Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. (English) Zbl 1398.65314
Summary: Tempered fractional diffusion equations (TFDEs) involving tempered fractional derivatives on the whole space were first introduced in [F. Sabzikar et al., J. Comput. Phys. 293, 14–28 (2015; Zbl 1349.26017)], but only the finite-difference approximation to a truncated problem on a finite interval was proposed therein. In this paper, we rigorously show the well-posedness of the models in [loc. cit.], and tackle them directly in infinite domains by using generalized Laguerre functions (GLFs) as basis functions. We define a family of GLFs and derive some useful formulas of tempered fractional integrals/derivatives. Moreover, we establish the related GLF-approximation results. In addition, we provide ample numerical evidences to demonstrate the efficiency and “tempered” effect of the underlying solutions of TFDEs.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 41A05 | Interpolation in approximation theory |
Keywords:
tempered fractional differential equations; singularity; Laguerre functions; generalized Laguerre functions; weighted Sobolev spaces; approximation results; spectral accuracyCitations:
Zbl 1349.26017References:
| [1] | Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972) · Zbl 0543.33001 |
| [2] | Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999) · Zbl 0920.33001 |
| [3] | Baeumer, B., Benson, D.A., Meerschaert, M.M., Wheatcraft, S.W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37(6), 1543-1550 (2001) · doi:10.1029/2000WR900409 |
| [4] | Baeumer, B., Kovács, M., Meerschaert, M.M.: Fractional reproduction-dispersal equations and heavy tail dispersal kernels. Bull. Math. Biol. 69(7), 2281-2297 (2007) · Zbl 1296.92195 · doi:10.1007/s11538-007-9220-2 |
| [5] | Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75(2), 305-333 (2002) · doi:10.1086/338705 |
| [6] | Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603-1638 (2016) · Zbl 1335.65066 · doi:10.1090/mcom3035 |
| [7] | Cushman, J.H., Ginn, T.R.: Fractional advection-dispersion equation: a classical mass balance with convolution-fickian flux. Water Resour. Res. 36(12), 3763-3766 (2000) · doi:10.1029/2000WR900261 |
| [8] | Deng, Z.Q., Bengtsson, L., Singh, V.P.: Parameter estimation for fractional dispersion model for rivers. Environ. Fluid Mech. 6(5), 451-475 (2006) · doi:10.1007/s10652-006-9004-5 |
| [9] | Diethelm, K.: The analysis of fractional differential equations. Lecture Notes in Math, vol. 2004. Springer, Berlin (2010) · Zbl 1323.34012 |
| [10] | Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M.: Fractional calculus and continuous-time finance III: the diffusion limit. Math. Financ. 171-180 (2001) · Zbl 1138.91444 |
| [11] | Huang, C., Song, Q., Zhang, Z.: Spectral collocation method for substantial fractional differential equations. arXiv:1408.5997 [math.NA] (2014) |
| [12] | Jeon, J.H., Monne, H.M.S., Javanainen, M., Metzler, R.: Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett. 109(18), 188103 (2012) · doi:10.1103/PhysRevLett.109.188103 |
| [13] | Li, C., Deng, W.H.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42(3), 543-572 (2016) · Zbl 1347.65136 · doi:10.1007/s10444-015-9434-z |
| [14] | Magdziarz, M., Weron, A., Weron, K.: Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75(1), 016708 (2007) · Zbl 1123.60045 · doi:10.1103/PhysRevE.75.016708 |
| [15] | Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2008). doi:10.1029/2008GL034899 |
| [16] | Meerschaert, M.M., Scalas, E.: Coupled continuous time random walks in finance. J. Phys. A Stat. Mech. Appl. 370(1), 114-118 (2006) · doi:10.1016/j.physa.2006.04.034 |
| [17] | Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 |
| [18] | Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37(31), R161 (2004) · Zbl 1075.82018 · doi:10.1088/0305-4470/37/31/R01 |
| [19] | Piryatinska, A., Saichev, A., Woyczynski, W.: Models of anomalous diffusion: the subdiffusive case. J. Phys. A Stat. Mech. A. 349(3), 375-420 (2005) · doi:10.1016/j.physa.2004.11.003 |
| [20] | Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press Inc., San Diego (1999) · Zbl 0924.34008 |
| [21] | Popov, G.Y.: Concentration of elastic stresses near punches, cuts, thin inclusions and supports. Nauka Mosc. 1, 982 (1982) · Zbl 0543.73017 |
| [22] | Sabzikar, F., Meerschaert, M.M., Chen, J.: Tempered fractional calculus. J. Comput. Phys. 293, 14-28 (2015) · Zbl 1349.26017 · doi:10.1016/j.jcp.2014.04.024 |
| [23] | Samko, S.G., Kilbas, A.A., Maričev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publ., Philadelphia (1993) · Zbl 0818.26003 |
| [24] | Scalas, E.: Five years of continuous-time random walks in econophysics. In: The Complex Networks of Economic Interactions, pp. 3-16. Springer, Berlin (2006) · Zbl 1183.91135 |
| [25] | Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Series in Computational Mathematics, vol. 41. Springer, Berlin (2011) · Zbl 1227.65117 |
| [26] | Stein, E.M., Weiss, G.L.: Introduction to Fourier Analysis on Euclidean Spaces, vol. 1. Princeton University Press, Princeton (1971) · Zbl 0232.42007 |
| [27] | Szegö, G.: Orthogonal Polynomials, 4th edn. AMS Coll. Publ., (1975) · Zbl 0305.42011 |
| [28] | Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: Tempered fractional Sturm-Liouville Eigen-problems. SIAM J. Sci. Comput. 37(4), A1777-A1800 (2015) · Zbl 1323.34012 · doi:10.1137/140985536 |
| [29] | Zhang, C., Guo, B.Y.: Domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations on unbounded domains. J. Sci. Comput. 53(2), 451-480 (2012) · Zbl 1273.65183 · doi:10.1007/s10915-012-9581-z |
| [30] | Zhang, Y., Meerschaert, M.M.: Gaussian setting time for solute transport in fluvial systems. Water Resour. Res. 47, W08601 (2011). doi:10.1029/2010WR010102 |
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