Abstract
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.
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Acknowledgements
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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Samiee, M., Kharazmi, E., Zayernouri, M. (2017). Fast Spectral Methods for Temporally-Distributed Fractional PDEs. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_47
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