Fractional differential equations: non-constant coefficients, simulation and model reduction

Abstract

We consider boundary value problems with Riemann-Liouville fractional derivatives of order $s\in (1, 2)$ with non-constant diffusion and reaction coefficients. A variational formulation is derived and analyzed leading to the well-posedness of the continuous problem and its Finite Element discretization. Then, the Reduced Basis Method through a greedy algorithm for parametric diffusion and reaction coefficients is analyzed. Its convergence properties, and in particular the decay of the Kolmogorov $n$-width, are seen to depend on the fractional order $s$. Finally, numerical results confirming our findings are presented.