On the fractional Laplace-Bessel operator

Borhen Halouani; Fethi Bouzeffour; Borhen Halouani; Fethi Bouzeffour

doi:10.3934/math.20241045

AIMS Mathematics

2024, Volume 9, Issue 8: 21524-21537. doi: 10.3934/math.20241045

Previous Article Next Article

Research article

On the fractional Laplace-Bessel operator

Borhen Halouani ^,,
Fethi Bouzeffour

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 24 March 2024 Revised: 11 May 2024 Accepted: 16 May 2024 Published: 05 July 2024
MSC : 43A32, 44A15

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator $ \Delta_{\nu} $, defined as
$ \Delta_{\nu} = \sum\limits_{i = 1}^{n}\frac{\partial^2}{\partial x_{i}^2} + \frac{\nu_i}{x_{i}}\frac{\partial}{\partial x_{i}}, \quad \nu_i\geq 0. $
The fractional power of this operator is introduced as a pseudo-differential operator through the multi-dimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations.
- Laplace-Bessel operator,
- fractional calculus,
- Bessel function
Citation: Borhen Halouani, Fethi Bouzeffour. On the fractional Laplace-Bessel operator[J]. AIMS Mathematics, 2024, 9(8): 21524-21537. doi: 10.3934/math.20241045

Related Papers:

Abstract

In this paper, we propose a novel approach to the fractional power of the Laplace-Bessel operator $ \Delta_{\nu} $, defined as

$ \Delta_{\nu} = \sum\limits_{i = 1}^{n}\frac{\partial^2}{\partial x_{i}^2} + \frac{\nu_i}{x_{i}}\frac{\partial}{\partial x_{i}}, \quad \nu_i\geq 0. $

The fractional power of this operator is introduced as a pseudo-differential operator through the multi-dimensional Bessel transform. Our primary contributions encompass a normalized singular integral representation, Bochner subordination, and intertwining relations.

References

[1]	K. B. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, New York and London: Academic Press, 1974.
[2]	B. Ross, The development of fractional calculus, New York: Academic Press, 1975.
[3]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[5]	O. Abu Arqub, Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates, J. Appl. Math. Comput., 59 (2019), 227–243. https://doi.org/10.1007/s12190-018-1176-x doi: 10.1007/s12190-018-1176-x
[6]	B. Maayah, A. Moussaoui, S. Bushnaq, O. Abu Arqub, The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach, Demonstr. Math., 55 (2022), 963–977. https://doi.org/10.1515/dema-2022-0183 doi: 10.1515/dema-2022-0183
[7]	O. Abu Arqub, B. Maayah, Adaptive the Dirichlet model of mobile/immobile advection/dispersion in a time-fractional sense with the reproducing kernel computational approach: Formulations and approximations, Int. J. Mod. Phys. B, 37 (2023), 2350179. https://doi.org/10.1142/S0217979223501795 doi: 10.1142/S0217979223501795
[8]	A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, et al., What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020), 109009. https://doi.org/10.1016/j.jcp.2019.109009 doi: 10.1016/j.jcp.2019.109009
[9]	M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7–51. https://doi.org/10.1515/fca-2017-0002 doi: 10.1515/fca-2017-0002
[10]	E. L. Shishkina, S. M. Sitnik, Transmutations, singular and fractional differential equations with applications to mathematical physics, Academic Press, 2020.
[11]	B. Muckenhoupt, E. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc., 118 (1965), 17–92.
[12]	I. A. Kipriyanov, Singular elliptic boundary value problems, 1997.
[13]	I. A. Kipriyanov, Fourier-Bessel transforms and imbedding theorems for weighted classes, Trudy Mat. Inst. Steklov., 89 (1967), 130–213.
[14]	I. A. Kipriyanov, M. I. Klyuchantsev, On singular integrals generated by the generalized shift operator Ⅱ, Sib. Math. J., 11 (1970), 787–804. https://doi.org/10.1007/BF00967838 doi: 10.1007/BF00967838
[15]	K. Trimeche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comput. Harmon. A., 4 (1997), 97–112. https://doi.org/10.1006/acha.1996.0206 doi: 10.1006/acha.1996.0206
[16]	L. N. Lyakhov, On classes of spherical functions and singular pseudodifferential operators, Dokl. Akad. Nauk. SSSR 272 (1983), 781–784.
[17]	L. N. Lyakhov, E. L. Shishkina, General $B$-hypersingular integrals with homogeneous characteristic, Dokl. Math., 75 (2007), 39–43. https://doi.org/10.1134/S1064562407010127 doi: 10.1134/S1064562407010127
[18]	K. Stempak, The Littlewood-Paley theory for the Fourier-Bessel transform, Mathematical Institute University of Wroclaw (Poland), 1985.
[19]	L. N. Lyakhov, A class of hypersingular integrals, Dokl. Akad. Nauk SSSR, 315 (1990), 291–296.
[20]	L. N. Lyakhov, E. L. Shishkina, General $B$-hypersingular integrals with homogeneous characteristic, Dokl. Math., 75, (2007), 39–43. https://doi.org/10.1134/S1064562407010127 doi: 10.1134/S1064562407010127
[21]	E. L. Shishkina, S. M. Sitnik, On fractional powers of the Bessel operator on semiaxis, Sib. Electron. Math. Rep., 15 (2018), 1–10. https://doi.org/10.17377/semi.2018.15.001 doi: 10.17377/semi.2018.15.001
[22]	A. Fitouhi, I. Jebabli, E. L. Shishkina, S. M. Sitnik, Applications of integral transforms composition method to wave-type singular differential equations and index shift transmutations, Electron. J. Differ. Equ., 2018 (2018), 1–27.
[23]	F. Bouzeffour, M. Garayev, On the fractional Bessel operator, Integr. Transf. Spec. F., 33 (2022), 230–246. https://doi.org/10.1080/10652469.2021.1925268 doi: 10.1080/10652469.2021.1925268
[24]	P. L. Butzer, G. Schmeisser, R. L. Stens, Sobolev spaces of fractional order, Lipschitz spaces, readapted modulation spaces and their interrelations; applications, J. Approx. Theory, 212 (2016), 1–40. https://doi.org/10.1016/j.jat.2016.08.001 doi: 10.1016/j.jat.2016.08.001
[25]	G. N. Watson, A treatise on the theory of Bessel functions, 1922.
[26]	T. H. Koornwinder, R. Wong, R. Koekoek, R. F. Swarttouw, W. P. Reinhardt, Orthogonal polynomials, NIST handbook of mathematical functions.
[27]	A. Marchaud, Sur les dérivées et sur les différences des fonctions de variables réelles, J. Math. Pure. Appl., 6 (1927), 337–425.
[28]	D. R. Coscia, The volume of the $n$-sphere, Suffolk Country Community College, 1973.
[29]	A. J. Castro, T. Z. Szarek, Calderón-Zygmund operators in the Bessel setting for all possible type indices, Acta Math. Sin. English Ser., 30 (2014), 637–648. https://doi.org/10.1007/s10114-014-2326-1 doi: 10.1007/s10114-014-2326-1

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)