Gegenbauer Collocation Discretization of Riesz Derivative of Variable-Order

Abstract

The fractional Laplacian has been extensively applied to model nonlocal phenomena such as anomalous diffusion. In this paper, we propose to employ the Gegenbauer collocation method to discretize one special case of the fractional Laplacian-Riesz fractional derivative of variable-order (VO). We analyze the features of the collocation method for both constant-order and VO cases and develop an efficient preconditioner. We present some approximation errors and several numerical tests to confirm the efficiency of the method.

Keywords

Gegenbauer orthogonal polynomial / Spectral collocation method / Riesz potential / Fractional derivative / Preconditioner / Fractional Laplacian / Variable-order (VO) / 65K05 / 90C46

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