CMSE Professor Mohsen Zayernouri, the director of FMATH group, is publishing a new peer-reviewed paper
- Nov 14, 2017
M. Samiee, E. Kharazmi, & M. Zayernouri, (2017). "Fast Spectral Methods for Temporally-Distributed Fractional PDEs",in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM16, Springer, Cham, pp. 651-667.
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. We develop a Petrov-Galerkin spectral method for high dimensional temporally-distributed fractional partial differential equations with two-sided derivatives in a space-time hyper-cube. 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.