Least squares finite-element solution of a fractional order two-point boundary value problem

In this paper, a theoretical framework for the least squares finite-element approximation of a fractional order differential equation is presented. Mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established. Using these properties existence and uniqueness of the least squares approximation is proven. Optimal error estimates are proven for piecewise linear trial elements. Numerical results are included which confirm the theoretical results.