Fast Spectral Methods for Temporally-Distributed Fractional PDEs

  • Mehdi Samiee
  • Ehsan Kharazmi
  • Mohsen ZayernouriEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


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).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Mehdi Samiee
    • 1
    • 2
  • Ehsan Kharazmi
    • 1
    • 2
  • Mohsen Zayernouri
    • 1
    • 2
    Email author
  1. 1.Department of Computational Mathematics, Science, and EngineeringMichigan State UniversityEast LansingUSA
  2. 2.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA

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