1. Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithms 75(1), 173–211 (2017) 2. Ainsworth, M., Glusa, C.: Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput. Methods Appl. Mech. Eng. 327, 4–35 (2017) 3. Ammi, M.R.S., Jamiai, I.: Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete Contin. Dyn. Syst. Ser. S 11(1), 103 (2018) 4. Ardakani, A.G.: Investigation of Brewster anomalies in one-dimensional disordered media having Lévy-type distribution. Eur. Phys. J. B 89(3), 76 (2016) 5. Armour, K.C., Marshall, J., Scott, J.R., Donohoe, A., Newsom, E.R.: Southern Ocean warming delayed by circumpolar upwelling and equatorward transport. Nat. Geosci. 9(7), 549–554 (2016) 6. Bazhlekova, E., Bazhlekov, I.: Subordination approach to multi-term time-fractional diffusion-wave equations. J. Comput. Appl. Math. 339, 179–192 (2018) 7. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403–1412 (2000) 8. Chechkin, A., Gorenflo, R., Sokolov, I.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(4), 046129 (2002) 9. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016) 10. Chen, S., Shen, J., Wang, L.L.: Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. J. Sci. Comput. 74, 1–28 (2017) 11. Cheng, A., Wang, H., Wang, K.: A Eulerian–Lagrangian control volume method for solute transport with anomalous diffusion. Numer. Methods Partial Differ. Equ. 31(1), 253–267 (2015) 12. Coronel-Escamilla, A., Gómez-Aguilar, J., Torres, L., Escobar-Jiménez, R.: A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A Stat. Mech. Appl. 491, 406–424 (2018) 13. Duan, B., Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Space-time Petrov–Galerkin FEM for fractional diffusion problems. Comput. Methods Appl. Math. 18, 1 (2017) 14. Duan, J.S., Baleanu, D.: Steady periodic response for a vibration system with distributed order derivatives to periodic excitation. J. Vib. Control 21, 3124 (2018) 15. Eab, C., Lim, S.: Fractional Langevin equations of distributed order. Phys. Rev. E 83(3), 031136 (2011) 16. Edery, Y., Dror, I., Scher, H., Berkowitz, B.: Anomalous reactive transport in porous media: experiments and modeling. Phys. Rev. E 91(5), 052130 (2015) 17. Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in ? d. Numer. Methods Partial Differ. Equ. 23(2), 256 (2007) 18. Fan, W., Liu, F.: A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl. Math. Lett. 77, 114–121 (2018) 19. Gorenflo, R., Luchko, Y., Yamamoto, M.: Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18(3), 799–820 (2015) 20. Goychuk, I.: Anomalous transport of subdiffusing cargos by single kinesin motors: the role of mechano-chemical coupling and anharmonicity of tether. Phys. Biol. 12(1), 016,013 (2015) 21. Iwayama, T., Murakami, S., Watanabe, T.: Anomalous eddy viscosity for two-dimensional turbulence. Phys. Fluids 27(4), 045104 (2015). https://doi.org/10.1063/1.4916956 22. Jin, B., Lazarov, R., Sheen, D., Zhou, Z.: Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fract. Calc. Appl. Anal. 19(1), 69–93 (2016) 23. Jin, B., Lazarov, R., Thomée, V., Zhou, Z.: On nonnegativity preservation in finite element methods for subdiffusion equations. Math. Comput. 86(307), 2239–2260 (2017) 24. Kharazmi, E., Zayernouri, M.: Fractional pseudo-spectral methods for distributed-order fractional PDEs. Int. J. Comput. Math. 95(6/7), 1340–1361 (2018) 25. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: Petrov–Galerkin and spectral collocation methods for distributed order differential equations. SIAM J. Sci. Comput. 39(3), A1003–A1037 (2017) 26. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: A Petrov–Galerkin spectral element method for fractional elliptic problems. Comput. Methods Appl. Mech. Eng. 324, 512–536 (2017) 27. Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications. WileyVCH, New York (2008) 28. Konjik, S., Oparnica, L., Zorica, D.: Distributed-order fractional constitutive stress-strain relation in wave propagation modeling. Zeitsch. Angew. Math. Phys. 70(2), 51 (2019) 29. Li, X., Rui, H.: Two temporal second-order H1-Galerkin mixed finite element schemes for distributedorder fractional sub-diffusion equations. Numer. Algorithms 79(4), 1107–1130 (2018) 30. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009) 31. Li, X., Xu, C.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016 (2010) 32. Liao, Hl, Lyu, P., Vong, S., Zhao, Y.: Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations. Numer. Algorithms 75(4), 845–878 (2017) 33. Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12(4), 409–422 (2009) 34. Macías-Díaz, J.: An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun. Nonlinear Sci. Numer. Simul. 59, 67–87 (2018) 35. Maday, Y.: Analysis of spectral projectors in one-dimensional domains. Math. Comput. 55(192), 537– 562 (1990) 36. Mainardi, F., Mura, A., Gorenflo, R., Stojanović, M.: The two forms of fractional relaxation of distributed order. J. Vib. Control 13(9/10), 1249–1268 (2007) 37. Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14(9/10), 1267–1290 (2008) 38. Mao, Z., Shen, J.: Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016) 39. Mao, Z., Shen, J.: Spectral element method with geometric mesh for two-sided fractional differential equations. Adv. Comput. Math. 44, 1–27 (2017) 40. Mashelkar, R., Marrucci, G.: Anomalous transport phenomena in rapid external flows of viscoelastic fluids. Rheol. Acta 19(4), 426–431 (1980) 41. Meerschaert, M.M.: Fractional Calculus, Anomalous Diffusion, and Probability. Fractional Dynamics: Recent Advances, pp. 265–284. World Scientific, Singapore (2012) 42. Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus, vol. 43. Walter de Gruyter, Berlin (2012) 43. Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014) 44. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000) 45. Naghibolhosseini, M.: Estimation of outer-middle ear transmission using DPOAEs and fractionalorder modeling of human middle ear. Ph.D. thesis, City University of New York, NY (2015) 46. Naghibolhosseini, M., Long, G.R.: Fractional-order modelling and simulation of human ear. Int. J. Comput. Math. 95, 1–17 (2017) 47. Perdikaris, P., Karniadakis, G.E.: Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42(5), 1012–1023 (2014) 48. Regner, B.M.: Randomness in biological transport. UC San Diego Electronic Theses and Dissertations, UC San Diego (2014) 49. Samiee, M., Akhavan-Safaei, A., Zayernouri, M.: A fractional subgrid-scale model for turbulent flows: theoretical formulation and a priori study. arXiv:1909.09943 (2019) 50. Samiee, M., Zayernouri, M., Meerschaert, M.M.: A unified spectral method for FPDES with two-sided derivatives; part I: a fast solver. J. Comput. Phys. 385, 225–243 (2019) 51. Samiee, M., Zayernouri, M., Meerschaert, M.M.: A unified spectral method for FPDES with two-sided derivatives; part II: stability, and error analysis. J. Comput. Phys. 385, 244–261 (2019) 52. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer Science & Business Media, New York (2011) 53. Sokolov, I., Chechkin, A., Klafter, J.: Distributed-order fractional kinetics. arXiv preprint condmat/0401146 (2004) 54. Suzuki, J., Zayernouri, M., Bittencourt, M., Karniadakis, G.: Fractional-order uniaxial visco-elastoplastic models for structural analysis. Comput. Methods Appl. Mech. Eng. 308, 443–467 (2016) 55. Suzuki, J.L., Zayernouri, M.: An automated singularity-capturing scheme for fractional differential equations. arXiv:1810.12219 (2018) 56. Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015) 57. Tomovski, Ž., Sandev, T.: Distributed-order wave equations with composite time fractional derivative. Int. J. Comput. Math. 95, 1–14 (2017) 58. Tomovski, Ž., Sandev, T.: Distributed-order wave equations with composite time fractional derivative. Int. J. Comput. Math. 95(6/7), 1100–1113 (2018) 59. Tyukhova, A., Dentz, M., Kinzelbach, W., Willmann, M.: Mechanisms of anomalous dispersion in flow through heterogeneous porous media. Phys. Rev. Fluids 1(7), 074,002 (2016) 60. Varghaei, P., Kharazmi, E., Suzuki, J.L., Zayernouri, M.: Vibration analysis of geometrically nonlinear and fractional viscoelastic cantilever beams. arXiv:1909.02142 (2019) 61. Yamamoto, M.: Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations. J. Math. Anal. Appl. 460(1), 365–381 (2018) 62. Zaky, M.A.: A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn. 2667, 1–15 (2018) 63. Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: Tempered fractional Sturm–Liouville eigenproblems. SIAM J. Sci. Comput. 37(4), A1777–A1800 (2015) 64. Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: A unified Petrov–Galerkin spectral method for fractional PDES. Comput. Methods Appl. Mech. Eng. 283, 1545–1569 (2015) 65. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013) 66. Zhang, Y., Meerschaert, M.M., Baeumer, B., LaBolle, E.M.: Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resour. Res. 51(8), 6311–6337 (2015) 67. Zhang, Y., Meerschaert, M.M., Neupauer, R.M.: Backward fractional advection dispersion model for contaminant source prediction. Water Resour. Res. 52(4), 2462–2473 (2016) |