Abstract
In this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.
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References
M.A. Al-Gwaiz, Sturm-Liouville Theory and its Applications. Springer Undergraduate Mathematics Series, Springer, London (2008).
E. Bayro-Corrochano, R. Vallejo, N. Arana-Daniel, Geometric preprocessing, geometric feedforward neural networks and Clifford support vector machines for visual learning. Neurocomputing 67, No 1-4 (2005), 54–105.
F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis. Research Notes in Mathematics. Vol.76, Pitman Advanced Publishing Program, Boston-London-Melbourne (1982).
R. Delanghe, F. Sommen, V. Souček, Clifford Algebras and Spinor-Valued Functions. A function Theory for the Dirac Operator. Mathematics and its Applications Vol. 53, Kluwer Academic Publishers, Dordrecht etc. (1992).
M. Ferreira, R. Kraußhar, M.M. Rodrigues, N. Vieira, A higher dimensional fractional Borel-Pompeiu formula and a related hypercomplex fractional operator calculus. Math. Methods Appl. Sci. 42, No 10 (2019), 3633–3653.
M. Ferreira, N. Vieira, Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators. J. Math. Anal. Appl. 447, No 1 (2017), 329–353.
I.M. Gelfand, S.V. Fomin, Calculus of Variations. Dover Publications Inc, New York (2000).
R.P. Gilbert, J.L. Buchanan, First Order Elliptic Systems. A Function Theoretic Approach. Mathematics in Science and Engineering. Vol. 163, Academic Press, New York-London etc. (1983).
R.B. Günther, J.W. Lee, Sturm-Liouville Problems. Theory and Numerical Implementation. Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL (2019).
K. Gürlebeck, W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers. Mathematical Methods in Practice, Wiley, Chichester (1997).
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Vol. 204, Elsevier, Amsterdam (2006).
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; https://www.degruyter.com/journal/key/FCA/21/1/html.
M. Klimek, Fractional Sturm-Liouville problem in terms of Riesz derivatives. In: Lecture Notes in Electrical Engineering - Vol. 357, 7th Conference on Non-Integer Order Calculus and Its Applications. Springer, Cham-Heidelberg-New York, etc. (2016), 3–16.
M. Klimek, T. Odzijewicz, A.B. Malinowska, Variational methods for the Sturm–Liouville problem. J. Math. Anal. Appl. 416, No 1 (2014), 402–426.
M. Klimek, O.P. Argawal, Fractional SturmLiouville problem. Comput. Math. Appl. 66, No 5 (2013), 795–812.
M. Klimek, O.P. Argawal, On a regular fractional Sturm-Liouville problem with derivatives of order in (0,1). In: Proc. of the 2012 13th International Carpathian Control Conference (ICCC 2012). I. Petrás, I. Podlubny, K. Kostúr, J. Kacur and A. Mojzisová (Eds.), Article No 6228655, 284–289.
I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Mathematics in Science and Engineering. Vol. 198, Academic Press, San Diego, CA (1999).
M. Rivero, J.J. Trujillo, M.P. Velasco, A fractional approach to the Sturm-Liouville problem. Centr. Eur. J. Phys. 11, No 10 (2013), 1246–1254.
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York, NY (1993).
V.E. Tarasov, Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science, Springer, Berlin (2010).
E. Whittaker, G. Robinson, The Calculus of Observations: A Treatise on Numerical Mathematics. Dover, New York (1967).
C. Xu, H. Liu, W. Cao, F.J. Wenming, Multispectral image edge detection via Clifford gradient. Sci. China, Inf. Sci. 55, No 2 (2012), 260–269.
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Ferreira, M., Manuela Rodrigues, M. & Vieira, N. A Fractional Analysis in Higher Dimensions for the Sturm-Liouville Problem. Fract Calc Appl Anal 24, 585–620 (2021). https://doi.org/10.1515/fca-2021-0026
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DOI: https://doi.org/10.1515/fca-2021-0026