On the generalized fractional Laplacian
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Chenkuan Li
Abstract
The objective of this paper is, for the first time, to extend the fractional Laplacian (−△)s u(x) over the space Ck(Rn) (which contains S(Rn) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn. We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.
Acknowledgements
This work is supported by NSERC under grant number 2019-03907. The author is grateful to the Editors and to the reviewers for the careful reading of the paper with productive suggestions, which improved its quality.
References
[1] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, North-Holland (2006).Search in Google Scholar
[2] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York (1997), 223–276.10.1007/978-3-7091-2664-6_5Search in Google Scholar
[3] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Search in Google Scholar
[4] H.M. Srivastava, R.G. Buschman, Theory and Applications of Convolution Integral Equations. Kluwer Academic Publ., Dordrecht-Boston-London (1992).10.1007/978-94-015-8092-2Search in Google Scholar
[5] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77; 10.1016/S0370-1573(00)00070-3.Search in Google Scholar
[6] R. Metzler, 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), page R161.10.1088/0305-4470/37/31/R01Search in Google Scholar
[7] B. Mandelbrot, The Fractal Geometry of Nature. Henry Holt and Company (1982).Search in Google Scholar
[8] A.I. Saichev, G.M. Zaslavsky, Fractional kinetic equations: solutions and applications. Chaos: An Interdisciplinary J. of Nonlinear Sci. 7 (1997), 753–764; 10.1063/1.166272.Search in Google Scholar PubMed
[9] S. Dipierro, O. Savin, E. Valdinoci, Definition of fractional Laplacian for functions with polynomial growth. Rev. Mat. Iberoam. 35 (2019), 1079–1122.10.4171/rmi/1079Search in Google Scholar
[10] T.M. Michelitsch, G.A. Maugin, S. Derogar, M. Rahman, A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot type fractal functions. IMA J. Appl. Math. 79 (2014), 753–777; 10.1093/imamat/hxu018.Search in Google Scholar
[11] I.M. Geľfand, G.E. Shilov, Generalized Functions. Vol. I, Academic Press, New York (1964).Search in Google Scholar
[12] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth, G.E. Karniadakis, What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404 (2020), Art. 109009.10.1016/j.jcp.2019.109009Search in Google Scholar
[13] J. Barros-Neto, An Introduction to the Theory of Distributions. Marcel Dekker, New York (1973).Search in Google Scholar
[14] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach (1993).Search in Google Scholar
[15] M. Cai, C.P. Li, On Riesz derivative. Fract. Calc. Appl. Anal. 22, No 2 (2019), 287–301; 10.1515/fca-2019-0019; https://www.degruyter.com/journal/key/fca/22/2/html.Search in Google Scholar
[16] M. Kwaśnicki, The equivalent definitions for the fractional Laplacian operator. Fract. Calc. Appl. Anal. 20, No 1 (2017), 7–51; 10.1515/fca-2017-0002; https://www.degruyter.com/journal/key/fca/20/1/html.Search in Google Scholar
[17] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.10.1016/j.bulsci.2011.12.004Search in Google Scholar
[18] R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. II. Interscience, New York (1962).10.1063/1.3057861Search in Google Scholar
[19] P. Pizzetti, Sulla media dei valori che una funzione del punti dello spazio assume alla superficie di una spera. Rend. Lincei 18 (1909), 182–185.Search in Google Scholar
[20] C. Li, B. Fisher, Examples of the neutrix product of distributions on Rm. Rad. Mat. 6 (1990), 129–137.Search in Google Scholar
[21] C. Li, A review on the products of distributions. In: Mathematical Methods in Engineering, Springer (2007), 71–96.10.1007/978-1-4020-5678-9_5Search in Google Scholar
[22] M. Aguirre, C. Li, The distributional products of particular distributions. Appl. Math. Comput. 187 (2007), 120–126.10.1016/j.amc.2006.08.098Search in Google Scholar
[23] M. Aguirre, C. Marinelli, The series expansion of δ(k)(r − c). Math. Notae 35 (1991), 53–61.Search in Google Scholar
[24] C. Li, Several asymptotic products of particular distributions. BJMCS 3 (2013), 291–303.10.9734/BJMCS/2013/2721Search in Google Scholar
[25] C. Li, An example of the generalized fractional Laplacian on Rn. Contemp. Math. 1 (2020), Art. 215.10.37256/cm.142020489Search in Google Scholar
[26] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products. Academic Press, New York (1980).Search in Google Scholar
[27] C. Li, Several results of fractional derivatives in 𝓓′(R+). Fract. Calc. Appl. Anal. 18, No 1 (2015), 192–207; 10.1515/fca-2015-0013; https://www.degruyter.com/journal/key/fca/18/1/html.Search in Google Scholar
[28] C. Li, C.P. Li, T. Humphries, H. Plowman, Remarks on the generalized fractional Laplacian operator. Mathematics 7 (2019), Art. 320; 10.3390/math7040320.Search in Google Scholar
[29] C. Li, J. Beaudin, On the generalized Riesz derivative. Mathematics 8 (2020), Art. 1089; 10.3390/math8071089.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–6–2021)
- Research Paper
- Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces
- B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison
- A strong maximum principle for the fractional laplace equation with mixed boundary condition
- Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ
- Some properties of the fractal convolution of functions
- Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations
- Discrete fractional boundary value problems and inequalities
- On the generalized fractional Laplacian
- Recent developments on the realization of fractance device
- Explicit representation of discrete fractional resolvent families in Banach spaces
- Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
- Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
- An inverse problem approach to determine possible memory length of fractional differential equations
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA–volume 24–6–2021)
- Research Paper
- Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces
- B-spline collocation discretizations of caputo and Riemann-Liouville derivatives: A matrix comparison
- A strong maximum principle for the fractional laplace equation with mixed boundary condition
- Difference between Riesz derivative and fractional Laplacian on the proper subset of ℝ
- Some properties of the fractal convolution of functions
- Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations
- Discrete fractional boundary value problems and inequalities
- On the generalized fractional Laplacian
- Recent developments on the realization of fractance device
- Explicit representation of discrete fractional resolvent families in Banach spaces
- Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems
- Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
- An inverse problem approach to determine possible memory length of fractional differential equations
