Fast Spectral Methods for Temporally-Distributed Fractional PDEs
Temporally-distributed fractional partial differential equations appear as rigorous mathematical models that solve the probability density function of non-Markovian processes coding multi-physics diffusion-to-wave and multi-rate ultra slow-to-super diffusion dynamics (Chechkin et al, Phys Rev E 66(4):046129, 2002). We develop a Petrov-Galerkin spectral method for high dimensional temporally-distributed fractional partial differential equations with two-sided derivatives in a space-time hypercube. We employ Jacobi poly-fractonomials given in (Zayernouri and Karniadakis, J Comput Phys 252:495–517, 2013) and Legendre polynomials as the temporal and spatial basis/test functions, respectively. Moreover, we formulate a fast linear solver for the corresponding Lyapunov system. Furthermore, we perform the corresponding discrete stability and error analysis of the numerical scheme. Finally, we carry out several numerical test cases to examine the efficiency and accuracy of the method.
This work was supported by the AFOSR Young Investigator Program (YIP) award on: “Data-Infused Fractional PDE Modeling and Simulation of Anomalous Transport” (FA9550-17-1-0150).
- 6.A. Chechkin, R. Gorenflo, I. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(4), 046129 (2002)Google Scholar
- 10.D. del Castillo-Negrete, B. Carreras, V. Lynch, Fractional diffusion in plasma turbulence, Phys. Plasmas (1994-present) 11(8), 3854–3864 (2004)Google Scholar
- 16.X. Li, C. Xu, 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)Google Scholar
- 18.R.L. Magin, Fractional Calculus in Bioengineering (Begell House Redding, West Redding, 2006)Google Scholar
- 23.M.M. Meerschaert, F. Sabzikar, M.S. Phanikumar, A. Zeleke, Tempered fractional time series model for turbulence in geophysical flows. J. Stat. Mech: Theory Exp. 2014(9), P09023 (2014)Google Scholar
- 24.M. Naghibolhosseini, Estimation of outer-middle ear transmission using DPOAEs and fractional-order modeling of human middle ear, Ph.D. thesis, City University of New York, NY, 2015Google Scholar
- 26.M. Samiee, M. Zayernouri, M.M. Meerschaert, A unified spectral method for FPDEs with two-sided derivatives; part I: a fast solver. J. Comput. Phys. (In Press)Google Scholar
- 27.T. Srokowski, Lévy flights in nonhomogeneous media: distributed-order fractional equation approach. Phys. Rev. E 78(3), 031135 (2008)Google Scholar
- 34.L. Zhao, W. Deng, J.S. Hesthaven, Spectral methods for tempered fractional differential equations. arXiv:1603.06511 (arXiv preprint)Google Scholar