A. Andersson, M. Kovács and S. Larsson, Weak error analysis for semilinear stochastic Volterra equations with additive noise. J. Math. Anal. Appl. 437 (2016) 1283–1304. [Google Scholar]
  3. A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev-Malliavin spaces and weak approximation of SPDE. Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016) 113–149. [Google Scholar]
  4. A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation. Math. Comp. 85 (2016) 1335–1358. [CrossRef] [Google Scholar]
  5. V.V. Anh, N.N. Leonenko and M. Ruiz-Medina, Space-time fractional stochastic equations on regular bounded open domains. Fract. Calc. Appl. Anal. 19 (2016) 1161–1199. [Google Scholar]
  6. W. Arendt, C.J. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. 2nd edition. Birkhäuser, Basel (2011) [CrossRef] [Google Scholar]
  7. D. Baffet and J.S. Hesthaven, A kernel compression scheme for fractional differential equations. SIAM J. Numer. Anal. 55 (2017) 496–520. [Google Scholar]
  8. E. Bazhlekova, B. Jin, R. Lazarov and Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid. Numer. Math. 131 (2015) 1–31. [Google Scholar]
  9. C.-E. Bréhier, M. Hairer and A.M. Stuart, Weak error estimates for trajectories of SPDEs under spectral Galerkin discretization. J. Comput. Math. 36 (2018) 159–182. [Google Scholar]
  10. L. Chen, Nonlinear stochastic time-fractional diffusion equations on R: moments, Hölder regularity and intermittency. Trans. Amer. Math. Soc. 369 (2017) 8497–8535. [CrossRef] [Google Scholar]
  11. L. Chen, Y. Hu and D. Nualart, Nonlinear stochastic time-fractional slow and fast diffusion equations on Formula . Preprint arXiv:1509.07763 (2015). [Google Scholar]
  12. Z.-Q. Chen, K.-H. Kim and P. Kim, Fractional time stochastic partial differential equations. Stochastic Process. Appl. 125 (2015) 1470–1499. [CrossRef] [Google Scholar]
  13. E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75 (2006) 673–696. [CrossRef] [Google Scholar]
  14. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd edition. Cambridge University Press, Cambridge (2014). [CrossRef] [Google Scholar]
  15. A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation. Math. Comp. 78 (2009) 845–863. [CrossRef] [Google Scholar]
  16. Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal. 40 (2002) 1421–1445. [Google Scholar]
  17. M. Foondun, Remarks on a fractional-time stochastic equation. Preprint arXiv:1811.05391 (2018). [Google Scholar]
  18. M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30 (2002) 1397–1465. [Google Scholar]
  19. H. Fujita and T. Suzuki, Evolution problems. In: Handbook of Numerical Analysis. Vol. II. NorthHolland, Amsterdam (1991) 789–928. [Google Scholar]
  20. M.B. Giles, Multilevel Monte Carlo methods. Acta Numer. 24 (2015) 259–328. [CrossRef] [Google Scholar]
  21. M. Gunzburger, B. Li and J. Wang, Convergence of finite element solution of stochastic partial integral-differential equations driven by white noise. Preprint arXiv:1711.01998 (2017). [Google Scholar]
  22. A. Jentzen and P.E. Kloeden, The numerical approximation of stochastic partial differential equations. Milan J. Math. 77 (2009) 205–244. [CrossRef] [Google Scholar]
  23. B. Jin, R. Lazarov and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51 (2013) 445–466. [Google Scholar]
  24. B. Jin, R. Lazarov and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38 (2016) A146–A170. [Google Scholar]
  25. B. Jin, R. Lazarov and Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview. Comput. Methods Appl. Mech. Eng. 346 (2019) 332–358. [Google Scholar]
  26. B. Jin, B. Li and Z. Zhou, Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39 (2017) A3129–A3152. [Google Scholar]
  27. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006). [Google Scholar]
  28. M. Kovács and J. Printems, Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation. Math. Comp. 83 (2014) 2325–2346. [CrossRef] [Google Scholar]
  29. M. Kovács and J. Printems, Weak convergence of a fully discrete approximation of a linear stochastic evolution equation with a positive-type memory term. J. Math. Anal. Appl. 413 (2014) 939–952. [Google Scholar]
  30. R. Kruse, Strong and Weak Approximation of Semilinear Stochastic Evolution Equations, In Vol. 2093, Lecture Notes in Mathematics. Springer, Heidelberg (2014). [CrossRef] [Google Scholar]
  31. X. Li and X. Yang, Error estimates of finite element methods for stochastic fractional differential equations. J. Comput. Math. 35 (2017) 346–362. [Google Scholar]
  32. W. Liu, M. Röckner and J.L. da Silva, Quasi-linear (stochastic) partial differential equations with time-fractional derivatives. SIAM J. Math. Anal. 50 (2018) 2588–2607. [CrossRef] [Google Scholar]
  33. S.V. Lototsky and B.L. Rozovsky, Classical and generalized solutions of fractional stochastic differential equations. Preprint. arXiv:1810.12951 (2018) . [Google Scholar]
  34. C. Lubich, Discretized fractional calculus. SIAM J. Math. Anal. 17 (1986) 704–719. [CrossRef] [MathSciNet] [Google Scholar]
  35. C. Lubich, I.H. Sloan and V. Thomée, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65 (1996) 1–17. [CrossRef] [Google Scholar]
  36. W. McLean and K. Mustapha, Time-stepping error bounds for fractional diffusion problems with non-smooth initial data. J. Comput. Phys. 293 (2015) 201–217. [Google Scholar]
  37. R. Metzler, J.H. Jeon, A.G. Cherstvy and E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16 (2014) 24128–24164. [CrossRef] [PubMed] [Google Scholar]
  38. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. 2nd edition. Springer-Verlag, Berlin (2006). [Google Scholar]
  39. Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43 (2005) 1363–1384. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you