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Sharp asymptotics in a fractional Sturm-Liouville problem

  • Pavel Chigansky EMAIL logo and Marina Kleptsyna


The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.


P. Chigansky’s research was funded by ISF 1383/18 Grant.


[1] M.S. Birman, M.Z. Solomyak, Spectral asymptotics of weakly polar integral operators. Izυ. AN SSSR 34 (1970), 1143–1158; DOI: 10.1070/im1970v004n05abeh000948.Search in Google Scholar

[2] T. Blaszczyk, M. Ciesielski, M. Klimek, J. Leszczynski, Numerical solution of fractional oscillator equation. Appl. Math. Comput. 218, No 6 (2011), 2480–2488; DOI: 10.1016/j.amc.2011.07.062.Search in Google Scholar

[3] T. Blaszczyk, M. Ciesielski, Numerical solution of fractional Sturm-Liouville equation in integral form. Fract. Calc. Appl. Anal. 17, No 2 (2014), 307–320; DOI: 10.2478/s13540-014-0170-8; in Google Scholar

[4] T. Blaszczyk, M. Ciesielski, Fractional oscillator equation: analytical solution and algorithm for its approximate computation. J. Vib. Control 22, No 8 (2016), 2045–2052; DOI: 10.1177/1077546314566836.10.1177/1077546314566836Search in Google Scholar

[5] J.C. Bronski, Asymptotics of Karhunen-Loeve eigenvalues and tight constants for probability distributions of passive scalar transport. Comm. Math. Phys. 238, No 3 (2003), 563–582; DOI: 10.1007/s00220-003-0835-3.Search in Google Scholar

[6] P. Chigansky, M. Kleptsyna, D. Marushkevych, Mixed fractional Brownian motion: a spectral take. J. Math. Anal. Appl. 482, No 2 (2020), 123558; DOI: 10.1016/j.jmaa.2019.123558.Search in Google Scholar

[7] P. Chigansky, Marina Kleptsyna, Exact asymptotics in eigenproblems for fractional Brownian covariance operators. Stochastic Process. Appl. 128, No 6 (2018), 2007–2059; DOI: 10.1016/ in Google Scholar

[8] M. Dehghan, A. B. Mingarelli, Fractional Sturm-Liouville eigenvalue problems, I. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RAC-SAM 114, No 2 (2020), Paper No. 46, 15; DOI: 10.1007/s13398-019-00756-8.Search in Google Scholar

[9] M-H. Derakhshan, A. Ansari, Fractional Sturm-Liouville problems for Weber fractional derivatives. Int. J. Comput. Math. 96, No 2 (2019), 217–237; DOI: 10.1080/00207160.2018.1425797.Search in Google Scholar

[10] F.D. Gakhov, Boundary Value Problems. Dover Publications, Inc., New York (1990).Search in Google Scholar

[11] A.G. Gibbs, Analytical solutions of the neutron transport equation in arbitrary convex geometry. J. Mathematical Phys. 10 (1969), 875–890; DOI: 10.1063/1.1664917.Search in Google Scholar

[12] H. Jin, W. Liu, Eigenvalue problem for fractional differential operator containing left and right fractional derivatives. Adv. Difference Equ. (2016), Paper No. 246, 12; DOI: 10.1186/s13662-016-0950-z.Search in Google Scholar

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

[14] M.L. Kleptsyna, D.A. Marushkevych, P.Yu. Chigansky, Asymptotic accuracy in estimation of a fractional signal in a small white noise. Automation and Remote Control81, No 3 (2020), 411–429; DOI: 10.1134/S0005117920030030.Search in Google Scholar

[15] M. Klimek, O.P. Agrawal, Fractional Sturm-Liouville problem. Comput. Math. Appl. 66, No 5 (2013), 795–812; DOI: 10.1016/j.camwa.2012.12.011.Search in Google Scholar

[16] M. Klimek, M. Blasik, Regular fractional Sturm-Liouville problem with discrete spectrum: Solutions and applications. In: ICFDA’14 International Conference on Fractional Differentiation and Its Applications (2014), 1–6; DOI: doi: 10.1109/ICFDA.2014.6967383.Search in Google Scholar

[17] M. Klimek, T. Odzijewicz, A.B. Malinowska, Variational methods for the fractional Sturm-Liouville problem. J. Math. Anal. Appl. 416, No 1 (2014), 402–426; DOI: 10.1016/j.jmaa.2014.02.009.Search in Google Scholar

[18] M. Klimek, Homogeneous Robin boundary conditions and discrete spectrum of fractional eigenvalue problem. Fract. Calc. Appl. Anal., 22, No 1 (2019), 78–94; DOI: 10.1515/fca-2019-0005; in Google Scholar

[19] M. Klimek, M. Blasik. Regular Sturm-Liouville problem with Riemann-Liouville derivatives of order in (1 2): discrete spectrum, solutions and applications. In: Advances in Modelling and Control of Non-integer Order Systems, Vol. 320 of Lect. Notes Electr. Eng., Springer, Cham (2015), 25–36; DOI: 10.1007/978-3-319-09900-2.Search in Google Scholar

[20] M. Klimek, M. Ciesielski, T. Blaszczyk, Exact and numerical solutions of the fractional Sturm-Liouville problem. Fract. Calc. Appl. Anal. 21, No 1 (2018), 45–71; DOI: 10.1515/fca-2018-0004 in Google Scholar

[21] J. Li, J. Qi, Eigenvalue problems for fractional differential equations with right and left fractional derivatives. Appl. Math. Comput. 256 (2015), 1–10; DOI: 10.1016/j.amc.2014.12.146.Search in Google Scholar

[22] J. Li, J. Qi, Note on a nonlocal Sturm-Liouville problem with both right and left fractional derivatives. Appl. Math. Lett. 97 (2019), 14– 19; DOI: 10.1016/j.aml.2019.05.011.Search in Google Scholar

[23] W.V. Li, Q.M. Shao, Gaussian processes: inequalities, small ball prob-abilities and applications. In: Stochastic Processes: Theory and Methods, Vol. 19 of Handbook of Statist., North-Holland, Amsterdam (2001), 533–597; DOI: 10.1016/S0169-7161(01)19019-X.Search in Google Scholar

[24] A.I. Nazarov, Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes. Commun. Contemp. Math. Online Ready (2020); doi: 10.1142/S0219199720500492; arXiv Preprint: 1908.10299 (2019).Search in Google Scholar

[25] R. Ozarslan, E. Bas, D. Baleanu, Representation of solutions for Sturm-Liouville eigenvalue problems with generalized fractional derivative. Chaos30, No 3 (2020), 033137, 11 pp.; DOI: 10.1063/1.5131167.Search in Google Scholar PubMed

[26] B. V. Pal'cev, Asymptotic behavior of the spectrum and eigenfunctions of convolution operators on a finite interval with the kernel having a homogeneous Fourier transform. Dokl. Akad. Nauk SSSR218 (1974), 28–31.Search in Google Scholar

[27] B. V. Pal'tsev, Asymptotics of the spectrum of integral convolution operators on a finite interval with homogeneous polar kernels. Izv. Ross. Akad. Nauk Ser. Mat. 67, No 4 (2003), 67–154; DOI: 10.1070/IM2003v067n04ABEH000443.Search in Google Scholar

[28] J. Qi, S. Chen, Eigenvalue problems of the model from nonlocal con-tinuum mechanics. J. Math. Phys. 52, No 7 (2011), 073516, 14 pp.; DOI: 10.1063/1.3610673.Search in Google Scholar

[29] S. Ukai, Asymptotic distribution of eigenvalues of the kernel in the Kirkwood-Riseman integral equation. J. Mathematical Phys. 12 (1971), 83–92; DOI: 10.1063/1.1665491.Search in Google Scholar

[30] M. Zayernouri, G E. Karniadakis, Fractional Sturm-Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252 (2013), 495–517; DOI: 10.1016/ in Google Scholar

Received: 2020-06-12
Revised: 2021-04-12
Published Online: 2021-06-23
Published in Print: 2021-06-25

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