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A simple finite element method for boundary value problems with a Riemann-Liouville derivative. (English) Zbl 1328.65172

Summary: We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order \(\alpha \in(3 / 2, 2)\) on the unit interval \((0, 1)\). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term \(x^{\alpha - 1}\) in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and \(L^2(D)\) error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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