Mellin definition of the fractional Laplacian

被引：0
作者：
Gianni Pagnini
Claudio Runfola
机构：
[1] BCAM – Basque Center for Applied Mathematics,
[2] Ikerbasque – Basque Foundation for Science,undefined
[3] Aix-Marseille University,undefined
[4] Inserm,undefined
来源：
Fractional Calculus and Applied Analysis | 2023年 / 26卷
关键词：
Fractional calculus (primary); Fractional Laplacian; Mellin transform; Radial functions; Riesz fractional derivative; Symmetric Riesz–Feller fractional derivative; Space-fractional diffusion equation; Lévy stable densities; Lévy flights; Anomalous diffusion; 26A33 (primary); 47G30; 35S05; 44A15; 35R11;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further equivalent definition that is based on the Mellin transform and it can be used when the fractional Laplacian is applied to radial functions. The main finding is tested in the case of the space-fractional diffusion equation. The one-dimensional case is also considered, such that the Mellin transform of the Riesz (namely the symmetric Riesz–Feller) fractional derivative is established. This one-dimensional result corrects an existing formula in literature. Further results for the Riesz fractional derivative are obtained when it is applied to symmetric functions, in particular its relation with the Caputo and the Riemann–Liouville fractional derivatives.
引用
收藏
页码：2101 / 2117
页数：16
相关论文
共 102 条
[1]  
Bardaro C(2015)The foundations of fractional calculus in the Mellin transform setting with applications J. Fourier Anal. Appl. 21 961-1017
[2]  
Butzer PL(2016)Definition of the Riesz derivative and its application to space fractional quantum mechanics J. Math. Phys. 57 368-370
[3]  
Mantellini I(1949)Diffusion equation and stochastic processes Proc. Natl. Acad. Sci. USA 35 127-293
[4]  
Bayın SŞ(1990)Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications Phys. Rep. 195 287-301
[5]  
Bochner S(2019)On Riesz derivative Fract. Calc. Appl. Anal. 22 1243-1272
[6]  
Bouchaud JP(2018)Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions SIAM J. Numer. Anal. 56 610-634
[7]  
Georges A(2020)Why fractional derivatives with nonsingular kernels should not be used Fract. Calc. Appl. Anal. 23 3245-3270
[8]  
Cai M(2022)Trends, directions for further research, and some open problems of fractional calculus Nonlinear Dyn. 107 957-966
[9]  
Li CP(2017)All functions are locally J. Eur. Math. Soc. 19 1428-1455
[10]  
Cusimano N(2019)-harmonic up to a small error J. Geom. Anal. 29 231-256
12345678910