Inversion formulas for space-fractional Bessel heat diffusion through Tikhonov regularization

Fethi Bouzeffour; Fethi Bouzeffour

doi:10.3934/math.20241013

AIMS Mathematics

2024, Volume 9, Issue 8: 20826-20842. doi: 10.3934/math.20241013

Previous Article Next Article

Research article

Inversion formulas for space-fractional Bessel heat diffusion through Tikhonov regularization

Fethi Bouzeffour ^,

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

Received: 05 May 2024 Revised: 14 June 2024 Accepted: 18 June 2024 Published: 27 June 2024
MSC : 44A15, 46E22, 35K05

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established.
- Weierstrass transform,
- best approximation,
- Hankel transform,
- reproducing kernel,
- Tikhonov regularization
Citation: Fethi Bouzeffour. Inversion formulas for space-fractional Bessel heat diffusion through Tikhonov regularization[J]. AIMS Mathematics, 2024, 9(8): 20826-20842. doi: 10.3934/math.20241013

Related Papers:

Abstract

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established.

References

[1]	S. Omri, L. T. Rachdi, Weierstrass transform associated with the Hankel operator, Bull. Math. Anal. Appl., 1 (2009), 1–16.
[2]	S. Saitoh, Best approximation, Tikhonov regularization and reproducing kernels, Kodai Math. J., 28 (2005), 359–367. https://doi.org/10.2996/kmj/1123767016 doi: 10.2996/kmj/1123767016
[3]	S. Saitoh, Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator equations on Hilbert spaces, Amer. Math. Soc. Transl. Series 2, 230 (2010).
[4]	S. Saitoh, Y. Sawano, Theory of reproducing kernels and applications, Devel. Math., 44 (2016). https://doi.org/10.1007/978-981-10-0530-5
[5]	T. Matsuura, S. Saitoh, Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces, Appl. Anal., 85 (2006), 901–915. https://doi.org/10.1080/00036810600643662 doi: 10.1080/00036810600643662
[6]	T. Matsuura, S. Saitoh, M. Yamada, Representations of inverse functions by the integral transform with the sign kernel, Frac. Calc. Appl. Anal., 2 (2007), 161–168.
[7]	F. Soltani, $L^p$-Fourier multipliers for the Dunkl operator on the real line, J. Funct. Anal., 209 (2004), 16–35.
[8]	F. Soltani, Extremal functions on Sobolev-Dunkl spaces, Integral Transf. Spec. Funct., 24 (2013), 582–595. https://doi.org/10.1080/10652469.2012.725167 doi: 10.1080/10652469.2012.725167
[9]	F. Soltani, Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator, Acta Math. Sci., 33 (2013), 430–442. https://doi.org/10.1016/S0252-9602(13)60010-7 doi: 10.1016/S0252-9602(13)60010-7
[10]	F. Soltani, Extremal functions on Sturm-Liouville hypergroups, Complex Anal. Oper. Theory, 8 (2014), 311–325. https://doi.org/10.1007/s11785-013-0303-9 doi: 10.1007/s11785-013-0303-9
[11]	F. Soltani, I. Maktouf, Dunkl-Weinstein multiplier operators and applications to reproducing kernel theory, Mediterr. J. Math., 21 (2024). https://doi.org/10.1007/s00009-024-02623-2
[12]	M. Dziri, A. Kroumi, Tikhonov regularisation for the Weierstrass transform associated with the Kontorovich-Lebedev transform, Complex Vari. Ellip. Equat., 68 (2022), 1119–1131. https://doi.org/10.1080/17476933.2022.2038584 doi: 10.1080/17476933.2022.2038584
[13]	S. Ghobber, H. Mejjaoli, Reproducing kernel theory associated with the generalized stockwell transform and applications, Complex Anal. Oper. Theory, 17 (2023). https://doi.org/10.1007/s11785-023-01407-y
[14]	H. Liu, M. Xie, B. Pan, N. Li, J. Zhang, M. Lu, et al., In-Situ intercalated pyrolytic Graphene/Serpentine Hybrid as an efficient lubricant additive in paraffin oil, Colloids Surf. A Phys. Eng. Aspects, 652 (2022), 129929. https://doi.org/10.1016/j.colsurfa.2022.129929 doi: 10.1016/j.colsurfa.2022.129929
[15]	M. Ding, H. Liu, G. H. Zheng, On inverse problems for several coupled PDE systems arising in mathematical biology, J. Math. Bio., 87 (2023). https://doi.org/10.1007/s00285-023-02021-4
[16]	W. Yin, B. Zhang, P. Meng, L. Zhou, D. Qi, A neural network method for inversion of turbulence strength, J. Nonlinear Math. Phys., 31 (2024), 22. https://doi.org/10.1007/s44198-024-00186-0 doi: 10.1007/s44198-024-00186-0
[17]	F. Bouzeffour, M. Garayev, On the fractional Bessel operator, Integr. Trans. Special Funct., 2001, 1–7. https://doi.org/10.1080/10652469.2021.1925268
[18]	F. Bouzeffour, Continuation of radial positive definite functions and their characterization, Fractal Fract., 7 (2023). https://doi.org/10.3390/fractalfract7080623
[19]	A. Bakushinsky, A. Goncharsky, Ill-Posed problems: Theory and applications, Springer Dordrecht, (1994). https://doi.org/10.1007/978-94-011-1026-6
[20]	J. Baumeister, Stable solution of inverse problems, Teubner Verlag Wiesbaden, 1987. https://doi.org/10.1007/978-3-322-83967-1
[21]	A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, Washington: Winston & Sons, 1977.
[22]	A. N. Tikhonov, A. S. Leonov, A. G. Yagola, Nonlinear Ill-Posed Problems, London: Chapman and Hall, 1998.
[23]	E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge: Cambridge University Press, 1952.
[24]	K. Trimeche, Generalized harmonic analysis and wavelet packets, CRC Press, 2001. https://doi.org/10.1201/9781482283174
[25]	W. R. Bloom, H. Heyer, Harmonic analysis of probability measures on hypergroups, De Gruyter Stu. Math., 20 (1995). https://doi.org/10.1017/S0013091500023130
[26]	E. M. Stein, W. Guido, Introduction to fourier analysis on euclidean spaces (PMS-32), Princeton University Press, 1971. Available from: http://www.jstor.org/stable/j.ctt1bpm9w6.12

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)