A spectral penalty method for two-sided fractional differential equations with general boundary conditions
This work provides a more accurate numerical method for solving fractional differential equations, which are important in modeling anomalous diffusion and other complex phenomena.
The authors developed a spectral penalty method (SPM) for solving two-sided fractional differential equations with general boundary conditions, achieving superior accuracy compared to an existing Petrov-Galerkin spectral tau method across all tested cases.
We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-$τ$, an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and demonstrate the superior accuracy of SPM for all cases considered.