Regularity of the solution to 1-D fractional order diffusion equations
HTML articles powered by AMS MathViewer
- by V. J. Ervin, N. Heuer and J. P. Roop PDF
- Math. Comp. 87 (2018), 2273-2294 Request permission
Abstract:
In this article we investigate the solution of the steady-state fractional diffusion equation on a bounded domain in $\mathbb {R}^{1}$. The diffusion operator investigated, motivated by physical considerations, is neither the Riemann-Liouville nor the Caputo fractional diffusion operator. We determine a closed form expression for the kernel of the fractional diffusion operator which, in most cases, determines the regularity of the solution. Next we establish that the Jacobi polynomials are pseudo eigenfunctions for the fractional diffusion operator. A spectral type approximation method for the solution of the steady-state fractional diffusion equation is then proposed and studied.References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- Dumitru Baleanu, Kai Diethelm, Enrico Scalas, and Juan J. Trujillo, Fractional calculus, Series on Complexity, Nonlinearity and Chaos, vol. 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. Models and numerical methods. MR 2894576, DOI 10.1142/9789814355216
- 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.
- Huanzhen Chen and Hong Wang, Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation, J. Comput. Appl. Math. 296 (2016), 480–498. MR 3430153, DOI 10.1016/j.cam.2015.09.022
- Sheng Chen, Jie Shen, and Li-Lian Wang, Generalized Jacobi functions and their applications to fractional differential equations, Math. Comp. 85 (2016), no. 300, 1603–1638. MR 3471102, DOI 10.1090/mcom3035
- Mingrong Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys. 228 (2009), no. 20, 7792–7804. MR 2561843, DOI 10.1016/j.jcp.2009.07.021
- Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. An application-oriented exposition using differential operators of Caputo type. MR 2680847, DOI 10.1007/978-3-642-14574-2
- Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev. 54 (2012), no. 4, 667–696. MR 3023366, DOI 10.1137/110833294
- V.J. Ervin, N. Heuer, and J.P. Roop. Regularity of the solution to 1-D fractional order diffusion equations. Preprint: \verb+http://arxiv.org/abs/1608.00128+, 2016.
- 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
- Jan S. Hesthaven, Sigal Gottlieb, and David Gottlieb, Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, Cambridge University Press, Cambridge, 2007. MR 2333926, DOI 10.1017/CBO9780511618352
- Bangti Jin, Raytcho Lazarov, Xiliang Lu, and Zhi Zhou, A simple finite element method for boundary value problems with a Riemann-Liouville derivative, J. Comput. Appl. Math. 293 (2016), 94–111. MR 3394205, DOI 10.1016/j.cam.2015.02.058
- Bangti Jin, Raytcho Lazarov, Joseph Pasciak, and William Rundell, Variational formulation of problems involving fractional order differential operators, Math. Comp. 84 (2015), no. 296, 2665–2700. MR 3378843, DOI 10.1090/mcom/2960
- 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
- N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
- Changpin Li, Fanhai Zeng, and Fawang Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal. 15 (2012), no. 3, 383–406. MR 2944106, DOI 10.2478/s13540-012-0028-x
- F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods (BAIL 2002), 2004, pp. 209–219. MR 2057973, DOI 10.1016/j.cam.2003.09.028
- Q. Liu, F. Liu, I. Turner, and V. Anh, Finite element approximation for a modified anomalous subdiffusion equation, Appl. Math. Model. 35 (2011), no. 8, 4103–4116. MR 2793693, DOI 10.1016/j.apm.2011.02.036
- F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 291–348. MR 1611587, DOI 10.1007/978-3-7091-2664-6_{7}
- Zhiping Mao, Sheng Chen, and Jie Shen, Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations, Appl. Numer. Math. 106 (2016), 165–181. MR 3499964, DOI 10.1016/j.apnum.2016.04.002
- Mark M. Meerschaert and Charles Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65–77. MR 2091131, DOI 10.1016/j.cam.2004.01.033
- 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
- 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
- M. F. Shlesinger, B. J. West, and J. Klafter, Lévy dynamics of enhanced diffusion: application to turbulence, Phys. Rev. Lett. 58 (1987), no. 11, 1100–1103. MR 884850, DOI 10.1103/PhysRevLett.58.1100
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- Charles Tadjeran and Mark M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys. 220 (2007), no. 2, 813–823. MR 2284325, DOI 10.1016/j.jcp.2006.05.030
- 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
- Hong Wang and Xuhao Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys. 281 (2015), 67–81. MR 3281961, DOI 10.1016/j.jcp.2014.10.018
- Qinwu Xu and Jan S. Hesthaven, Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM J. Numer. Anal. 52 (2014), no. 1, 405–423. MR 3164558, DOI 10.1137/130918174
- G. M. Zaslavsky, D. Stevens, and H. Weitzner, Self-similar transport in incomplete chaos, Phys. Rev. E (3) 48 (1993), no. 3, 1683–1694. MR 1377915, DOI 10.1103/PhysRevE.48.1683
- Mohsen Zayernouri, Mark Ainsworth, and George Em Karniadakis, A unified Petrov-Galerkin spectral method for fractional PDEs, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1545–1569. MR 3283821, DOI 10.1016/j.cma.2014.10.051
Additional Information
- V. J. Ervin
- Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
- MR Author ID: 64070
- Email: vjervin@clemson.edu
- N. Heuer
- Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile
- MR Author ID: 314970
- Email: nheuer@mat.puc.cl
- J. P. Roop
- Affiliation: Department of Mathematics, North Carolina A & T State University, Greensboro, North Carolina 27411
- Email: jproop@ncat.edu
- Received by editor(s): August 14, 2016
- Received by editor(s) in revised form: January 7, 2017, March 15, 2017, and May 4, 2017
- Published electronically: January 26, 2018
- Additional Notes: The first author was partially support by CONICYT through FONDECYT project 1150056
The second author was partially support by CONICYT through FONDECYT projects 1150056, and Anillo ACT1118 (ANANUM) - © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 2273-2294
- MSC (2010): Primary 65N30, 35B65, 41A10, 33C45
- DOI: https://doi.org/10.1090/mcom/3295
- MathSciNet review: 3802435