Free Access
Issue
ESAIM: M2AN
Volume 54, Number 1, January-February 2020
Page(s) 59 - 78
DOI https://doi.org/10.1051/m2an/2019052
Published online 14 January 2020
  1. M. Ahmadinia, Z. Safari and S. Fouladi, Analysis of LDG method for time-space fractional convection–diffusion equations. BIT 58 (2018) 533–554. [CrossRef] [Google Scholar]
  2. B. Baeumer and M.M. Meerschaert, Tempered stable lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233 (2010) 2438–2448. [Google Scholar]
  3. B. Baeumer, D.A. Benson, M.M. Meerschaert and S.W. Wheatcraft, Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37 (2001) 1543–1550. [Google Scholar]
  4. B. Baeumer, M. Kovács and M.M. Meerschaert, Fractional reproduction-dispersal equations and heavy tail dispersal kernels. Bull. Math. Biol. 69 (2007) 2281–2297. [Google Scholar]
  5. D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000) 1403–1412. [Google Scholar]
  6. D.A. Benson, R. Schumer, M.M. Meerschaert, S.W. Wheatcraft, Fractional dispersion, lévy motion, and the made tracer tests. Transp. Porous Media 42 (2001) 211–240. [Google Scholar]
  7. P. Carr, H. Geman, D.B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation. J. Bus. 75 (2002) 305–332. [Google Scholar]
  8. P. Carr, H. Geman, D.B. Madan and M. Yor, Stochastic volatility for lévy processes. Math. Finance 13 (2003) 345–382. [CrossRef] [MathSciNet] [Google Scholar]
  9. Á. Cartea and D. del Castillo-Negrete, Fluid limit of the continuous-time random walk with general lévy jump distribution functions. Phys. Rev. E 76 (2007) 041105. [Google Scholar]
  10. P. Castillo, B. Cockburn, D. Schötzau and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection–diffusion problems. Math. Comput. 71 (2002) 455–478. [Google Scholar]
  11. M. Chen and W. Deng, Discretized fractional substantial calculus. ESAIM: M2AN 49 (2015) 373–394. [CrossRef] [EDP Sciences] [Google Scholar]
  12. M. Chen and W. Deng, A second-order accurate numerical method for the space–time tempered fractional diffusion-wave equation. Appl. Math. Lett. 68 (2017) 87–93. [Google Scholar]
  13. M. Chen and W. Deng, High order algorithm for the time-tempered fractional Feynman-Kac equation. J. Sci. Comput. 76 (2018) 867–887. [Google Scholar]
  14. S. Chen, J. Shen and L.-L. Wang, Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. J. Sci. Comput. 74 (2018) 1286–1313. [Google Scholar]
  15. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). [Google Scholar]
  16. B. Cockburn and K. Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer. Math. 130 (2015) 293–314. [Google Scholar]
  17. B. Cockburn, G. Kanschat, I. Perugia and D. Schotzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on cartesian grids. SIAM J. Numer. Anal. 39 (2001) 264–285. [Google Scholar]
  18. J.H. Cushman and T.R. Ginn, Fractional advection-dispersion equation: A classical mass balance with convolution-fickian flux. Water Resour. Res. 36 (2000) 3763–3766. [Google Scholar]
  19. M. Dehghan and M. Abbaszadeh, A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation. Comput. Math. App. 75 (2018) 2903–2914. [Google Scholar]
  20. W. Deng and J.S. Hesthaven, Local discontinuous Galerkin methods for fractional diffusion equations. ESAIM: M2AN 47 (2013) 1845–1864. [CrossRef] [EDP Sciences] [Google Scholar]
  21. W. Deng and J.S. Hesthaven, Local discontinuous Galerkin methods for fractional ordinary differential equations. BIT 55 (2015) 967–985. [CrossRef] [Google Scholar]
  22. Z. Deng, L. Bengtsson and V.P. Singh, Parameter estimation for fractional dispersion model for rivers. Environ. Fluid Mech. 6 (2006) 451–475. [CrossRef] [Google Scholar]
  23. J. Deng, L. Zhao and Y. Wu, Fast predictor-corrector approach for the tempered fractional differential equations. Numer. Algorithms 74 (2017) 717–754. [Google Scholar]
  24. V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22 (2006) 558–576. [Google Scholar]
  25. R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and continuous-time finance III: the diffusion limit Mathematical Finance. Springer (2001) 171–180. [Google Scholar]
  26. X. Guo, Y. Li and H. Wang, A high order finite difference method for tempered fractional diffusion equations with applications to the cgmy model. SIAM J. Sci. Comput. 40 (2018) A3322–A3343. [Google Scholar]
  27. E. Hanert and C. Piret, A chebyshev pseudospectral method to solve the space-time tempered fractional diffusion equation. SIAM J. Sci. Comput. 36 (2014) A1797–A1812. [Google Scholar]
  28. A. Hanyga, Wave propagation in media with singular memory. Math. Comput. Model. 34 (2001) 1399–1421. [Google Scholar]
  29. J.-H. Jeon, H.M.-S. Monne, M. Javanainen and R. Metzler, Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins. Phys. Rev. Lett. 109 (2012) 188103. [CrossRef] [PubMed] [Google Scholar]
  30. A.A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Limited. 204 (2006). [Google Scholar]
  31. C. Li and W. Deng, High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42 (2016) 543–572. [Google Scholar]
  32. R.L. Magin, Fractional Calculus in Bioengineering. Begell House Redding (2006). [Google Scholar]
  33. F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas, Fractional calculus and continuous-time finance ii: the waiting-time distribution. Phys. A: Stat. Mech. App. 287 (2000) 468–481. [CrossRef] [Google Scholar]
  34. O. Marom and E. Momoniat, A comparison of numerical solutions of fractional diffusion models in finance. Nonlinear Anal.: Real World App. 10 (2009) 3435–3442. [CrossRef] [Google Scholar]
  35. W. McLean and K. Mustapha, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51 (2013) 491–515. [Google Scholar]
  36. M.M. Meerschaert and E. Scalas, Coupled continuous time random walks in finance. Phys. A: Stat. Mech. App. 370 (2006) 114–118. [CrossRef] [Google Scholar]
  37. M.M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35 (2008). [CrossRef] [PubMed] [Google Scholar]
  38. R. Metzler and J. Klafter, 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 (2004) R161. [NASA ADS] [CrossRef] [Google Scholar]
  39. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, 1993. [Google Scholar]
  40. K.K. Mustapha, B. Abdallah and K.M. Furati, A discontinuous Petrov-Galerkin method for time-fractinal diffusion equations. SIAM J. Numer. Anal. 52 (2014) 2512–2529. [Google Scholar]
  41. I. Podlubny, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. In Vol. 198 ofMathematics in Science and Engineering (1999). [Google Scholar]
  42. B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia, PA (2008). [CrossRef] [Google Scholar]
  43. F. Sabzikar, M.M. Meerschaert and J. Chen, Tempered fractional calculus. J. Comput. Phys. 293 (2015) 14–28. [CrossRef] [PubMed] [Google Scholar]
  44. E. Scalas, Five years of continuous-time random walks in econophysics. In: The Complex Networks of Economic Interactions, Springer (2006) 3–16. [CrossRef] [Google Scholar]
  45. R. Schumer, D.A. Benson, M.M. Meerschaert and S.W. Wheatcraft, Eulerian derivation of the fractional advection–dispersion equation. J. Contam. Hydrol. 48 (2001) 69–88. [CrossRef] [PubMed] [Google Scholar]
  46. X. Wang and W. Deng, Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations. J. Comput. Appl. Math In Press (2019). [Google Scholar]
  47. S. Wang, J. Yuan, W. Deng, Y. Wu, A hybridized discontinuous Galerkin method for 2d fractional convection–diffusion equations. J. Sci. Comput. 68 (2016) 826–847. [Google Scholar]
  48. Q. Xu, J.S. Hesthaven, Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52 (2014) 405–423. [Google Scholar]
  49. Y. Yu, W. Deng, Y. Wu and J. Wu, Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Appl. Numer. Math. 112 (2017) 126–145. [Google Scholar]
  50. M. Zayernouri, M. Ainsworth and G.E. Karniadakis, Tempered fractional sturm–liouville eigenproblems. SIAM J. Sci. Comput. 37 (2015) A1777–A1800. [Google Scholar]
  51. Y. Zhang and M.M. Meerschaert, Gaussian setting time for solute transport in fluvial systems. Water Resour. Res. 47 (2011). [CrossRef] [PubMed] [Google Scholar]
  52. Y. Zhang, M.M. Meerschaert and A.I. Packman, Linking fluvial bed sediment transport across scales. Geophys. Res. Lett. 39 (2012). [Google Scholar]
  53. L. Zhao, W. Deng, J.S. Hesthaven, Spectral methods for tempered fractional differential equations. Preprint arXiv:1603.06511 (2016). [Google Scholar]

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