Free Access
Issue
ESAIM: M2AN
Volume 54, Number 3, May-June 2020
Page(s) 751 - 774
DOI https://doi.org/10.1051/m2an/2019076
Published online 02 April 2020
  1. N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian. Ann. Inst. Henri Poincaré AN 34 (2017) 439–467. [CrossRef] [Google Scholar]
  2. G. Acosta and J.P. Borthagaray, A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55 (2015) 472–495. [Google Scholar]
  3. G. Acosta, J.P. Borthagaray, O. Bruno and M. Maas, Regularity theory and high order numerical methods for the (1D)-fractional Laplacian. Math. Comput. 87 (2018) 1821–1857. [Google Scholar]
  4. H. Antil and E. Otárola, A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53 (2015) 3432–3456. [Google Scholar]
  5. H. Antil, J. Pfefferer and S. Rogovs, Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization. Commun. Math. Sci. 16 (2018) 1395–1426. [Google Scholar]
  6. C. Bacuta, J.H. Bramble and J. Xu, Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains. J. Numer. Math. 11 (2003) 75–94. [CrossRef] [Google Scholar]
  7. O.G. Bakaunin, Turbulence and Diffusion. Springer Series in Synergetics. Springer-Verlag, Berlin (2008). [Google Scholar]
  8. P. Bates, On some nonlocal evolution equations arising in materials science. Nonlinear dynamics and evolution equations. Fields Inst. Comm. 48 (2006) 13–52. [Google Scholar]
  9. K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes. Probab. Theory Relat. Fields 127 (2003) 89–152. [Google Scholar]
  10. M. Bonforte, A. Figalli and J.L. Vázquez, Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations. Calc. Var. Part. Diff. Equ. 57 (2018) 34. [CrossRef] [Google Scholar]
  11. M. Bonforte, A. Figalli and J.L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. Part. Differ. Equ. 11 (2018) 945–982. [Google Scholar]
  12. A. Bonito and J.E. Pasciak, Numerical approximation of fractional powers of elliptic operators. Math. Comput. 84 (2015) 2083–2110. [Google Scholar]
  13. A. Bonito and J.E. Pasciak, Corrigendum to the paper “Numerical approximation of fractional powers of regularly accretive operators”. IMA J. Numer. Anal. 37 (2017) 2170. [CrossRef] [Google Scholar]
  14. A. Bonito and J.E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators. IMA J. Numer. Anal. 37 (2017) 1245–1273. [CrossRef] [Google Scholar]
  15. A. Bonito, W. Lei and J.E. Pasciak, The approximation of parabolic equations involving fractional powers of elliptic operators. J. Comput. Appl. Math. 315 (2017) 32–48. [Google Scholar]
  16. A. Bonito, J.P. Borthagaray, R.H. Nochetto, E. Otárola and A.J. Salgado, Numerical methods for fractional diffision. Comput. Visual Sci. 19 (2018) 19–46. [CrossRef] [Google Scholar]
  17. A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez and K. Burrage, Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization. J. R. Soc. Interf. 11 (2014) 20140352. [CrossRef] [PubMed] [Google Scholar]
  18. X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224 (2010) 2052–2093. [CrossRef] [MathSciNet] [Google Scholar]
  19. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Part. Differ. Equ. 32 (2007) 1245–1260. [CrossRef] [Google Scholar]
  20. L.A. Caffarelli and P.R. Stinga, Fractional elliptic equation, Cacioppoli estimates and regularity. Ann. Inst. Henri Poincaré AN 33 (2016) 767–807. [CrossRef] [MathSciNet] [Google Scholar]
  21. P. Chatzipantelidis, R.D. Lazarov, V. Thomée and L.B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains. BIT 46 (2006) S113–S143. [CrossRef] [Google Scholar]
  22. L. Chen, R.H. Nochetto, E. Otárola and A.J. Salgado, A PDE approach to fractional diffusion: a posteriori error analysis. J. Comput. Phys. 293 (2015) 339–358. [Google Scholar]
  23. Ó. Ciaurri, L. Roncal, P.R. Stinga, J.L. Torrea and J.L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications. Adv. Math. 330 (2018) 688–738. [CrossRef] [Google Scholar]
  24. S. Cifani and E.R. Jakobsen, On numerical methods and error estimates for degenerate fractional convection-diffusion equations. Numer. Math. 127 (2014) 447–483. [Google Scholar]
  25. N. Cusimano and L. Gerardo-Giorda, A space-fractional Monodomain model for cardiac electrophysiology combining anisotropy and heterogeneity on realistic geometries. J. Comput. Phys. 362 (2018) 409–424. [Google Scholar]
  26. N. Cusimano, F. del Teso, L. Gerardo-Giorda and G. Pagnini, Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions. SIAM J. Numer. Anal. 56 (2018) 1243–1272. [Google Scholar]
  27. D. del Castillo-Negrete and L. Chacón, Parallel heat transport in integrable and chaotic magnetic fields. Phys. Plasmas 19 (2012) 056112. [Google Scholar]
  28. F. del Teso, Finite difference method for a fractional porous medium equation. Calcolo 51 (2014) 615–638. [CrossRef] [Google Scholar]
  29. F. del Teso, J. Endal and E.R. Jakobsen, Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments. SIAM J. Numer. Anal. 56 (2018) 3611–3647. [Google Scholar]
  30. F. del Teso, J. Endal and E.R. Jakobsen, Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory. SIAM J. Numer. Anal. 57 (2019) 2266–2299. [Google Scholar]
  31. M. D’Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comp. Math. Appl. 66 (2013) 1245–1260. [CrossRef] [Google Scholar]
  32. G. Di Blasio and B. Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods. J. Differ. Equ. 253 (2012) 2593–2615. [Google Scholar]
  33. S. Dohr, C. Kahle, S. Rogovs and P. Swierczynski, A FEM for an optimal control problem subject to the fractional Laplace equation. Calcolo 56 (2019) 37. [CrossRef] [Google Scholar]
  34. J. Droniou, A numerical method for fractal conservation laws. Math. Comput. 79 (2010) 95–124. [Google Scholar]
  35. J. Droniou and E.R. Jakobsen, A uniformly converging scheme for fractal conservation laws, In: Vol. 77 of Finite volumes for complex applications. VII. Methods and theoretical aspects. Springer Proc. Math. Stat. Springer, Cham (2014) 237–245. [Google Scholar]
  36. P. Garbaczewski and V.A. Stephanovich, Fractional laplacians in bounded domains: killed, reflected, censored and taboo Lévy flights. Phys. Rev. E 99 (2019) 042126. [Google Scholar]
  37. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. 2nd ed. Springer-Verlag, Berlin Heidelberg (2001). [Google Scholar]
  38. G. Gilboa and S. Osher, Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7 (2008) 1005–1028. [Google Scholar]
  39. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing (1985). [Google Scholar]
  40. G. Grubb Regularity of spectral fractional Dirichlet and Neumann problems. Math. Nachr. 289 (2016) 831–844. [CrossRef] [Google Scholar]
  41. Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: a finite difference quadrature approach. SIAM J. Numer. Anal. 52 (2014) 3056–3084. [Google Scholar]
  42. Y. Huang and A. Oberman, Finite difference methods for fractional laplacians. Preprint arXiv:1611.00164v1 (2016) . [Google Scholar]
  43. M. Ilić, F. Liu, I. Turner and V. Anh, Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation (I). Fract. Calc. Appl. Anal. 8 (2005) 323–341. [Google Scholar]
  44. M. Ilić, F. Liu, I. Turner and V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) – with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal. 9 (2006) 333–349. [Google Scholar]
  45. A. Iserles, A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge, 2008. [CrossRef] [Google Scholar]
  46. T. Kato, Perturbation theory for linear operators. In: Classics in Mathematics, Reprint of the 1980 edition. Springer-Verlag, Berlin (1995). [CrossRef] [Google Scholar]
  47. J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris 1 (1968). [Google Scholar]
  48. A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth and G.E. Karniadakis, What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404 (2020) 109009. [Google Scholar]
  49. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010). [Google Scholar]
  50. 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. [Google Scholar]
  51. R. Musina and A.I. Nazarov, On fractional Laplacians. Comm. Part. Differ. Equ. 39 (2014) 1780–1790. [CrossRef] [Google Scholar]
  52. R.H. Nochetto, E. Otárola and A.J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15 (2015) 733–791. [CrossRef] [MathSciNet] [Google Scholar]
  53. A. Pazy, Semigroups of operators in Banach spaces, In: Vol. 1017 of Equadiff 82 (Würzburg, 1982)., Springer, Berlin (1983) 508–524. [Google Scholar]
  54. Ł Płociniczak, Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting. Commun. Nonlinear Sci. Numer. Simul. 76 (2019) 66–70. [Google Scholar]
  55. A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Series in Computational Mathematics. Springer-Verlag, Berlin Heidelberg (2008). [Google Scholar]
  56. Y.A. Rossikhin and M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63 (2010) 010801. [Google Scholar]
  57. R. Seeley, Singular integrals and boundary value problems. Am. J. Math. 88 (1966) 781–809. [CrossRef] [Google Scholar]
  58. R. Seeley, Complex powers of an elliptic operator. In: Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Am. Math. Soc., Providence, RI (1967) 288–307. [Google Scholar]
  59. R. Seeley, The resolvent of an elliptic boundary problem. Amer. J. Math. 91 (1969) 889–920. [CrossRef] [Google Scholar]
  60. R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh Sect. A 144 (2014) 831–855. [CrossRef] [Google Scholar]
  61. F. Song, C. Xu and G.E. Karniadakis, Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39 (2017) A1320–A1344. [Google Scholar]
  62. P.R. Stinga and J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Part. Differ. Equ. 35 (2010) 2092–2122. [CrossRef] [Google Scholar]
  63. V. Thomée, Galerkin finite element methods for parabolic problems. In: Vol. 25 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin (1997). [CrossRef] [Google Scholar]
  64. P.N. Vabishchevich, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. 282 (2015) 289–302. [Google Scholar]
  65. P.N. Vabishchevich, A splitting scheme to solve an equation for fractional powers of elliptic operators. Comput. Methods Appl. Math. 16 (2016) 161–174. [CrossRef] [Google Scholar]
  66. J.L. Vázquez, Nonlinear diffusion with fractional Laplacian operators. In: Vol. 7 of Nonlinear Partial Differential Equations. Abel Symp. Springer, Heidelberg (2012) 271–298. [CrossRef] [Google Scholar]
  67. J.L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Disc. Cont. Dyn. Sys. – Ser. S 7 (2014) 857–885. [Google Scholar]
  68. J.L. Vázquez, The mathematical theories of diffusion: nonlinear and fractional diffusion. In: Vol. 2186 of Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Math. Springer, Cham (2017) 205–278. [CrossRef] [Google Scholar]
  69. B. Volzone, Symmetrization for fractional Neumann problems. Nonlinear Anal. 147 (2016) 1–25. [CrossRef] [Google Scholar]
  70. M. Wrobel, Mathematical and numerical analysis of initial boundary value problem for a linear nonlocal equation. Math. Comput. Simul. (2019). [Google Scholar]
  71. K. Yosida, Functional analysis. In: Classics in Mathematics. Reprint of the sixth (1980) edition. Springer-Verlag, Berlin (1995). [CrossRef] [Google Scholar]
  72. Y. Zhang, M.M. Meerschaert and R.M. Neupauer, Backward fractional advection dispersion model for contaminant source prediction. Water Resour. Res. 52 (2016) 2462–2473. [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