A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative

@article{Mao2018ASM,
title={A Spectral Method (of Exponential Convergence) for Singular Solutions of the Diffusion Equation with General Two-Sided Fractional Derivative},
author={Zhiping Mao and George Em Karniadakis},
journal={SIAM J. Numerical Analysis},
year={2018},
volume={56},
pages={24-49}
}

An open problem in the numerical solution of fractional partial differential equations (FPDEs) is how to obtain high-order accuracy for singular solutions; even for smooth right-hand sides solutions of FPDEs are singular. Here, we consider the one-dimensional diffusion equation with general two-sided fractional derivative characterized by a parameter p ∈ [0, 1]; for p = 1/2 we recover the Riesz fractional derivative, while for p = 1, 0 we obtain the one-sided fractional derivative. We employ a… CONTINUE READING