Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation
Provides theoretical foundations and a numerical method for a class of fractional diffusion equations with variable coefficients, which is incremental for the fractional PDE community.
The authors prove wellposedness and regularity of a two-sided variable coefficient fractional diffusion equation by transforming it to a constant coefficient equation, and develop a spectral approximation scheme with error estimates validated by numerical experiments.
In this article a two-sided variable coefficient fractional diffusion equation (FDE) is investigated, where the variable coefficient occurs outside of the fractional integral operator. Under a suitable transformation the variable coefficient equation is transformed to a constant coefficient equation. Then, using the spectral decomposition approach with Jacobi polynomials, we proved the wellposedness of the model and the regularity of its solution. A spectral approximation scheme is proposed and the accuracy of its approximation studied. Two numerical experiments are presented to demonstrate the derived error estimates.