## Variational formulation of problems involving fractional order differential operators

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- by Bangti Jin, Raytcho Lazarov, Joseph Pasciak and William Rundell PDF
- Math. Comp.
**84**(2015), 2665-2700 Request permission

## Abstract:

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order $\alpha \in (1,2)$ on the unit interval $(0,1)$. These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space $H_0^{\alpha /2}(0,1)$ but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem.## References

- Robert A. Adams and John J. F. Fournier,
*Sobolev spaces*, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR**2424078** - Qasem M. Al-Mdallal,
*An efficient method for solving fractional Sturm-Liouville problems*, Chaos Solitons Fractals**40**(2009), no. 1, 183–189. MR**2517924**, DOI 10.1016/j.chaos.2007.07.041 - I. Babuška and J. Osborn,
*Eigenvalue problems*, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR**1115240** - D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert,
*The fractional-order governing equation of Lévy motion*, Water Resour. Res.**36**(2000), no. 6, 1413–1424. - A. S. Chaves,
*A fractional diffusion equation to describe Lévy flights*, Phys. Lett. A**239**(1998), no. 1-2, 13–16. MR**1616103**, DOI 10.1016/S0375-9601(97)00947-X - Klaus Deimling,
*Nonlinear functional analysis*, Springer-Verlag, Berlin, 1985. MR**787404**, DOI 10.1007/978-3-662-00547-7 - Diego del-Castillo-Negrete,
*Chaotic transport in zonal flows in analogous geophysical and plasma systems*, Phys. Plasmas**7**(2000), no. 5, 1702–1711. 41st Annual Meeting of the Division of Plasma Physics of the American Physical Society (Seattle, WA, 1999). MR**1784245**, DOI 10.1063/1.873988 - D. del-Castillo-Negrete, B. A. Carreras, and V. E. Lynch,
*Front dynamics in reaction-diffusion systems with Levy flights*, Phys. Rev. Lett.**91**(2003), (1):018302, 4 pp. - D. del-Castillo-Negrete, B. A. Carreras, and V. E. Lynch,
*Nondiffusive transport in plasma turbulence: a fractional diffusion approach*, Phys. Rev. Lett.**94**(2005), (6):065003, 4 pp. - Mkhitar M. Djrbashian,
*Harmonic analysis and boundary value problems in the complex domain*, Operator Theory: Advances and Applications, vol. 65, Birkhäuser Verlag, Basel, 1993. Translated from the manuscript by H. M. Jerbashian and A. M. Jerbashian [A. M. Dzhrbashyan]. MR**1249271**, DOI 10.1007/978-3-0348-8549-2 - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - M. M. Džrbašjan,
*A boundary value problem for a Sturm-Liouville type differential operator of fractional order*, Izv. Akad. Nauk Armjan. SSR Ser. Mat.**5**(1970), no. 2, 71–96 (Russian, with Armenian and English summaries). MR**0414982** - Alexandre Ern and Jean-Luc Guermond,
*Theory and practice of finite elements*, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR**2050138**, DOI 10.1007/978-1-4757-4355-5 - Vincent J. Ervin and John Paul Roop,
*Variational formulation for the stationary fractional advection dispersion equation*, Numer. Methods Partial Differential Equations**22**(2006), no. 3, 558–576. MR**2212226**, DOI 10.1002/num.20112 - Vincent J. Ervin and John Paul Roop,
*Variational solution of fractional advection dispersion equations on bounded domains in $\Bbb R^d$*, Numer. Methods Partial Differential Equations**23**(2007), no. 2, 256–281. MR**2289452**, DOI 10.1002/num.20169 - Rudolf Gorenflo, Joulia Loutchko, and Yuri Luchko,
*Computation of the Mittag-Leffler function $E_{\alpha ,\beta }(z)$ and its derivative*, Fract. Calc. Appl. Anal.**5**(2002), no. 4, 491–518. Dedicated to the 60th anniversary of Prof. Francesco Mainardi. MR**1967847** - R. Gorenflo, J. Loutchko, and Y. Luchko,
*Correction: “Computation of the Mittag-Leffler function $E_{\alpha ,\beta }(z)$ and its derivative” [Fract. Calc. Appl. Anal. 5 (2002), no. 4, 491–518; MR1967847 (2004d:33020a)]*, Fract. Calc. Appl. Anal.**6**(2003), no. 1, 111–112. MR**1992470** - P. Grisvard,
*Elliptic Problems in Nonsmooth Domains*. Pitman, Boston, MA, 1985. - B. Jin, R. Lazarov, J. Pasciak, and W. Rundell,
*A finite element method for the fractional Sturm-Liouville problem*, preprint, available as arXiv:1307.5114, 2013. - Bangti Jin and William Rundell,
*An inverse Sturm-Liouville problem with a fractional derivative*, J. Comput. Phys.**231**(2012), no. 14, 4954–4966. MR**2927980**, DOI 10.1016/j.jcp.2012.04.005 - Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo,
*Theory and applications of fractional differential equations*, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR**2218073** - J.-L. Lions and E. Magenes,
*Problèmes aux limites non homogènes et applications. Vol. 1*, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR**0247243** - Ralf Metzler and Joseph Klafter,
*The random walk’s guide to anomalous diffusion: a fractional dynamics approach*, Phys. Rep.**339**(2000), no. 1, 77. MR**1809268**, DOI 10.1016/S0370-1573(00)00070-3 - A. M. Nahušev,
*The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms*, Dokl. Akad. Nauk SSSR**234**(1977), no. 2, 308–311 (Russian). MR**0454145** - Igor Podlubny,
*Fractional differential equations*, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR**1658022** - Igor Podlubny, Aleksei Chechkin, Tomas Skovranek, YangQuan Chen, and Blas M. Vinagre Jara,
*Matrix approach to discrete fractional calculus. II. Partial fractional differential equations*, J. Comput. Phys.**228**(2009), no. 8, 3137–3153. MR**2509311**, DOI 10.1016/j.jcp.2009.01.014 - A. Yu. Popov and A. M. Sedletskiĭ,
*Distribution of roots of Mittag-Leffler functions*, Sovrem. Mat. Fundam. Napravl.**40**(2011), 3–171 (Russian); English transl., J. Math. Sci. (N.Y.)**190**(2013), no. 2, 209–409. MR**2883249**, DOI 10.1007/s10958-013-1255-3 - Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev,
*Fractional integrals and derivatives*, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR**1347689** - Alfred H. Schatz,
*An observation concerning Ritz-Galerkin methods with indefinite bilinear forms*, Math. Comp.**28**(1974), 959–962. MR**373326**, DOI 10.1090/S0025-5718-1974-0373326-0 - Hansjörg Seybold and Rudolf Hilfer,
*Numerical algorithm for calculating the generalized Mittag-Leffler function*, SIAM J. Numer. Anal.**47**(2008/09), no. 1, 69–88. MR**2452852**, DOI 10.1137/070700280 - C. Shen and M. S. Phanikumar,
*An efficient space-fractional dispersion approximation for stream solute transport modeling*, Adv. Water Res.**32**(2009), no. 10, 1482–1494. - M. Kh. Shkhanukov,
*On the convergence of difference schemes for differential equations with a fractional derivative*, Dokl. Akad. Nauk**348**(1996), no. 6, 746–748 (Russian). MR**1440740** - T. H. Solomon, E. R. Weeks, and H. L. Swinney,
*Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow*, Phys. Rev. Lett.**71**(1993), no. 24, 3975–3978. - Ercília Sousa,
*Finite difference approximations for a fractional advection diffusion problem*, J. Comput. Phys.**228**(2009), no. 11, 4038–4054. MR**2524510**, DOI 10.1016/j.jcp.2009.02.011 - Charles Tadjeran, Mark M. Meerschaert, and Hans-Peter Scheffler,
*A second-order accurate numerical approximation for the fractional diffusion equation*, J. Comput. Phys.**213**(2006), no. 1, 205–213. MR**2203439**, DOI 10.1016/j.jcp.2005.08.008 - Hong Wang and Treena S. Basu,
*A fast finite difference method for two-dimensional space-fractional diffusion equations*, SIAM J. Sci. Comput.**34**(2012), no. 5, A2444–A2458. MR**3023711**, DOI 10.1137/12086491X - Hong Wang and Danping Yang,
*Wellposedness of variable-coefficient conservative fractional elliptic differential equations*, SIAM J. Numer. Anal.**51**(2013), no. 2, 1088–1107. MR**3036999**, DOI 10.1137/120892295 - X. Zhang, M. Lv, J. W. Crawford, and I. M. Young,
*The impact of boundary on the fractional advection-dispersion equation for solute transport in soil: Defining the fractional dispersive flux with the caputo derivatives*, Adv. Water Res.**30**(2007), no. 5, 1205–1217.

## Additional Information

**Bangti Jin**- Affiliation: Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, California 92521
- Address at time of publication: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom
- MR Author ID: 741824
- Email: bangti.jin@gmail.com
**Raytcho Lazarov**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 111240
- Email: lazarov@math.tamu.edu
**Joseph Pasciak**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: pasciak@math.tamu.edu
**William Rundell**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: rundell@math.tamu.edu
- Received by editor(s): August 20, 2013
- Received by editor(s) in revised form: February 12, 2014, and April 6, 2014
- Published electronically: April 30, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 2665-2700 - MSC (2010): Primary 65L60, 65N12, 65N30
- DOI: https://doi.org/10.1090/mcom/2960
- MathSciNet review: 3378843