Free Access
Issue
ESAIM: M2AN
Volume 55, Number 1, January-February 2021
Page(s) 171 - 207
DOI https://doi.org/10.1051/m2an/2020072
Published online 18 February 2021
  1. G. Akrivis, B. Li and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comput. 86 (2017) 1527–1552. [Google Scholar]
  2. L. Banjai and M. López-Fernández, Efficient high order algorithms for fractional integrals and fractional differential equations. Numer. Math. 141 (2019) 289–317. [Google Scholar]
  3. P. Baveye, P. Vandevivere, B.L. Hoyle, P.C. DeLeo and D.S. de Lozada, Environmental impact and mechanisms of the biological clogging of saturated soils and aquifer materials. Crit. Rev. Environ. Sci. Technol. 28 (2006) 123–191. [Google Scholar]
  4. J. Bear, Dynamics of Fluids in Porous Media. Elsevier, New York (1972). [Google Scholar]
  5. L.C. Becker, Resolvents and solutions of weakly singular linear Volterra integral equations. Nonlinear Anal. 74 (2011) 1892–1912. [Google Scholar]
  6. D. Benson, R. Schumer, M.M. Meerschaert and S.W. Wheatcraft, Fractional dispersion, Lévy motions, and the MADE tracer tests. Transp. Porous Media 42 (2001) 211–240. [Google Scholar]
  7. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction. Springer, Berlin-Heidleberg (1976). [Google Scholar]
  8. V. Bogachev, Measure Theory I, Springer, Berlin-Heidelberg (1978). [Google Scholar]
  9. T.A. Burton and D.P. Dwiggins, Resolvents of integral equations with continuous kernels. Nonlinear Stud. 18 (2011) 293–305. [Google Scholar]
  10. C. Cheng, V. Thomée and L. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math. Comput. 58 (1992) 587–602. [Google Scholar]
  11. E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75 (2006) 673–696. [Google Scholar]
  12. W. Deng, Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47 (2008) 204–226. [Google Scholar]
  13. P. Embrechts and M. Maejima, Selfsimilar Processes. In: Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ (2002). [Google Scholar]
  14. L.C. Evans, Partial Differential Equations, 2nd edition. In: Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2010). [Google Scholar]
  15. L. Gandossi and U.V. Estorff, An overview of hydraulic fracturing and other formation stimulation technologies for shale gas production. Scientific and Technical Research Reports. Joint Research Centre of the European Commission; Publications Office of the European Union (2015). . DOI: 10.2790/379646. [Google Scholar]
  16. L. Grafakos, Classical Fourier Analysis, 3rd edition. In: Vol. 249 of Graduate Texts in Mathematics. Springer, New York (2014). [Google Scholar]
  17. 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]
  18. 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]
  19. B. Jin, B. Li and Z. Zhou, An analysis of the Crank-Nicolson method for subdiffusion. IMA J. Numer. Anal. 38 (2018) 518–541. [Google Scholar]
  20. B. Jin, B. Li and Z. Zhou, Subdiffusion with a time-dependent coefficient: analysis and numerical solution. Math. Comput. 88 (2019) 2157–2186. [Google Scholar]
  21. Y. Kian, E. Soccorsi and M. Yamamoto, Finite difference/spectral approximations for the time-fractional diffusion equation. Ann. Henri Poincaré 19 (2018) 3955–3881. [Google Scholar]
  22. G.E. King, Hydraulic fracturing 101: What every representative, environmentalist, regulator, reporter, investor, university researcher, neighbor and engineer should know about estimating frac risk and improving frac performance in unconventional gas and oil wells. Society of Petroleum Engineers, SPE 152596 (2012). [Google Scholar]
  23. N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88 (2019) 2135–2155. [Google Scholar]
  24. A. Kubica and M. Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients. Fract. Calc. Appl. Anal. 21 (2018) 276–311. [Google Scholar]
  25. T. Lee, L. Bocquet and B. Coasne, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Nat. Commun. 29 (2016) 11890. [Google Scholar]
  26. L. Li and J.-G. Liu, A discretization of caputo derivatives with application to time fractional SDEs and gradient flows. SIAM J. Numer. Anal. 57 (2019) 2095–2120. [Google Scholar]
  27. H. Liao, W. McLean and J. Zhang, A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57 (2019) 218–237. [Google Scholar]
  28. Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007) 1533–1552. [Google Scholar]
  29. C.F. Lorenzo and T.T. Hartley, Variable order and distributed order fractional operators. Nonlinear Dyn. 29 (2002) 57–98. [Google Scholar]
  30. C. Lubich, Convolution quadrature and discretized operational calculus. I and II. Numer. Math. 52 (1988) 129–145 and 413–425. [Google Scholar]
  31. C. Lubich, Convolution quadrature revisited. BIT 44 (2004) 503–514. [Google Scholar]
  32. 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. Comput. 65 (1996) 1–17. [Google Scholar]
  33. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, 3rd edition, Birkhäuser Verlag, Basel (1995). [Google Scholar]
  34. M. Luskin and R. Rannacher, On the smoothing property of the Galerkin method for parabolic equations. SIAM J. Numer. Anal. 19 (1982) 93–113. [Google Scholar]
  35. C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38 (2016) A2699–A2724. [Google Scholar]
  36. W. McLean and K. Mustapha, Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer. Algorithms 52 (2009) 69–88. [Google Scholar]
  37. M.M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus. In: De Gruyter Studies in Mathematics (2011). [Google Scholar]
  38. R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1–77. [Google Scholar]
  39. R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004) R161–R208. [Google Scholar]
  40. K. Mustapha, Time-stepping discontinuous Galerkin methods for fractional diffusion problems. Numer. Math. 130 (2015) 497–516. [Google Scholar]
  41. K. Mustapha, FEM for time-fractional diffusion equations, novel optimal error analyses. Math. Comput. 87 (2018) 2259–2272. [Google Scholar]
  42. K. Mustapha and W. McLean, Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation. Numer. Algorithms 56 (2011) 159–184. [Google Scholar]
  43. K. Mustapha and W. McLean, Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation. IMA J. Numer. Anal. 32 (2011) 906–925. [Google Scholar]
  44. K. Mustapha and D. Schötzau, Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equation. IMA J. Numer. Anal. 34 (2013) 1426–1446. [Google Scholar]
  45. K. Mustapha, B. Abdallah and K.M. Furati, A discontinuous Petrov-Galerkin method for time-fractional diffusion equations. SIAM J. Numer. Anal. 52 (2014) 2512–2529. [Google Scholar]
  46. E.M. Ouhabaz, Gaussian estimates and holomorphy of semigroups. Proc. Amer. Math. Soc. 123 (1995) 1465–1474. [Google Scholar]
  47. R. Schumer, D.A. Benson, M.M. Meerschaert and B. Baeumer, Fractal mobile/immobile solute transport. Water Resour. Res. 39 (2003) 1–12. [Google Scholar]
  48. M. Stynes, E. O’Riordan and J.L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55 (2017) 1057–1079. [Google Scholar]
  49. Z.-Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006) 193–209. [Google Scholar]
  50. H. Sun, W. Chen and Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A: Stat. Mech. Appl. 388 (2009) 4586–4592. [Google Scholar]
  51. H. Sun, A. Chang, Y. Zhang and W. Chen, A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications. Fract. Calc. Appl. Anal. 22 (2019) 27–59. [Google Scholar]
  52. J. Sun, D. Nie and W. Deng, Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data. J. Sci. Comput. 84 (2020) 1–23. [Google Scholar]
  53. H. Wang and X. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equations. J. Math. Anal. App. 475 (2019) 1778–1802. [Google Scholar]
  54. L. Weis, A new approach to maximal Lp-regularity, edited by B. Herrenalb. In: Vol. 215 of Lecture Notes in Pure and Applied Mathematics. Evolution Equations and their Applications in Physical and Life Sciences. Dekker, New York (2001) 195–214. [Google Scholar]
  55. Y. Xian, M. Jin, H. Zhan and Y. Liu, Reactive transport of nutrients and bioclogging during dynamic disconnection process of stream and groundwater. Water Resour. Res. 55 (2019) 3882–3903. [Google Scholar]
  56. Y. Zhang, C. Green and B. Baeumer, Linking aquifer spatial properties and non-Fickian transport in mobile–immobile like alluvial settings. J. Hydrol. 512 (2014) 315–331. [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