On the decomposition of solutions: From fractional diffusion to fractional Laplacian

References

[1] S.S. Bayin, On the consistency of the solutions of the space fractional Schrödinger equation. J. Math. Phys. 53, No 4 (2012), 1–9; DOI: 10.1063/1.4705268.10.1063/1.4705268Search in Google Scholar

[2] S.S. Bayin, Comment on "On the consistency of the solutions of the space fractional Schrödinger equation" [J. Math. Phys. 53, 042105 (2012)] [MR2953264]. J. Math. Phys. 54, No 7 (2013), 1–4; DOI: 10.1063/1.481600710.1063/1.4816007Search in Google Scholar

[3] D.A. Benson, R. Schumer, M.M. Meerschaert, and S.W. Wheatcraft, Fractional dispersion, Lévy motion, and the MADE tracer tests. Transp. Porous Media 42, No 1-2 (2001), 211–240; DOI: 10.1023/A:1006733002131.10.1023/A:1006733002131Search in Google Scholar

[4] D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, The fractional-order governing equation of Lévy motion. Water Resources Research 36, No 6 (2000), 1413–1423.10.1029/2000WR900032Search in Google Scholar

[5] R.M. Blumenthal and R.K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators. Pacific J. Math. 9 (1959), 399–408.10.2140/pjm.1959.9.399Search in Google Scholar

[6] A. Buades, B. Coll, and J.M. Morel, Image denoising methods. A new nonlocal principle. SIAM Rev. 52 (2010), 113–147; DOI: 10.1137/090773908.10.1137/090773908Search in Google Scholar

[7] B. Carmichael, H. Babahosseini, S. Mahmoodi, and M. Agah, The fractional viscoelastic response of human breast tissue cells. Physical Biology 12, No 4 (2015).10.1088/1478-3975/12/4/046001Search in Google Scholar PubMed

[8] H. Chen and H. Wang, Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation. J. Comput. Appl. Math. 296 (2016), 480–498; DOI: 10.1016/j.cam.2015.09.022.10.1016/j.cam.2015.09.022Search in Google Scholar

[9] Z.-Q. Chen, M.M. Meerschaert, and E. Nane, Space-time fractional diffusion on bounded domains. J. Math. Anal. Appl. 393, No 2 (2012), 479–488; DOI: 10.1016/j.jmaa.2012.04.032.10.1016/j.jmaa.2012.04.032Search in Google Scholar

[10] Z.-Q. Chen, and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains. J. Funct. Anal. 226, No 1 (2005), 90–113; DOI: 10.1016/j.jfa.2005.05.004.10.1016/j.jfa.2005.05.004Search in Google Scholar

[11] Z.-Q. Chen, and R. Song, Continuity of eigenvalues of subordinate processes in domains. Math. Z. 252, No 1 (2006), 71–89; DOI: 10.1007/s00209-005-0845-2.10.1007/s00209-005-0845-2Search in Google Scholar

[12] D. Del Castillo-Negrete, B. Carreras, and V. Lynch, Fractional diffusion in plasma turbulence. Physics of Plasmas 11, No 8 (2004), 3854–3864.10.1063/1.1767097Search in Google Scholar

[13] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, No 15 (2012), 521–573; DOI: 10.1016/j.bulsci.2011.12.004.10.1016/j.bulsci.2011.12.004Search in Google Scholar

[14] B.P. Epps and B. Cushman-Roisin, Turbulence modeling via the fractional Laplacian. arXiv:1803.05286.Search in Google Scholar

[15] V.J. Ervin, Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces. J. Differential Equations 278 (2021), 294–325; DOI: 10.1016/j.jde.2020.12.034.10.1016/j.jde.2020.12.034Search in Google Scholar

[16] V.J. Ervin, N. Heuer, and J.P. Roop, Regularity of the solution to 1-D fractional order diffusion equations. Math. Comp. 87, No 313 (2018), 2273–2294; DOI: 10.1090/mcom/3295.10.1090/mcom/3295Search in Google Scholar

[17] V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, No 3 (2006), 558–576.10.1002/num.20112Search in Google Scholar

[18] V.J. Ervin and J.P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in ℝ^d. Numer. Methods Partial Differential Equations 23, No 2 (2007), 256–281; DOI: 10.1002/num.20169.10.1002/num.20169Search in Google Scholar

[19] L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, RI (2010).Search in Google Scholar

[20] G.B. Folland, Real Analysis. John Wiley & Sons, Inc., New York (1999).Search in Google Scholar

[21] F.D. Gakhov, Boundary Value Problems. Transl. Ed. by I.N. Sneddon. Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London (1999).Search in Google Scholar

[22] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, No 3 (2008), 1005–1028; DOI: 10.1137/070698592.10.1137/070698592Search in Google Scholar

[23] V. Ginting and Y. Li, On the fractional diffusion-advection-reaction equation in ℝ. Fract. Calc. Appl. Anal. 22, No 4 (2019), 1039–1062; DOI: 10.1515/fca-2019-0055; https://www.degruyter.com/journal/key/fca/22/4/html.10.1515/fca-2019-0055Search in Google Scholar

[24] G. Grubb, Distributions and Operators. Springer, New York (2009).Search in Google Scholar

[25] Z. Hao, G. Lin, and Z. Zhang, Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations. Appl. Math. Comput. 374 (2020), 1–13; DOI: 10.1016/j.amc.2020.125045.10.1016/j.amc.2020.125045Search in Google Scholar

[26] Z. Hao and Z. Zhang, Optimal regularity and error estimates of a spectral Galerkin method for fractional advection-diffusion-reaction equations. SIAM J. Numer. Anal. 58, No 1 (2020), 211–233; DOI: 10.1137/18M1234679.10.1137/18M1234679Search in Google Scholar

[27] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles. Water Resources Research 34, No 5 (1998), 1027–1033.10.1029/98WR00214Search in Google Scholar

[28] F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells. Rep. Progr. Phys. 76, No 4 (2013), 1–50; DOI: 10.1088/0034-4885/76/4/046602.10.1088/0034-4885/76/4/046602Search in Google Scholar PubMed

[29] M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J.M. Schwarz, On the nonlocality of the fractional Schrödinger equation. J. Math. Phys. 51, No 6, 062102 (2010), 1–6; DOI: 10.1063/1.3430552.10.1063/1.3430552Search in Google Scholar

[30] B. Jin, R. Lazarov, X. Lu, and Z. Zhou, A simple finite element method for boundary value problems with a Riemann-Liouville derivative. J. Comput. Appl. Math. 293 (2016), 94–111; DOI: 10.1016/j.cam.2015.02.058.10.1016/j.cam.2015.02.058Search in Google Scholar

[31] B. Jin, R. Lazarov, X. Lu, J. Pasciak, and W. Rundell, Variational formulation of problems involving fractional order differential operators. Math. Comp. 84, No 296 (2015), 2665–2700; DOI: 10.1090/mcom/2960.10.1090/mcom/2960Search in Google Scholar

[32] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam (2006).Search in Google Scholar

[33] M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262, No 5 (2012), 2379–2402; DOI: 10.1016/j.jfa.2011.12.004.10.1016/j.jfa.2011.12.004Search in Google Scholar

[34] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20, No 1 (2017), 7–51; DOI: 10.1515/fca-2017-0002; https://www.degruyter.com/journal/key/fca/20/1/html.10.1515/fca-2017-0002Search in Google Scholar

[35] N.S. Landkof, Foundations of Modern Potential Theory. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg (1972).10.1007/978-3-642-65183-0Search in Google Scholar

[36] Y. Li, Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation. Rend. Circ. Mat. Palermo, II. Ser (2021), DOI: 10.1007/s12215-021-00592-z.10.1007/s12215-021-00592-zSearch in Google Scholar

[37] Y. Li, Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions. J. Integral Equations Applications (In press).10.1216/jie.2021.33.327Search in Google Scholar

[38] Y. Li, On Fractional Differential Equations and Related Questions, Thesis (Ph.D.) – University of Wyoming. ProQuest LLC, Ann Arbor, MI (2019).Search in Google Scholar

[39] Y. Li, A note on generalized Abel equations with constant coefficients. Rocky Mountain J. Math. (In press).10.1216/rmj.2021.51.1749Search in Google Scholar

[40] Y. Li, On the skewed fractional diffusion advection reaction equation on the interval. arXiv:2005.04405.Search in Google Scholar

[41] Y. Li, H. Chen, and H. Wang, A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations. Math. Methods Appl. Sci. 40, No 14 (2017), 5018–5034; DOI: 10.1002/mma.4367.10.1002/mma.4367Search in Google Scholar

[42] A. Lischke, G. Pang, M. Gulian, and et al., What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009 (2020), 1–62; DOI: 10.1016/j.jcp.2019.109009.10.1016/j.jcp.2019.109009Search in Google Scholar

[43] Y. Luchko, Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys. 54, No 1, 012111 (2013), 1–10; DOI: 10.1063/1.4777472.10.1063/1.4777472Search in Google Scholar

[44] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996) 378, Springer, Vienna (1997), 291–348; DOI: 10.1007/978-3-7091-2664-6_7.10.1007/978-3-7091-2664-6_7Search in Google Scholar

[45] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (An Introduction to Mathematical Models). Worls Sci. - Imperial College Press, London (2010).10.1142/p614Search in Google Scholar

[46] Z. Mao and G.E. Karniadakis, A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56, No 1 (2018), 24–49; DOI: 10.1137/16M1103622.10.1137/16M1103622Search in Google Scholar

[47] B.M. McCay, M.N.L. Narasimhan, Theory of nonlocal electromagnetic fluids. Arch. Mech. (Arch. Mech. Stos.) 33, No 3 (1981), 365–384.Search in Google Scholar

[48] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1–77; DOI: 10.1016/S0370-1573(00)00070-3.10.1016/S0370-1573(00)00070-3Search in Google Scholar

[49] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60, No 1 (2016), 3–26.10.5565/PUBLMAT_60116_01Search in Google Scholar

[50] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101, No 3 (2014), 275–302; DOI: 10.1016/j.matpur.2013.06.003.10.1016/j.matpur.2013.06.003Search in Google Scholar

[51] S. G. Samko, A new approach to the inversion of the Riesz potential operator. Fract. Calc. Appl. Anal. 1, No 3 (1998), 225–245.Search in Google Scholar

[52] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993).Search in Google Scholar

[53] K. Saĭevand and K. Pichagkhi, Reanalysis of an open problem associated with the fractional Schrödinger equation. Teoret. Mat. Fiz. 192, No 1 (2017), 103–114; DOI: 10.4213/tmf9224.10.4213/tmf9224Search in Google Scholar

[54] M.F. Shlesinger, B.J. West and J. Klafter, Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, No 11 (1987), 1100–1103; DOI: 10.1103/PhysRevLett.58.1100.10.1103/PhysRevLett.58.1100Search in Google Scholar PubMed

[55] D.W. Sims, E.J. Southford, N.E. Humphries, G.C. Hays, C.J. Bradshaw, J.W. Pitchford, A. James, M.Z. Ahmed, A.S. Brierley, M.A. Hindell, et al., Scaling laws of marine predator search behaviour. Nature 451, No 7182 (2008), 1098–1102.10.1038/nature06518Search in Google Scholar PubMed

[56] E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Ser. No. 30, Princeton University Press, Princeton, N.J. (1970).Search in Google Scholar

[57] M. Stynes, Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1554–1562; DOI: 10.1515/fca-2016-0080; https://www.degruyter.com/journal/key/fca/19/6/html.10.1515/fca-2016-0080Search in Google Scholar

[58] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin; UMI, Bologna (2007).Search in Google Scholar

[59] H. Wang and D. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51, No 2 (2013), 1088-1107; DOI: 10.1137/120892295.10.1137/120892295Search in Google Scholar

[60] H. Wang, D. Yang, and S. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations. J. Sci. Comput. 70, No 1 (2017), 429–449; DOI: 10.1007/s10915-016-0196-7.10.1007/s10915-016-0196-7Search in Google Scholar

[61] H. Wang and X. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations. J. Comput. Phys. 281 (2017), 67–81; DOI: 10.1016/j.jcp.2014.10.018.10.1016/j.jcp.2014.10.018Search in Google Scholar

[62] G.M. Zaslavsky, D. Stevens, and H. Weitzner, Self-similar transport in incomplete chaos. Phys. Rev. E (3) 48, No 3 (1993), 1683–1694; DOI: 10.1103/PhysRevE.48.1683.10.1103/PhysRevE.48.1683Search in Google Scholar

[63] Y. Zhang, D.A. Benson, M.M. Meerschaert, and E.M. LaBolle, Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data. Water Resources Research 43, No 5 (2007).10.1029/2006WR004912Search in Google Scholar

[64] Z. Zhang, Error estimates of spectral Galerkin methods for a linear fractional reaction-diffusion equation. J. Sci. Comput. 78, No 2 (2019), 1087–1110; DOI: 10.1007/s10915-018-0800-0.10.1007/s10915-018-0800-0Search in Google Scholar

[65] X. Zheng, V. Ervin, and H. Wang, Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval. J. Sci. Comput. 86, No 3 (2021), DOI: 10.1007/s10915-020-01366-y.10.1007/s10915-020-01366-ySearch in Google Scholar

[66] X. Zheng, V. Ervin, and H. Wang, Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation. Appl. Numer. Math. 153 (2019), 234–247; DOI: 10.1016/j.apnum.2020.02.019.10.1016/j.apnum.2020.02.019Search in Google Scholar

[67] X. Zheng, V. Ervin, and H. Wang, Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension. Appl. Math. Comput. 361 (2019), 98–111; DOI: 10.1016/j.amc.2019.05.017.10.1016/j.amc.2019.05.017Search in Google Scholar