Variational approximation for fractional Sturm–Liouville problem
-
Prashant K. Pandey
and
Om P. Agrawal
Abstract
In this paper, we consider a regular Fractional Sturm–Liouville Problem (FSLP) of order μ (0 < μ < 1). We approximate the eigenvalues and eigenfunctions of the problem using a fractional variational approach. Recently, Klimek et al. [16] presented the variational approach for FSLPs defined in terms of Caputo derivatives and obtained eigenvalues, eigenfunctions for a special range of fractional order 1/2 < μ < 1. Here, we extend the variational approach for the FSLPs and approximate the eigenvalues and eigenfunctions of the FSLP for fractional-order μ (0 < μ < 1). We also prove that the FSLP has countably infinite eigenvalues and corresponding eigenfunctions.
Acknowledgments
The authors sincerely thank the Editor and Reviewers for their valuable comments to improve the manuscript. The second author acknowledges the financial support provided under the Core Research Grant scheme (CRG/2018/002654) of SERB, Govt. of India.
References
[1] S. Abbasbandy, A. Shirzadi, Homotopy analysis method for multiple solutions of the fractional Sturm–Liouville problems. Numer. Algorithms54, No 4 (2010), 521–532; 10.1007/s11075-009-9351-7.Search in Google Scholar
[2] O.P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, No 1 (2002), 368–379; 10.1016/S0022-247X(02)00180-4.Search in Google Scholar
[3] O.P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A40, No 24 (2007), 6287; 10.1088/1751-8113/40/24/003.Search in Google Scholar
[4] O.P. Agrawal, Generalized multiparameters fractional variational calculus. Int. J. Differ. Equ. 2012 (2012); 10.1155/2012/521750.Search in Google Scholar
[5] Q.M. Al-Mdallal, An efficient method for solving fractional Sturm–Liouville problems. Chaos Solitons Fractals40, No 1 (2009), 183–189; 10.1016/j.chaos.2007.07.041.Search in Google Scholar
[6] Q.M. Al-Mdallal, On the numerical solution of fractional Sturm–Liouville problems. Int. J. Comput. Math. 87, No 12 (2010), 2837–2845; 10.1080/00207160802562549.Search in Google Scholar
[7] R. Almeida, D.F.M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16, No 3 (2011), 1490–1500; 10.1016/j.cnsns.2010.07.016.Search in Google Scholar
[8] W.O. Amrein, A.M. Hinz, D.B. Pearson, Sturm–Liouville Theory: Past and Present. Birkhäuser, Basel (2005).10.1007/3-7643-7359-8Search in Google Scholar
[9] W. Deng, Z. Zhang, Variational formulation and efficient implementation for solving the tempered fractional problems. Numer. Methods Partial Differential Equations34, No 4 (2018), 1224–1257; 10.1002/num.22254.Search in Google Scholar
[10] I.M. Gelfand, S.V. Fomin, Calculus of Variations. Dover Publications Inc., New York (2000).Search in Google Scholar
[11] M.A. Hajji, Q.M. Al-Mdallal, F.M. Allan, An efficient algorithm for solving higher-order fractional Sturm–Liouville eigenvalue problems. J. Comput. Phys. 272, (2014), 550–558; 10.1016/j.jcp.2014.04.048.Search in Google Scholar
[12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar
[13] M. Klimek, Fractional sequential mechanics-models with symmetric fractional derivative Czech. J. Phys. 51, No 12 (2001), 1348–1354; 10.1023/A:1013378221617.Search in Google Scholar
[14] M. Klimek, O.P. Agrawal, On a regular fractional Sturm—Liouville problem with derivatives of order in (0, 1). In: Proc. of the 13th International Carpathian Control Conf., Vysoke Tatry (2012), 284–289; 10.1109/CarpathianCC.2012.6228655.Search in Google Scholar
[15] M. Klimek, O.P. Agrawal, Fractional Sturm–Liouville problem. Comput. Math. Appl. 66 (2013), 795–812; 10.1016/j.camwa.2012.12.011.Search in Google Scholar
[16] M. Klimek, T. Odzijewicz, A.B. Malinowska, Variational methods for the fractional Sturm–Liouville problem. J. Math. Anal. Appl. 416, No 1 (2014), 402–426; 10.1016/j.jmaa.2014.02.009.Search in Google Scholar
[17] M. Klimek, A.B. Malinowska, T. Odzijewicz, Applications of the fractional Sturm–Liouville problem to the space-time fractional diffusion in a finite domain. Fract. Calc. Appl. Anal. 19, No 2 (2016), 516–550; 10.1515/fca-2016-0027; https://www.degruyter.com/view/journals/fca/19/2/fca.19.issue-2.xml.Search in Google Scholar
[18] M. Klimek, M. Ciesielski, T. Blaszczyk, Exact and numerical solutions of the fractional Sturm–Liouville problem. Fract. Calc. Appl. Anal. 21, No 1 (2018), 45–71; 10.1515/fca-2018-0004; https://www.degruyter.com/view/journals/fca/21/1/fca.21.issue-1.xml.Search in Google Scholar
[19] W. McLean, W.C.H. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000).Search in Google Scholar
[20] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York (1993).Search in Google Scholar
[21] T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fractional calculus of variations in terms of a generalized fractional integral with applications to physics. Abstr. Appl. Anal. 2012 (2012); 10.1155/2012/871912.Search in Google Scholar
[22] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar
[23] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E53, No 2 (1996), 1890–1899; 10.1103/PhysRevE.53.1890.Search in Google Scholar PubMed
[24] M. Rivero, J.J. Trujillo, M.P. Velasco, A fractional approach to the Sturm–Liouville problem. Cent. Eur. J. Phys. 11, No 10 (2013), 1246–1254; 10.2478/s11534-013-0216-2.Search in Google Scholar
[25] Y. Tien, Z. Du, W. Ge, Existence results for discrete Sturm–Liouville problem via variational methods. J. Difference Equ. Appl. 13, No 6 (2007), 467–478; 10.1080/10236190601086451.Search in Google Scholar
[26] B.V. Brunt, The Calculus of Variations. Springer, New York (2004).10.1007/b97436Search in Google Scholar
[27] M. Zayernouri, G.E. Karniadakis, Fractional Sturm–Liouville eigenproblems: Theory and numerical approximation. J. Comput. Phys. 252 (2013), 495–517; 10.1016/j.jcp.2013.06.031.Search in Google Scholar
[28] M. Zayernouri, M. Ainsworth, G.E. Karniadakis, Tempered fractional Sturm–Liouville eigenproblems. SIAM J. Sci. Computing37, No 4 (2015), A1777–A1800; 10.1137/140985536.Search in Google Scholar
[29] A. Zettl, Sturm–Liouville Theory. Math. Surveys Monogr. 121, AMS, Providence, RI (2010).10.1090/surv/121Search in Google Scholar
© 2020 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–3–2020)
- Survey Paper
- Why fractional derivatives with nonsingular kernels should not be used
- Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease
- Generalized fractional Poisson process and related stochastic dynamics
- Research Paper
- Determination of the fractional order in semilinear subdiffusion equations
- Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
- Solution of linear fractional order systems with variable coefficients
- “Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
- The green function for a class of Caputo fractional differential equations with a convection term
- Inverse problem for a multi-term fractional differential equation
- Maximum principles for a class of generalized time-fractional diffusion equations
- Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method
- Variational approximation for fractional Sturm–Liouville problem
- The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field
- Construction of fixed point operators for nonlinear difference equations of non integer order with impulses
- An averaging principle for stochastic differential equations of fractional order 0 < α < 1
- Weak solvability of the variable-order subdiffusion equation
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–3–2020)
- Survey Paper
- Why fractional derivatives with nonsingular kernels should not be used
- Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease
- Generalized fractional Poisson process and related stochastic dynamics
- Research Paper
- Determination of the fractional order in semilinear subdiffusion equations
- Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
- Solution of linear fractional order systems with variable coefficients
- “Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
- The green function for a class of Caputo fractional differential equations with a convection term
- Inverse problem for a multi-term fractional differential equation
- Maximum principles for a class of generalized time-fractional diffusion equations
- Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method
- Variational approximation for fractional Sturm–Liouville problem
- The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field
- Construction of fixed point operators for nonlinear difference equations of non integer order with impulses
- An averaging principle for stochastic differential equations of fractional order 0 < α < 1
- Weak solvability of the variable-order subdiffusion equation
