Fractional Sturm-Liouville eigenvalue problems. I. (English) Zbl 1443.34009
The authors introduce and obtaine the general solution of three two-term fractional differential equations of mixed Caputo/Riemann-Liouville type. The 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 endpoints. 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 \(\alpha\to 1^-\), and that the fractional operator converges to an ordinary two term Sturm-Liouville operator as \(\alpha\to 1^-\) with Dirichlet boundary conditions.
Reviewer: Krishnan Balachandran (Coimbatore)
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 33E12 | Mittag-Leffler functions and generalizations |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
Keywords:
fractional; Sturm-Liouville; Caputo; Laplace transform; Mittag-Leffler functions; Riemann-Liouville; eigenvaluesReferences:
| [1] | Al-Mdallal, QM, An efficient method for solving fractional Sturm-Liouville problems, Chaos Solitons Fractals, 40, 183-189 (2009) · Zbl 1197.65097 · doi:10.1016/j.chaos.2007.07.041 |
| [2] | Al-Mdallal, QM, On the numerical solution of fractional Sturm-Liouville problems, Int. J. Comput. Math., 87, 2837-2845 (2010) · Zbl 1202.65100 · doi:10.1080/00207160802562549 |
| [3] | Blaszczyk, T.; Ciesielski, M., Numerical solution of fractional Sturm-Liouville equation in integral form, Fract. Calculus Appl. Anal., 17, 2, 307-320 (2014) · Zbl 1305.34008 |
| [4] | Bochner, S.; Chandrasekharan, K., Fourier Transforms (1949), Princeton: Princeton University Press, Princeton · Zbl 0065.34101 |
| [5] | Bouchaud, JP; Georges, A., Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep., 195, 4-5, 127-293 (1990) · doi:10.1016/0370-1573(90)90099-N |
| [6] | Erdélyi, A., Higher Transcendental Functions (1953), New York: McGraw-Hill Book Company Inc., New York · Zbl 0051.30303 |
| [7] | 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) · Zbl 0916.34011 |
| [8] | 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 · Zbl 1291.30034 |
| [9] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and Application of Fractional Differential Equations (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003 |
| [10] | Klimek, M.; Agrawal, OP, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66, 5, 795-812 (2013) · Zbl 1348.34018 · doi:10.1016/j.camwa.2012.12.011 |
| [11] | Klimek, M., Agrawal, O.P.: On a regular fractional Sturm-Liouville problem with derivatives of order \((0,1)\). IEEE, pp. 284-289 (2012). 10.1109/CarpathianCC.2012.6228655 |
| [12] | Klimek, M.; Odzijewicz, T.; Malinowska, AB, Variational methods for the fractional Sturm-Liouville problem, J. Math. Anal. Appl., 416, 402-426 (2014) · Zbl 1297.65087 · doi:10.1016/j.jmaa.2014.02.009 |
| [13] | Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 368 (2010), Singapore: World Scientific, Singapore · Zbl 1210.26004 |
| [14] | Metzler, R.; Nonnenmacher, TF, Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations and physical motivations, Chem. Phys., 284, 1-2, 1-77 (2000) |
| [15] | Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1, 67-90 (2002) |
| [16] | Mittag-Leffler, GM, Sur la nouvelle function \(E_{\alpha }\), C.R. Acad. Sci. Paris, 137, 554-558 (1903) · JFM 34.0435.01 |
| [17] | Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1998), New York: Academic Press, New York · Zbl 0922.45001 |
| [18] | Pooseh, S.; Rodrigues, HS; Torres, DFM, Fractional derivatives in Dengue epidemics, AIP Conf. Proc., 1389, 739-742 (2011) · doi:10.1063/1.3636838 |
| [19] | Rossikhin, YA; Shitikova, MV, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, Appl. Mech. Rev., 63, 1, 010801 (2009) · doi:10.1115/1.4000563 |
| [20] | 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) · Zbl 0818.26003 |
| [21] | Shlesinger, MF; West, BJ; Klafter, J., Levy dynamics of enhanced diffusion: application to turbulence, Phys. Rev. Lett., 58, 11, 1100-1103 (1987) · doi:10.1103/PhysRevLett.58.1100 |
| [22] | Zayernouri, M.; Karniadakis, GE, Fractional Sturm-Liouville eigen-problems: theory and numerical approximation, J. Comput. Phys., 252, 495-517 (2013) · Zbl 1349.34095 · doi:10.1016/j.jcp.2013.06.031 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
