206
Views
13
CrossRef citations to date
0
Altmetric
Original Article

Fractional Sturm–Liouville problems for Weber fractional derivatives

&
Pages 217-237 | Received 14 Mar 2017, Accepted 21 Nov 2017, Published online: 29 Jan 2018

References

  • S. Abbasbandy and A. Shirzadi, Homotopy analysis method for multiple solutions of the fractional Sturm–Liouville problems, Numer. Algorithms 54(4) (2010), pp. 521–532. doi: 10.1007/s11075-009-9351-7
  • M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1972.
  • Q.A. Al-Mdallal, An efficient method for solving fractional Sturm–Liouville problems, Chaos Solitons Fractals 40(15) (2009), pp. 183–189. doi: 10.1016/j.chaos.2007.07.041
  • A. Ansari, On finite fractional Sturm–Liouville transforms, Integral Transforms Spec. Funct. 26(1) (2015), pp. 51–64. doi: 10.1080/10652469.2014.966102
  • A. Ansari, Some inverse fractional Legendre transforms of gamma function form, Kodai Math. J. 38(3) (2015), pp. 658–671. doi: 10.2996/kmj/1446210600
  • H. Askari and A. Ansari, Fractional calculus of variations with a generalized fractional derivative, Fract. Differ. Calc. 6(1) (2016), pp. 57–72. doi: 10.7153/fdc-06-04
  • E. Bas and F. Metin, Fractional singular Sturm–Liouville operator for coulomb potential, Adv. Difference Equ. doi: 10.1186/1687-1847-2013-300.
  • YuA. Brychkov, Hand Book of Special Functions, Computing Center of the Russian Academy of Sciences, Moscow, 2008.
  • L. Changpin and Z. Fanhai, Numerical Methods for Fractional Calculus, Chapman & Hall/CRC, Boca Raton, 2015.
  • M. Ciesielski, M. Klimek, and T. Blaszczyk, The fractional Sturm–Liouville problem-numerical approximation and application in fractional diffusion, J. Comput. Appl. Math. (2016). doi: 10.1016/j.cam.2016.12.014.
  • M. D'Ovidio and F. Polito, Fractional diffusion-telegraph equations and their associated stochastic solutions, preprint (2013). Available at arXiv:1307.1696v3 [math.PR], 22 pages.
  • L. Debnath and D. Bhatta, Integral Transforms and their Applications, 2nd ed., Chapman & Hall/CRC, Taylor & Francis Group, New York, 2007.
  • A. Elbert and M.E. Muldoon, Inequalities and monotonicity properties for zeros of Hermite functions, Proc. Roy. Soc. Edinburgh 129(1) (1999), pp. 57–75. doi: 10.1017/S0308210500027463
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York–Toronto–London, 1953.
  • V.S. Erturk, Computing eigenelements of Sturm–Liouville problems of fractional order via fractional differential transform method, Math. Comput. Appl. 16(3) (2011), pp. 712–720.
  • S. Eshaghi and A. Ansari, Finite fractional Sturm–Liouville transforms for generalized fractional derivatives, Iran. J. Sci. Technol. 41(4) (2017), pp. 931–937. doi: 10.1007/s40995-017-0311-0
  • S. Eshaghi and A. Ansari, Lyapunov inequality for fractional differential equations with Prabhakar derivative, Math. Inequal. Appl. 19(1) (2016), pp. 349–358.
  • R. Garra, R. Gorenflo, F. Polito, and Z. Tomovski, Hilfer–Prabhakar derivatives and some applications, Appl. Math. Comput. 242 (2014), pp. 576–589.
  • H.A. Helfgot, Major arcs for Goldbach's problem, arXiv:1305.2897v4 [math.NT], 2014.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000.
  • R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons and I. M. Sokolov eds., Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2008, pp. 17–73.
  • R. Hilfer, Y. Luchko, and Z. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann–Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318.
  • A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional, Elsevier, New York, 2006.
  • M. Klimek, On Solutions of a Linear Fractional Differential Equations of a Variational type, The Publishing Office of the Czestochowa University of Technology, Czestochowa, 2009.
  • M. Klimek and O.P. Agrawal, On a regular fractional Sturm–Liouville problem with derivatives of order in (0,1), in Proceedings of the 13th International Carpathian Control Conference, Vysoke Tatry (Podbanske), Slovakia, 2831 May 2012.
  • M. Klimek and O.P. Agrawal, Fractional Sturm–Liouville problem, Comput. Math. Appl. 66 (2013), pp. 795–812. doi: 10.1016/j.camwa.2012.12.011
  • M. Klimek, A.B. Malinowska, and T. Odzijewicz, Applications of the fractional Sturm–Liouville problem to the space-time fractional diffusion in finite domain, Fract. Calc. Appl. Anal. 19(2) (2016), pp. 516–550. doi: 10.1515/fca-2016-0027
  • M. Klimek, T. Odzijewicz, and A.B. Malinowska, Variational methods for the fractional Sturm–Liouville problem, J. Math. Anal. Appl. 416 (2014), pp. 402–426. doi: 10.1016/j.jmaa.2014.02.009
  • Y. Luchko and M. Yamamoto, General time-fractional diffusion equation: Some uniqueness and existence results for the initial-boundary-value problems, Fract. Calc. Appl. Anal. 19(2) (2016), pp. 676–695.
  • K.B. Oldham and J. Spanier, An Atlas of Functions, Academic Press, New York, 1987.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
  • S.Y. Reutskiy, A new numerical method for solving high-order fractional eigenvalue problems, J. Comput. Appl. Math. (2016). doi:10.1016/j.cam.2016.12.027.
  • M. Rivero, J.J. Trujillo, and M.P. Velasco, A fractional approach to the Sturm–Liouville problem, Cent. Eur. J. Phys. 11(10) (2013), pp. 1246–1254.
  • I.N. Sneddon, The Use of Integral Transforms, Mac Graw-Hill, New York, 1979.
  • M.N. Temme, Numerical and asymptotic aspects of parabolic cylinder functions, J. Comput. Appl. Math. 121(1) (2000), pp. 221–246. doi: 10.1016/S0377-0427(00)00347-2
  • M.N. Temme, Parabolic cylinder functions, in Nist Handbook of Mathematical Functions, F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, eds., Cambridge University Press, New York, 2010, pp. 303–319.
  • M. Zayernouri and G.E. Karniadakis, Fractional Sturm–Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys. 252(1) (2013), pp. 495–517. doi: 10.1016/j.jcp.2013.06.031
  • M. Zayernouri and G.E. Karniadakis, Discontinuous spectral element methods for time and space-fractional advection equations, SIAM. J. Sci. Comput. 36(4) (2014), pp. 684–707. doi: 10.1137/130940967
  • M. Zayernouri and G.E. Karniadakis, Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput Phys 257 (2014), pp. 460–480. doi: 10.1016/j.jcp.2013.09.039
  • M. Zayernouri and G.E. Karniadakis, Fractional spectral collocation method, SIAM. J. Sci. Comput. 36(1) (2014), pp. 40–62. doi: 10.1137/130933216
  • A. Zettl, Sturm–Liouville Theory, Mathematical Surveys and Monographs, Vol. 121, American Mathematical Society, Providence, 2005.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.