Abstract
We are going to present a suitable bases to treat the space- and timefractional diffusion equation with the Galerkin method to obtain spectral convergence in both, time and space. Furthermore, by carefully choosing a Fourier ansatz in space, we can guarantee the resulting matrices to be sparse, even though fractional order differential equations are global operator. This is due to the fact that the chosen basis consists of eigenfunctions of the given fractional differential operator. Numerical experiments validate the theoretically predicted spectral convergence for smooth problems. Additionally, we show that this method is also capable of computing approximation of the solution of the wave equation by letting the order of the spatial and temporal derivative approach two arbitrarily close.
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