Error estimates with spectral approximation for higher order fractional derivatives using Jacobi poly fractonomials collocation method

This paper presents a stable and efficient spectral collocation method for solving fractional differential equations. The proposed approach utilizes Jacobi polyfractonomials as basis functions, which exhibit orthogonality and are expressed through fractional Lagrange interpolating polynomials. These basis functions satisfy the Kronecker delta property at the collocation points. Various types of grid points derived from the roots of orthogonal polynomials and their derivatives are explored for interpolation and collocation. A general derivative matrix is formulated for higher-order fractional derivatives, applicable to both same-order and arbitrary-order basis functions. This formulation transforms the original problem into a system of linear equations solvable by standard numerical techniques. A key contribution of this work lies in establishing the stability and deriving error bounds for the proposed method in the context of higher-order fractional derivatives. Several model problems are used to validate the method’s accuracy and efficiency, revealing significant improvements over existing approaches.