FAST FOURIER-LIKE MAPPED CHEBYSHEV

Abstract. In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving 6 integral fractional Laplacian in Rd, which is built upon two essential components: (i) the Dunford7 Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev 8 functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, 9 and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., O((N log2N) d) with 10 d ≥ 2 and N being the number of modes in each spatial direction. Ample numerical tests for various 11 decaying exact solutions show that the convergence of the fast solver matches the order of theoretical 12 error estimates. With a suitable time-discretisation, the fast solver can be directly applied to a large 13 class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schrödinger 14 equation by using the fourth-order time-splitting method together with the proposed MCF-spectral15 Galerkin method. 16

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