References
- A. Bonito and J.E. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comput. 84(295) (2015), pp. 2083–2110. doi: 10.1090/S0025-5718-2015-02937-8
- C. Braendle, E. Colorado, A.D. Pablo, and U. Snchez, A concave–convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. 143(1) (2013), pp. 39–71. doi: 10.1017/S0308210511000175
- X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire. 31(1) (2014), pp. 23–53. doi: 10.1016/j.anihpc.2013.02.001
- X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc. 367(2) (2015), pp. 911–941. doi: 10.1090/S0002-9947-2014-05906-0
- X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224(5) (2010), pp. 2052–2093. doi: 10.1016/j.aim.2010.01.025
- L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partaial Differential Equations 32(7–9) (2007), pp. 1245–1260. doi: 10.1080/03605300600987306
- A. Capella, J. Davila, L. Dupaigne, and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partaial Differential Equations 36(8) (2011), pp. 1353–1384. doi: 10.1080/03605302.2011.562954
- R.L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in R, Acta. Math. 210(2) (2013), pp. 261–318. doi: 10.1007/s11511-013-0095-9
- Y. Hu, C.P. Li, and H.F. Li, The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case, Chaos Solitons Fractals 102 (2017), pp. 319–326. doi: 10.1016/j.chaos.2017.03.038
- C.P. Li, Q. Yi, and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys. 316 (2016), pp. 614–631. doi: 10.1016/j.jcp.2016.04.039
- C.P. Li and F.H. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim. 34(2) (2013), pp. 149–179. doi: 10.1080/01630563.2012.706673
- C.P. Li and F.H. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, 2015.
- F. Liu, P. Zhuang, and Q. Liu, Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press, China (in Chinese), 2015.
- E.D. Nezza, G. Palatucci, and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012), pp. 521–573. doi: 10.1016/j.bulsci.2011.12.004
- R.H. Nochetto, E. Otarola, and A.J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math. 15(3) (2015), pp. 733–791. doi: 10.1007/s10208-014-9208-x
- R.H. Nochetto, E. Otarola, and A.J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal. 54(2) (2016), pp. 848–873. doi: 10.1137/14096308X
- K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974, renewed 2006.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
- F.D. Teso and J.L. Vázquez, Finite difference method for a general fractional porous medium equation, Math. 51(4) (2013), pp. 615–638.
- Q. Yang, I. Turner, T. Moroney, and F. Liu, A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction-diffusion equations, Appl. Math. Model. 38(15-16) (2014), pp. 3755–3762. doi: 10.1016/j.apm.2014.02.005
- Q. Yang, I. Turner, F. Liu, and M. Ilis, Novel numerical methods for solving the time-space fractional diffusion equation in 2D, SIAM J. Sci. Comput. 33 (2011), pp. 1159–1180. doi: 10.1137/100800634
- Q. Yu, F. Liu, I. Turner, and K. Burrage, Numerical investigation of three types of space and time fractional Bloch–Torrey equations in 2D, Cent. Eur. J. Phys. 11(6) (2013), pp. 646–665.