A spectral framework for fractional variational problems based on fractional Jacobi functions

A family of orthogonal systems of fractional functions is introduced. The proposed orthogonal systems are based on Jacobi polynomials through a fractional coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of fractional differential models. Approximation errors by the basic orthogonal projection are established. Three new kinds of fractional Jacobi-Gauss-type interpolations are introduced. As an example of application, an efficient approximation based on the proposed fractional functions to a fractional variational problem is presented and implemented. This approximation takes into account the potential irregularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. Implementation details are provided for the scheme, together with a series of numerical examples to show the efficiency of the proposed method.