Research ArticleOpen AccessGoogle Scholar indexed
Spectral Analysis for Fractional Hydrogen Atom Equation
Department of Mathematics, Faculty of Science, Elazig, Turkey
Department of Mathematics, Faculty of Science, Elazig, Turkey
- 1 Department of Mathematics, Faculty of Science, Elazig, Turkey
- 2 Department of Mathematics, Faculty of Science, Elazig, Turkey
Advances in Pure Mathematics·Volume 05 (2015)·Pages 767–773·Published 9 November 2015·DOI10.4236/apm.2015.513070
Copy link · social · email
Abstract
In this paper, spectral analysis of fractional Sturm Liouville problem defined on (0, 1], having the singularity of type at zero and researched the fundamental properties of the eigenfunctions and eigenvalues for the operator. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively.
KeywordsSturm-LiouvilleFractionalHydrogen AtomSingularSpectral
- Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York.
- Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
- Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
- Klimek, M. (2009) On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publishing Office of Czestochowa, University of Technology, Czestochowa.
- Al-Mdallal, Q.M. (2009) An Efficient Method for Solving Fractional Sturm-Liouville Problems. Chaos Solitons and Fractals, 40, 183-189. http://dx.doi.org/10.1016/j.chaos.2007.07.041
- Erturk, V.S. (2011) Computing Eigenelements of Sturm-Liouville Problems of Fractional Order via Fractional Differential Transform Method. Mathematical and Computational Applications, 16, 712-720.
- Klimek, M. and Argawal, O.P. (2012) On a Regular Fractional Sturm-Liouville Problem with Derivatives of Order in (0,1). Proceedings of the 13th International Carpathian Control Conference, Vysoke Tatry (Podbanske), Slovakia, 28-31 May 2012. http://dx.doi.org/10.1109/carpathiancc.2012.6228655
- Baleanu, D., Octavian, M.G. and Agarwal, P.R. (2011) Asymptotic Integration of (1+Alpha)-Order Fractional Differential Equations. Computers & Mathematics with Applications, 62-63, 1492-1500. http://dx.doi.org/10.1016/j.camwa.2011.03.021
- Yilmazer, R. (2010) N-Fractional Calculus Operator N^{μ} Method to a Modified Hydrogen Atom Equation. Mathematical Communications, 15, 489-501.
- Bas, E. (2013) Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator. Journal of Function Spaces and Applications, Article ID: 915830.
- Samko, S.G., Kilbass, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Philadelphia.
- Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
- Carpinteri, A. and Mainardi, F., Eds. (1998) Fractals and Fractional Calculus in Continum Mechanics. Springer-Verlag, Telos.
- Johnson, R.S. (2006) An Introduction to Sturm-Liouville Theory. University of Newcastle, Upon Tyne Publishing, Newcastle upon Tyne.