A fractional approach to the Sturm-Liouville problem

• Authors: Margarita Rivero , Juan Trujillo , M. Velasco
• Languages of publication: EN

Abstracts

EN

The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem, by using different fractional operators that return to the ordinary operator for integer order. For each fractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a particular example that evidences the applicability of the fractional Sturm-Liouville theory.

Keywords

EN

fractional operators; fractional spatial derivatives; Sturm-Liouville theory

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 10
• Pages: 1246-1254

Dates

published

1 - 10 - 2013

online

19 - 12 - 2013

Contributors

author

Margarita Rivero

• Facultad de Matemáticas, Universidad de La Laguna, 38271, La Laguna, Tenerife, Spain

author

Juan Trujillo

• Facultad de Matemáticas, Universidad de La Laguna, 38271, La Laguna, Tenerife, Spain

author

M. Velasco

• Estadística e Investigación Operativa, Centro Universitario de la Defensa Área de Matemáticas, 50090, Zaragoza, Spain

References

• [1] W.E. Boyce, R.C. DiPrima, Elementary differential equations and boundary value problems (John Wiley and Sons, USA, 2005)
• [2] R.V. Churchill, J.W. Brown, Fourier series and boundary value problems (McGraw-Hill, New York, 1993)
• [3] E.A. Coddington, R. Carlson, Linear ordinary differential equations (Siam, Philadelphia, 1997) http://dx.doi.org/10.1137/1.9781611971439[Crossref]
• [4] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods, In: Series on Complexity, Nonlinearity and Chaos (World Scientific, Singapore, 2012)
• [5] R.L. Magin, Fractional Calculus in Bioengineering (Begell House, Connecticut, 2006)
• [6] F. Mainardi, Fractional Calculus and waves in linear viscoelasticity: An introduction to mathematical models (Imperial College Press, London, 2010) http://dx.doi.org/10.1142/9781848163300[Crossref]
• [7] J. Sabatier, O.P. Agrawal, J.A. Tenreiro (Eds.), Advances in fractional calculus: Theoretical developments and applications in physics and engineering (Springer, Netherlands 2007)
• [8] J. Tenreiro, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. 16, 1140 (2011) http://dx.doi.org/10.1016/j.cnsns.2010.05.027[Crossref]
• [9] D. Baleanu, Signal Process. 86, 2632 (2006) http://dx.doi.org/10.1016/j.sigpro.2006.02.008[Crossref]
• [10] D. Baleanu, Commun. Nonlinear Sci. 14, 2520 (2009) http://dx.doi.org/10.1016/j.cnsns.2008.10.002[Crossref]
• [11] H. Delavari, D.M. Senejohnny, D. Baleanu, Cent. Eur. J. Phys. 10, 1095 (2012) http://dx.doi.org/10.2478/s11534-012-0073-4[Crossref]
• [12] D. Baleanu, O.G. Mustafa, R.P. Agarwal, J. Phys. A - Math. Theor. 43, 385209 (2010) http://dx.doi.org/10.1088/1751-8113/43/38/385209[Crossref]
• [13] S. Abbasbandy, A. Shizardi, Numer. Algorithms 54, 521 (2010) http://dx.doi.org/10.1007/s11075-009-9351-7[Crossref]
• [14] Q.M. Al-Mdallal, Chaos Soliton. Fract. 40, 138 (2009) http://dx.doi.org/10.1016/j.chaos.2007.07.022[Crossref]
• [15] M.M. Djrbashian, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 5, 71 (1970)
• [16] M. D’Ovidio, Stoch. Proc. Appl. 122, 3513 (2012) http://dx.doi.org/10.1016/j.spa.2012.06.002[Crossref]
• [17] V.S. Erturk, Math. Comput. Appl. 16, 712 (2011)
• [18] A.K. Golmankhaneh, T. Khatuni, N.A. Porghoveh, D. Baleanu, Cent. Eur. J. Phys. 10, 966 (2012) http://dx.doi.org/10.2478/s11534-012-0038-7[Crossref]
• [19] B. Jin, W. Rundell, J. Comput. Phys. 231, 4954 (2012) http://dx.doi.org/10.1016/j.jcp.2012.04.005[Crossref]
• [20] Y. Liu, T. He, H. Shi, U.P.B. Sci. Bull. 74, 93 (2012)
• [21] A.M. Nahusev, Dokl. Akad. Nauk SSSR, 234, 308 (1977)
• [22] M. Klimek, On solutions of linear fractional differential equations of a variational type (Layout, Czestochowa, 2009)
• [23] M. Klimek, O.P. Agrawal, On a regular fractional Sturm-Liouville problem with derivatives of order in (0,1), in: I. Petrás, I. Podlubny, K. Kostúr, Ján Kacur, A. Mojzisová (Eds.), Proceedings of the 13th International Carpathian Control Conference, May. 28–31, 2012, Vysoke Tatry, Slovakia (IEEE Explore Digital Library, 2012)
• [24] K. Diethelm, The analysis of fractional differential equations (Springer, New York, 2010) http://dx.doi.org/10.1007/978-3-642-14574-2[Crossref]
• [25] A.A. Kilbas, H.M. Srivastava, J.J Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
• [26] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley and Sons, New York, 1993)
• [27] K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974)
• [28] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
• [29] G.M. Zaslavsky, D. Baleanu, J.A. Tenreiro (Eds.), Fractional Differentiation and its Applications (Physica Scripta, 2009) [WoS]
• [30] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Switzerland, 1993)

Identifiers

DOI

10.2478/s11534-013-0216-2