Abstract
We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann–Liouville type. We then solve a Dirichlet type Sturm–Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann–Liouville operator on a finite interval where the boundary conditions are induced by evaluating Riemann–Liouville integrals at those end-points. For each \(1/2<\alpha <1\) it is shown that there is a finite number of real eigenvalues, an infinite number of non-real eigenvalues, that the number of such real eigenvalues grows without bound as \(\alpha \rightarrow 1^-\), and that the fractional operator converges to an ordinary two term Sturm–Liouville operator as \(\alpha \rightarrow 1^-\) with Dirichlet boundary conditions. Finally, two-sided estimates as to their location are provided as is their asymptotic behavior as a function of \(\alpha \).
Similar content being viewed by others
References
Al-Mdallal, Q.M.: An efficient method for solving fractional Sturm–Liouville problems. Chaos Solitons Fractals 40, 183–189 (2009)
Al-Mdallal, Q.M.: On the numerical solution of fractional Sturm–Liouville problems. Int. J. Comput. Math. 87, 2837–2845 (2010)
Blaszczyk, T., Ciesielski, M.: Numerical solution of fractional Sturm–Liouville equation in integral form. Fract. Calculus Appl. Anal. 17(2), 307–320 (2014)
Bochner, S., Chandrasekharan, K.: Fourier Transforms. Princeton University Press, Princeton (1949)
Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195(4–5), 127–293 (1990)
Erdélyi, A., et al.: Higher Transcendental Functions, vol. 3. McGraw-Hill Book Company Inc., New York (1953)
Gorenflo, R., Mainardi, F.: Fractional oscillations and Mittag–Leffler functions. In Proceeding of the International workshop on the Recent Advances in Applied Mathematics (RAAM 96). Kuwait University, Kuwait City (1996)
Hanneken, J.W., Narahari Achar, B.N., Vaught, D.M.: An alpha-beta phase diagram representation of the zeros and properties of the Mittag–Leffler function. Adv. Math. Phys. 2013, Article ID 421685
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Klimek, M., Agrawal, O.P.: Fractional Sturm–Liouville problem. Comput. Math. Appl. 66(5), 795–812 (2013)
Klimek, M., Agrawal, O.P.: On a regular fractional Sturm–Liouville problem with derivatives of order \((0,1)\). IEEE, pp. 284–289 (2012). https://doi.org/10.1109/CarpathianCC.2012.6228655
Klimek, M., Odzijewicz, T., Malinowska, A.B.: Variational methods for the fractional Sturm–Liouville problem. J. Math. Anal. Appl. 416, 402–426 (2014)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, p. 368. World Scientific, Singapore (2010)
Metzler, R., Nonnenmacher, T.F.: Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations and physical motivations. Chem. Phys. 284(1–2), 1–77 (2000)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 67–90 (2002)
Mittag-Leffler, G.M.: Sur la nouvelle function \(E_{\alpha }\). C.R. Acad. Sci. Paris 137, 554–558 (1903)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, New York (1998)
Pooseh, S., Rodrigues, H.S., Torres, D.F.M.: Fractional derivatives in Dengue epidemics. AIP Conf. Proc. 1389, 739–742 (2011)
Rossikhin, Y.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63(1), 010801 (2009)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Philadelphia, xxvii, 1006 pp. (1993)
Shlesinger, M.F., West, B.J., Klafter, J.: Levy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987)
Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dehghan, M., Mingarelli, A.B. Fractional Sturm–Liouville eigenvalue problems, I. RACSAM 114, 46 (2020). https://doi.org/10.1007/s13398-019-00756-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-019-00756-8
Keywords
- Fractional
- Sturm–Liouville
- Caputo
- Laplace transform
- Mittag–Leffler functions
- Riemann–Liouville
- Eigenvalues