A spectral penalty method for two-sided fractional differential equations with general boundary conditions

TL;DR

This paper introduces a spectral penalty method (SPM) using Jacobi poly-fractonomials and polynomial approximations for solving two-sided fractional differential equations with various boundary conditions, demonstrating high accuracy and stability.

Contribution

The paper develops a novel spectral penalty method tailored for two-sided fractional differential equations with general boundary conditions, including new analysis for coercivity and penalty parameters.

Findings

SPM achieves high accuracy in stationary and time-dependent FDEs.

The method outperforms the Petrov-Galerkin spectral tau method in numerical experiments.

Theoretical analysis ensures well-posedness and stability of the proposed approach.

Abstract

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-$\tau$, an extension of [Z. Mao, G.E. Karniadakis, SIAM J. Numer. Anal., 2018]) and demonstrate the superior accuracy of SPM for all cases considered.